Preprints & selected publications:
2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1994 1992 1980-1990
A.N. Gorban,
G.S.Yablonsky
Extended detailed
balance for systems with irreversible reactions, Chemical
Engineering Science 66 (2011) 53885399.
The
principle of detailed balance states that in equilibrium each elementary
process is equilibrated by its reverse process. For many real physico-chemical complex systems (e.g. homogeneous
combustion, heterogeneous catalytic oxidation, most enzyme reactions etc), detailed mechanisms include both reversible and
irreversible reactions. In this case, the principle of detailed balance cannot
be applied directly. We represent irreversible reactions as limits of
reversible steps and obtain the principle of detailed balance for complex
mechanisms with some irreversible elementary processes. We proved two
consequences of the detailed balance for these mechanisms: the structural
condition and the algebraic condition that form together the extended form of
detailed balance. The algebraic condition is the principle of detailed balance
for the reversible part. The structural condition is: the convex hull of the
stoichiometric vectors of the irreversible reactions has empty intersection
with the linear span of the stoichiometric vectors of the reversible reaction.
Physically, this means that the irreversible reactions cannot be included in
oriented pathways.
The systems with the extended form of detailed balance
are also the limits of the reversible systems with detailed balance when some
of the equilibrium concentrations (or activities) tend to zero. Surprisingly,
the structure of the limit reaction mechanism crucially depends on the relative
speeds of this tendency to zero.
A.N. Gorban,
H.P. Sargsyan and H.A. Wahab
Quasichemical Models of Multicomponent Nonlinear Diffusion, Mathematical
Modelling of Natural Phenomena, Volume 6 / Issue 05, (2011), 184−262.
Diffusion preserves the positivity of concentrations, therefore, multicomponent
diffusion should be nonlinear if there exist non-diagonal terms. The vast
variety of nonlinear multicomponent diffusion equations should be ordered and
special tools are needed to provide the systematic construction of the
nonlinear diffusion equations for multicomponent mixtures with significant
interaction between components. We develop an approach to nonlinear multicomponent
diffusion based on the idea of the reaction mechanism borrowed from chemical
kinetics.
Chemical kinetics gave rise to
very seminal tools for the modeling of processes.
This is the stoichiometric algebra supplemented by the simple kinetic law. The results
of this invention are now applied in many areas of science, from particle
physics to sociology. In our work we extend the area of applications onto
nonlinear multicomponent diffusion.
We demonstrate, how the mechanism
based approach to multicomponent diffusion can be included into the general
thermodynamic framework, and prove the corresponding dissipation inequalities.
To satisfy thermodynamic restrictions, the kinetic law of an elementary process
cannot have an arbitrary form. For the general kinetic law (the generalized
Mass Action Law), additional conditions are proved. The celljump formalism
gives an intuitively clear representation of the elementary transport processes
and, at the same time, produces kinetic finite elements, a tool for numerical
simulation
A. Gorban
and S. Petrovskii
Collective
dynamics: when one plus one does not make two, Mathematical
Medicine and Biology (2011) 28, 85−88.
A brief introduction into the
interdisciplinary field of collective dynamics is given, followed by an
overview of Mathematical Models of Collective Dynamics in Biology and
Evolution (University of Leicester, 1113 May 2009). Collective
dynamicsunderstood as the dynamics arising from the interplay between the
constituting elementary argents or parts of a more complex systemhas been one
of the main paradigms of the natural sciences over the last several decades.
A.N. Gorban and M. Shahzad
The Michaelis-Menten-Stueckelberg
Theorem. Entropy 2011, 13,
966-1019
We
study chemical reactions with complex mechanisms under two assumptions: (i) intermediates are present in small amounts (this is the
quasi-steady-state hypothesis or QSS) and (ii) they are in equilibrium
relations with substrates (this is the quasiequilibrium
hypothesis or QE). Under these assumptions, we prove the generalized mass
action law together with the basic relations between kinetic factors, which are
sufficient for the positivity of the entropy production but hold even without microreversibility, when the detailed balance is not
applicable. Even though QE and QSS produce useful approximations by themselves,
only the combination of these assumptions can render the possibility beyond the
rarefied gas limit or the molecular chaos hypotheses. We do not use any a
priori form of the kinetic law for the chemical reactions and describe their equilibria by thermodynamic relations. The transformations
of the intermediate compounds can be described by the Markov kinetics because
of their low density (low density of elementary events). This
combination of assumptions was introduced by Michaelis
and Menten in 1913. In 1952, Stueckelberg
used the same assumptions for the gas kinetics and produced the remarkable
semi-detailed balance relations between collision rates in the Boltzmann
equation that are weaker than the detailed balance conditions but are still
sufficient for the Boltzmann H-theorem to be valid. Our results are
obtained within the Michaelis-Menten-Stueckelbeg
conceptual framework.
G. S. Yablonsky,
A. N. Gorban, D. Constales,
V. V. Galvita and G. B. Marin
Reciprocal
relations between kinetic curves, EPL, 93
(2011) 20004.
We study coupled irreversible
processes. For linear or linearized kinetics with microreversibility, 
,
the kinetic operator K is symmetric in the entropic inner
product. This form of Onsager's reciprocal relations implies that the shift in
time, exp(Kt), is
also a symmetric operator. This generates the reciprocity relations between the
kinetic curves. For example, for the Master equation, if we start the process
from the i-th pure state and measure the
probability pj(t) of
the j-th state (j≠i),
and, similarly, measure pi(t) for the process,
which starts at the j-th pure state, then
the ratio of these two probabilities pj(t)/pi(t)
is constant in time and coincides with the ratio of the equilibrium
probabilities. We study similar and more general reciprocal relations between
the kinetic curves. The experimental evidence provided as an example is from
the reversible water gas shift reaction over iron oxide catalyst. The
experimental data are obtained using Temporal Analysis of Products (TAP)
pulse-response studies. These offer excellent confirmation within the
experimental error.
A.N. Gorban,
D. Roose
Preface, In:
Coping with Complexity: Model Reduction and Data Analysis, A.N. Gorban and D. Roose (eds.),
Lecture Notes in Computational Science and Engineering, 75, Springer:
Heidelberg Dordrecht - London -New York, 2011, pp. V-VI.
A
mathematical model is an intellectual device that works.
A.N. Gorban
Self-simplification
in Darwins Systems, In: Coping with Complexity:
Model Reduction and Data Analysis, A.N. Gorban and D.
Roose (eds.), Lecture Notes in Computational Science
and Engineering, 75, Springer: Heidelberg Dordrecht - London -New York, 2011,
pp. 311-344
We prove that a non-linear kinetic
system with conservation of supports for distributions has
generically limit distributions with final support only. The conservation of
support has a biological interpretation: inheritance. We call
systems with inheritance Darwins systems. Such systems are apparent in many
areas of biology, physics (the theory of parametric wave interaction),
chemistry and economics. The finite dimension of limit distributions
demonstrates effects of natural selection. Estimations of the
asymptotic dimension are presented. After some initial time, solution of a
kinetic equation with conservation of support becomes a finite set of narrow
peaks that become increasingly narrow over time and move increasingly slowly.
It is possible that these peaks do not tend to fixed positions, and the path
covered tends to infinity as t → ∞. The drift
equations for peak motion are obtained. They describe the asymptotic
layer near the omega-limit distributions with finite support .
D.J. Packwood, J. Levesley, and A.N. Gorban
Time step
expansions and the invariant manifold approach to lattice Boltzmann models,
In: Coping with Complexity: Model Reduction and Data Analysis, A.N. Gorban and D. Roose (eds.),
Lecture Notes in Computational Science and Engineering, 75, Springer:
Heidelberg Dordrecht - London -New York, 2011, pp. 169-206.
The classical method for deriving
the macroscopic dynamics of a lattice Boltzmann system is to use a combination
of different approximations and expansions. Usually a Chapman-Enskog analysis is performed, either on the continuous
Boltzmann system, or its discrete velocity counterpart. Separately a discrete
time approximation is introduced to the discrete velocity Boltzmann system, to
achieve a practically useful approximation to the continuous system, for use in
computation. Thereafter, with some additional arguments, the dynamics of the
Chapman-Enskog expansion are linked to the discrete
time system to produce the dynamics of the completely discrete scheme. In this
paper we put forward a different route to the macroscopic dynamics. We begin
with the system discrete in both velocity space and time. We hypothesize that
the alternating steps of advection and relaxation, common to all lattice
Boltzmann schemes, give rise to a slow invariant manifold. We perform a time
step expansion of the discrete time dynamics using the invariance of the
manifold. Finally we calculate the dynamics arising from this system. By
choosing the fully discrete scheme as a starting point we avoid mixing
approximations and arrive at a general form of the microscopic dynamics up to
the second order in the time step. We calculate the macroscopic dynamics of two
commonly used lattice schemes up to the first order, and hence find the precise
form of the deviation from the Navier-Stokes
equations in the dissipative term, arising from the discretization of velocity
space.
Finally we perform a short wave
perturbation on the dynamics of these example systems, to find the necessary
conditions for their stability.
A.N. Gorban
Kinetic path
summation, multi-sheeted extension of master equation, and evaluation of ergodicity coefficient, Physica A 390 (2011)
10091025.
We study the master equation with
time-dependent coefficients, a linear kinetic equation for the Markov chains or
for the monomolecular chemical kinetics. For the solution of this equation a
path summation formula is proved. This formula represents the solution as a sum
of solutions for simple kinetic schemes (kinetic paths), which are available in
explicit analytical form. The relaxation rate is studied and a family of
estimates for the relaxation time and the ergodicity
coefficient is developed. To calculate the estimates we introduce the
multi-sheeted extensions of the initial kinetics. This approach allows us to
exploit the internal (micro) structure of the extended kinetics without
perturbation of the base kinetics.
A.N. Gorban,
L.I. Pokidysheva,E,V. Smirnova,
T.A. Tyukina.
Law of the
Minimum Paradoxes, Bull Math Biol 73(9)
(2011), 2013-2044; Online first 19.11.2010,
The Law of the Minimum states
that growth is controlled by the scarcest resource (limiting factor). This
concept was originally applied to plant or crop growth (Justus von Liebig,
1840) and quantitatively supported by many experiments. Some generalizations
based on more complicated dose-response curves were proposed. Violations of
this law in natural and experimental ecosystems were also reported. We study
models of adaptation in ensembles of similar organisms under load of
environmental factors and prove that violation of Liebigs law follows from
adaptation effects. If the fitness of an organism in a fixed environment
satisfies the Law of the Minimum then adaptation equalizes the pressure of
essential factors and, therefore, acts against the Liebigs law. This is the the Law of the Minimum paradox: if for a
randomly chosen pair organismenvironment the Law of the Minimum typically
holds, then in a well-adapted system, we have to expect violations of this law.
For the opposite interaction of
factors (a synergistic system of factors which amplify each other), adaptation
leads from factor equivalence to limitations by a smaller number of factors.
For analysis of adaptation, we
develop a system of models based on Selyes idea of
the universal adaptation resource (adaptation energy). These models predict
that under the load of an environmental factor a population separates into two
groups (phases): a less correlated, well adapted group and a highly correlated
group with a larger variance of attributes, which experiences problems with
adaptation. Some empirical data are presented and evidences of
interdisciplinary applications to econometrics are discussed.
A.N. Gorban, E.V. Smirnova, T.A. Tyukina,
Correlations,
risk and crisis: From physiology to finance, Physica A, Vol. 389, Issue 16, 2010, 3193-3217. Number 9 in the Top Hottest Articles in the Journal, April
to June 2010
We study
the dynamics of correlation and variance in systems under the load of
environmental factors. A universal effect in ensembles of similar systems under
the load of similar factors is described: in crisis, typically, even before
obvious symptoms of crisis appear, correlation increases, and, at the same
time, variance (and volatility) increases too. This effect is supported by many
experiments and observations of groups of humans, mice, trees, grassy plants,
and on financial time series.
A general
approach to the explanation of the effect through dynamics of individual
adaptation of similar non-interactive individuals to a similar system of
external factors is developed. Qualitatively, this approach follows Selyes idea about adaptation energy.
A.N. Gorban
We
study the Master equation with time--dependent coefficients, a linear kinetic
equation for the Markov chains or for the monomolecular chemical kinetics. For
the solution of this equation a paths summation formula is proved. This formula
represents the solution as a sum of solutions for simple kinetic schemes
(kinetic paths), which are available in explicit analytical form. The
relaxation rate is studied and a family of estimates for the relaxation time
and the ergodicity coefficient is developed. To
calculate the estimates we introduce the multi--sheeted extensions} of the
initial kinetics. This approach allows us to exploit the internal
("micro")structure of the extended kinetics without perturbation of
the base kinetics.
A. N. Gorban, A. Zinovyev.
Principal manifolds and graphs in practice: from molecular
biology to dynamical systems, International
Journal of Neural Systems, Vol. 20, No. 3 (2010) 219232.
We present several applications of
non-linear data modeling, using principal manifolds
and principal graphs constructed using the metaphor of elasticity (elastic
principal graph approach). These approaches are generalizations of the Kohonens self-organizing maps, a class of artificial
neural networks. On several examples we show advantages of using non-linear
objects for data approximation in comparison to the linear ones. We propose
four numerical criteria for comparing linear and non-linear mappings of datasets
into the spaces of lower dimension. The examples are taken from
comparative political science, from analysis of high-throughput data
in molecular biology, from analysis of dynamical systems.
E. Chiavazzo,
I.V. Karlin, A.N. Gorban,
K. Boulouchos,
Coupling of the model reduction technique with the lattice
Boltzmann method, Combustion and Flame 157 (2010) 18331849 
doi:10.1016/j.combustflame.2010.06.009
A new
framework of simulation of reactive flows is proposed based on a coupling
between accurate reduced reaction mechanism and the lattice Boltzmann
representation of the flow phenomena. The model reduction is developed in the
setting of slow invariant manifold construction, and the simplest lattice
Boltzmann equation is used in order to work out the procedure of coupling of
the reduced model with the flow solver. Practical details of constructing slow
invariant manifolds of a reaction system under various thermodynamic conditions
are reported. The proposed method is validated with the two-dimensional
simulation of a premixed counterflow flame in the
hydrogen-air mixture.
Gorban A.N., Gorban P.A., Judge G.
Entropy: The Markov Ordering
Approach. Entropy. 2010; 12(5):1145-1193. GorbanGorbanJudgeEntropy2010.pdf
The
focus of this article is on entropy and Markov processes. We study the
properties of functionals which are invariant with
respect to monotonic transformations and analyze two invariant additivity properties: (i)
existence of a monotonic transformation which makes the functional additive
with respect to the joining of independent systems and (ii) existence of a
monotonic transformation which makes the functional additive with respect to
the partitioning of the space of states. All Lyapunov
functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general
ordering of the distribution space, with respect to which all continuous-time
Markov processes are monotonic (the Markov order). The solution
differs significantly from the ordering given by the inequality of entropy
growth. For inference, this approach results in a convex compact set of
conditionally most random distributions.
A. N. Gorban
and V. M. Cheresiz,
Slow Relaxations and Bifurcations of the Limit Sets of
Dynamical Systems. I. Bifurcations of Limit Sets, Journal of Applied and Industrial Mathematics, 2010,
Vol. 4, No. 1, pp. 5464.
We consider one-parameter semigroups of homeomorphisms depending continuously on the
parameters. We study the phenomenon of slow relaxation that consists in
anomalously slow motion to the limit sets. We investigate the connection
between slow relaxations and bifurcations of limit sets and other singularities
of the dynamics. The statements of some of the problems stem from mathematical
chemistry.
A. N. Gorban
and V. M. Cheresiz,
Slow Relaxations and Bifurcations of the Limit Sets of
Dynamical Systems. II. Slow Relaxations of a Family of Semiflows, Journal of Applied and Industrial Mathematics, 2010,
Vol. 4, No. 2, pp. 182190.
We propose a number of approaches
to the notion of the relaxation time of a dynamical system which are motivated
by the problems of chemical kinetics, give exact mathematical definitions of
slow relaxations, study their possible reasons, among which an important role
is played by bifurcations of limit sets.
E. Chiavazzo,
I.V. Karlin, and A.N. Gorban,
The Role of Thermodynamics in Model Reduction when Using
Invariant Grids, Commun.
Comput. Phys., Vol. 8, No. 4
(2010), pp. 701-734.
In the present work, we develop in
detail the process leading to reduction of models in chemical kinetics when
using the Method of Invariant Grids (MIG). To this end, reduced models
(invariant grids) are obtained by refining initial approximations of slow invariant
manifolds, and used for integrating smaller and less stiff systems of equations
capable to recover the detailed description with high accuracy. Moreover, we
clarify the role played by thermodynamics in model reduction, and carry out a
comparison between detailed and reduced solutions for a model hydrogen
oxidation reaction.
Andrei Zinovyev,
Nadya Morozova, Nora Nonne, Emmanuel Barillot, Annick Harel-Bellan, Alexander N Gorban
Dynamical modeling of microRNA
action on the protein translation process, BMC Systems Biology 2010, 4:13
(24 February 2010)
Background
Protein translation is a multistep
process which can be represented as a cascade of biochemical reactions
(initiation, ribosome assembly, elongation, etc.), the rate of which can be
regulated by small non-coding microRNAs through multiple mechanisms. It remains
unclear what mechanisms of microRNA action are the most dominant: moreover,
many experimental reports deliver controversal
messages on what is the concrete mechanism actually observed in the experiment.
Nissan and Parker have recently demonstrated that it might be impossible to
distinguish alternative biological hypotheses using the steady state data on
the rate of protein synthesis. For their analysis they used two simple kinetic
models of protein translation.
Results
In contrary to the study by Nissan
and Parker, we show that dynamical data allow to discriminate some of the
mechanisms of microRNA action. We demonstrate this using the same models as
developed by Nissan and Parker for the sake of comparison but the methods
developed (asymptotology of biochemical networks) can
be used for other models. We formulate a hypothesis that the effect of microRNA
action is measurable and observable only if it affects the dominant system
(generalization of the limiting step notion for complex networks) of the
protein translation machinery. The dominant system can vary in different
experimental conditions that can partially explain the existing controversy of
some of the experimental data.
Conclusions
Our analysis of the transient protein
translation dynamics shows that it gives enough information to verify or reject
a hypothesis about a particular molecular mechanism of microRNA action on
protein translation. For multiscale systems only that
action of microRNA is distinguishable which affects the parameters of dominant
system (critical parameters), or changes the dominant system itself. Dominant
systems generalize and further develop the old and very popular idea of
limiting step. Algorithms for identifying dominant systems in multiscale kinetic models are straightforward but not
trivial and depend only on the ordering of the model parameters but not on
their concrete values. Asymptotic approach to kinetic models allows to put in
order diverse experimental observations in complex situations when many
alternative hypotheses co-exist.
A. N. Gorban, O. Radulescu, A. Y. Zinovyev,
Asymptotology of chemical reaction networks, Chemical
Engineering Science 65 (2010) 23102324 GorbRadZinCES2010Rev.pdf
The concept of the limiting step
is extended to the asymptotology of multiscale reaction networks. Complete theory for linear
networks with well separated reaction rate constants is developed. We present
algorithms for explicit approximations of eigenvalues and eigenvectors of
kinetic matrix. Accuracy of estimates is proven. Performance of the algorithms
is demonstrated on simple examples. Application of algorithms to nonlinear
systems is discussed.
A.N. Gorban, E.V. Smirnova, T.A. Tyukina
General Laws of Adaptation to Environmental Factors: from
Ecological Stress to Financial Crisis. Math. Model.
Nat. Phenom. Vol. 4, No. 6, 2009, pp. 1-53
We study ensembles of similar
systems under load of environmental factors. The phenomenon of adaptation has
similar properties for systems of different nature. Typically, when the load
increases above some threshold, then the adapting systems become more different
(variance increases), but the correlation increases too. If the stress
continues to increase then the second threshold appears: the correlation
achieves maximal value, and start to decrease, but the variance continue to
increase. In many applications this second threshold is a signal of approaching
of fatal outcome. This effect is supported by many experiments and observation
of groups of humans, mice, trees, grassy plants, and on financial time series.
A general approach to explanation
of the effect through dynamics of adaptation is developed. H. Selye introduced adaptation energy for explanation of
adaptation phenomena. We formalize this approach in factors resource models
and develop hierarchy of models of adaptation. Different organization of
interaction between factors (Liebigs versus synergistic systems) lead to
different adaptation dynamics. This gives an explanation to qualitatively
different dynamics of correlation under different types of load and to some
deviation from the typical reaction to stress. In addition to the quasistatic optimization factor resource models,
dynamical models of adaptation are developed, and a simple model (three
variables) for adaptation to one factor load is formulated explicitly.
A. N. Gorban, A. Y. Zinovyev
Principal Graphs and Manifolds, Chapter
2 in: Handbook of Research on Machine Learning Applications and Trends:
Algorithms, Methods, and Techniques, Emilio Soria Olivas et al. (eds), IGI Global, Hershey, PA, USA, 2009, pp. 28-59.
In many physical, statistical,
biological and other investigations it is desirable to approximate a system of
points by objects of lower dimension and/or complexity. For this purpose, Karl
Pearson invented principal component analysis in 1901 and found lines and
planes of closest fit to system of points. The famous k-means algorithm solves
the approximation problem too, but by finite sets instead of lines and planes.
This chapter gives a brief practical introduction into the methods of
construction of general principal objects (i.e., objects embedded in the
middle of the multidimensional data set). As a basis, the unifying framework
of mean squared distance approximation of finite datasets is selected. Principal
graphs and manifolds are constructed as generalisations of principal components
and k-means principal points. For this purpose, the family of
expectation/maximisation algorithms with nearest generalisations is presented.
Construction of principal graphs with controlled complexity is based on the
graph grammar approach.
A.N. Gorban, L.I. Pokidysheva, E.V. Smirnova, T.A. Tyukina
Law of the Minimum Paradoxes, e-print http://arxiv.org/abs/0907.1965
The "law of the minimum" states that growth is controlled by the
scarcest resource (limiting factor) (Justus von Liebig (1840)). This concept
was originally applied to plant or crop growth and quantitatively supported by
many experiments. Some generalizations based on more complicated
"dose-response" curves were proposed. Violations of this law in
natural and experimental ecosystems were also reported. We study models of
adaptation in ensembles of similar organisms under load of environmental
factors and prove that violation of the Liebig law follows from adaptation
effects. If the fitness of an organism in fixed environment satisfies the law
of the minimum then adaptation equalizes the pressure of essential factors and
therefore acts against the law. This is the the law
of the minimum paradox: if for a randomly chosen pair
"organism--environment" the law of the minimum typically holds, then,
in a well-adapted system, we have to expect violations of this law. For the
opposite interaction of factors (a synergistic system of factors which amplify
each other) adaptation leads from factor equivalence to limitations by a
smaller number of factors. For analysis of adaptation we develop a system of
models based on Selye's idea of the universal
adaptation resource (adaptation energy). These models predict that under the
load of an environmental factor a population separates into two groups
(phases): a less correlated, well adapted group and a highly correlated group
with a larger variance of attributes, which experiences problems with
adaptation. Some empirical data are presented and some evidences of
interdisciplinary applications to econometrics are discussed.
E. Chiavazzo,
I. V. Karlin, A. N. Gorban
and K Boulouchos,
Combustion simulation via
lattice Boltzmann and reduced chemical kinetics, J. Stat. Mech. (2009) P06013, MIG-LB_StatMech_2009.pdf
We present and validate a
methodology for coupling reduced models of detailed combustion mechanisms
within the lattice Boltzmann framework. A detailed mechanism (9 species, 21
elementary reactions) for modeling reacting mixtures
of air and hydrogen is considered and reduced using the method of invariant
grids (MIG). In particular, a 2D quasi-equilibrium grid is constructed, further
refined via the MIG method, stored in the form of tables and used to simulate a
1D flame propagating freely through a homogeneous premixed mixture. Comparisons
between the detailed and reduced models show that the technique presented
enables one to achieve a remarkable speedup in the computations with excellent
accuracy.
A. N. Gorban, E. V. Smirnova, T. A. Tyukina,
Correlations,
Risk and Crisis: from Physiology to Finance, e-print: http://arxiv.org/abs/0905.0129. Available at SSRN: http://ssrn.com/abstract=1397677.
We
study the dynamics of correlation and variance in systems under the load of
environmental factors. A universal effect in ensembles of similar systems under
load of similar factors is described: in crisis, typically, even before obvious
symptoms of crisis appear, correlation increases, and, at the same time,
variance (and volatility) increases too. After the crisis achieves its bottom,
it can develop into two directions: recovering (both correlations and variance
decrease) or fatal catastrophe (correlations decrease, but variance not). This
effect is supported by many experiments and observation of groups of humans,
mice, trees, grassy plants, and on financial time series. A general approach to
explanation of the effect through dynamics of adaptation is developed.
Different organization of interaction between factors (Liebig's versus
synergistic systems) lead to different adaptation dynamics. This gives an
explanation to qualitatively different dynamics of correlation under different
types of load.
A. N. Gorban, O. Radulescu, A. Y. Zinovyev,
Limitation
and Asymptotology of Chemical Reaction Networks, e-print: http://arxiv.org/abs/0903.5072
The
concept of the limiting step is extended to the asymptotology
of multiscale reaction networks. Complete theory for
linear networks with well separated reaction rate constants is developed. We
present algorithms for explicit approximations of eigenvalues and eigenvectors
of kinetic matrix. Accuracy of estimates is proven. Performance of the
algorithms is demonstrated on simple examples. Application of algorithms to nonlinear
systems is discussed.
A. Gorban, I. Tyukin, E. Steur, H. Nijmeijer
Positive
Invariance Lemmas for Control Problems with Convergence to Lyapunov-unstable
Sets, e-print http://arxiv.org/abs/0901.3577
We
provide Lyapunov-like characterizations of positive
invariance, boundedness and convergence of
non-trivial solutions for a class of systems with unstable invariant sets. The
systems of this class comprise of a stable part coupled with a one-dimensional
unstable or critically stable subsystem. Examples of these systems appear in
the problems of nonlinear output regulation, parameter estimation and adaptive
control. We demonstrate that, for a large class of systems with unstable equilibria and solutions that might escape to infinity in
finite time, it is always possible to determine simple criteria for positive
invariance and boundedness of the system's nontrivial
solutions. Conversely, it is possible to characterize domains of initial
conditions that lead to solutions escaping from the origin. In contrast to
other works addressing convergence issues in unstable systems, our results do
not rely on the availability of input-output gains or contraction rates that
are usually required for the stable compartment.
Principal Graphs and Manifolds, e-print: http://arxiv.org/abs/0809.0490
In
many physical statistical, biological and other investigations it is desirable
to approximate a system of points by objects of lower dimension and/or
complexity. For this purpose, Karl Pearson invented principal component
analysis in 1901 and found "lines and planes of closest fit to system of
points". The famous k-means algorithm solves the approximation problem
too, but by finite sets instead of lines and planes. This chapter gives a brief
practical introduction into the methods of construction of general principal
objects, i.e. objects embedded in the "middle" of the
multidimensional data set. As a basis, the unifying framework of mean squared
distance approximation of finite datasets is selected. Principal graphs and
manifolds are constructed as generalisations of
principal components and k-means principal points. For this purpose, the family
of expectation/maximisation algorithms with nearest generalisations is presented. Construction of principal
graphs with controlled complexity is based on the graph grammar approach.
Ovidiu Radulescu, Alexander N Gorban, Andrei
Zinovyev, and Alain Lilienbaum
Robust simplifications of multiscale biochemical networks, BMC Systems Biology 2008, 2:86 doi:10.1186/1752-0509-2-86
The most accessed paper in BMC Systems Biology in November 2008
Background
Cellular processes such as metabolism, decision making in
development and differentiation, signalling, etc., can be modeled
as large networks of biochemical reactions. In order to understand the
functioning of these systems, there is a strong need for general model
reduction techniques allowing to simplify models without loosing
their main properties. In systems biology we also need to compare models or to
couple them as parts of larger models. In these situations reduction to a
common level of complexity is needed.
Results
We propose a systematic treatment of model reduction of multiscale biochemical networks. First, we consider linear
kinetic models, which appear as "pseudo-monomolecular" subsystems of multiscale nonlinear reaction networks. For such linear
models, we propose a reduction algorithm which is based on a generalized theory
of the limiting step that we have developed in (Gorban
and Radulescu 2008). Second, for non-linear systems
we develop an algorithm based on dominant solutions of quasi-stationarity equations. For oscillating systems, quasi-stationarity and averaging are combined to eliminate time
scales much faster and much slower than the period of the oscillations. In all
cases, we obtain robust simplifications and also identify the critical
parameters of the model. The methods are demonstrated for simple examples and
for a more complex model of NFkB pathway.
Conclusions
Our approach allows critical parameter identification and
produces hierarchies of models. Hierarchical modeling
is important in "middle-out" approaches when there is need to zoom in
and out several levels of complexity. Critical parameter identification is an
important issue in systems biology with potential applications to biological
control and therapeutics. Our approach also deals naturally with the presence
of multiple time scales, which is a general property of systems biology models.
A.N. Gorban and O. Radulescu,
Dynamic and Static Limitation in Multiscale
Reaction Networks, Revisited, Advances
in Chemical Engineering 34, 103-173. GorbanRadulescuAdvChemEng2008.pdf
Local copy
The concept of the limiting step gives the limit simplification: the whole network behaves as a single step. This is the most popular approach for model simplification in chemical kinetics. However, in its elementary form this idea is applicable only to the simplest linear cycles in steady states. For simple cycles the nonstationary behavior is also limited by a single step, but not the same step that limits the stationary rate. In this chapter, we develop a general theory of static and dynamic limitation for all linear multiscale networks. Our main mathematical tools are auxiliary discrete dynamical systems on finite sets and specially developed algorithms of cycles surgery for reaction graphs. New estimates of eigenvectors for diagonally dominant matrices are used.
Multiscale ensembles of reaction networks with well-separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors (modes) is presented. In particular, we prove that for systems with well-separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose the selection of such modules
that it is possible to solve the kinetic equation for every module in the explicit form. All such solvable networks are described. The obtained multiscale approximations, that we call dominant systems are computationally cheap and robust. These dominant systems can be used for direct computation of steady states and relaxation dynamics, especially when kinetic information is incomplete, for design of experiments and mining of experimental data, and could serve as a robust first approximation in perturbation theory or for preconditioning.
A. N. Gorban,
Selection Theorem for
Systems with Inheritance,
Math. Model. Nat. Phenom., Vol. 2, No. 4,
2007, pp. 1-45. GOtborMMNP2(4)2007.pdf Local copy. The original publication is
available at www.edpsciences.org
The problem of finite-dimensional asymptotics
of infinite-dimensional dynamic systems is studied. A non-linear kinetic system
with conservation of supports for distributions has generically
finite-dimensional asymptotics. Such systems are
apparent in many areas of biology, physics (the theory of parametric wave
interaction), chemistry and economics. This conservation of support has a
biological interpretation: inheritance. The finite-dimensional asymptotics demonstrates effects of natural selection.
Estimations of the asymptotic dimension are presented. After some initial time,
solution of a kinetic equation with conservation of support becomes a finite
set of narrow peaks that become increasingly narrow over time and move
increasingly slowly. It is possible that these peaks do not tend to fixed
positions, and the path covered tends to infinity as t→∞.
The drift equations for peak motion are obtained. Various types of
distribution stability are studied: internal stability (stability with respect
to perturbations that do not extend the support), external stability or uninvadability (stability with respect to strongly small
perturbations that extend the support), and stable realizability
(stability with respect to small shifts and extensions of the density peaks).
Models of self-synchronization of cell division are studied, as an example of
selection in systems with additional symmetry. Appropriate construction of the
notion of typicalness in infinite-dimensional space
is discussed, and the notion of completely thin sets is introduced.
R. A. Brownlee, A. N. Gorban, and
J. Levesley,
Nonequilibrium entropy limiters in lattice Boltzmann methods, Physica A: Statistical Mechanics
and its Applications
Volume 387, Issues 2-3,
We construct a system of nonequilibrium entropy limiters for the lattice Boltzmann methods (LBM). These limiters erase spurious oscillations without blurring of shocks, and do not affect smooth solutions. In general, they do the same work for LBM as flux limiters do for finite differences, finite volumes and finite elements methods, but for LBM the main idea behind the construction of nonequilibrium entropy limiter schemes is to transform a field of a scalar quantity nonequilibrium entropy. There are two families of limiters: (i) based on restriction of nonequilibrium entropy (entropy trimming) and (ii) based on filtering of nonequilibrium entropy (entropy filtering). The physical properties of LBM provide some additional benefits: the control of entropy production and accurate estimation of introduced artificial dissipation are possible. The constructed limiters are tested on classical numerical examples: 1D athermal shock tubes with an initial density ratio 1:2 and the 2D lid-driven cavity for Reynolds numbers between 2000 and 7500 on a coarse 100×100 grid. All limiter constructions are applicable both for entropic and for non-entropic equilibria.
A.N. Gorban
and O. Radulescu
Dynamical
robustness of biological networks with hierarchical distribution of time scales,
IET Syst. Biol., 2007,
1, (4), pp. 238246 Gorban2007IEESystemsBiology.pdf
Concepts of distributed robustness and
r-robustness proposed by biologists to explain a variety of stability phenomena
in molecular biology are analysed. Then, the robustness of the relaxation time
using a chemical reaction description of genetic and signalling networks is
discussed. First, the following result for linear networks is obtained: for
large multiscale systems with hierarchical
distribution of time scales, the variance of the inverse relaxation time (as
well as the variance of the stationary rate) is much lower than the variance of
the separate constants. Moreover, it can tend to 0 faster than 1/n, where n is
the number of reactions. Similar phenomena are valid in the nonlinear case as
well. As a numerical illustration, a model of signalling network is used for
the important transcription factor NFkB.
A.N. Gorban and A.Y. Zinovyev
The Mystery of Two Straight Lines in
Bacterial Genome Statistics, Bulletin
of Mathematical Biology (2007) DOI
10.1007/s11538-007-9229-6 (Online First) GorbanZinovyev2007BMB1.pdf
In special coordinates (codon position-specific nucleotide frequencies),
bacterial genomes form two straight lines in 9-dimensional space: one line for eubacterial genomes, another for archaeal
genomes. All the 348 distinct bacterial genomes available in Genbank in April 2007, belong to these lines with high
accuracy. The main challenge now is to explain the observed high accuracy. The
new phenomenon of complementary symmetry for codon position-specific nucleotide
frequencies is observed. The results of analysis of several codon usage models
are presented.We demonstrate that the mean-field
approximation, which is also known as context-free, or complete independence
model, or Segre variety, can serve as a reasonable approximation to the real
codon usage. The first two principal components of codon usage correlate
strongly with genomic G+C content and
the optimal growth temperature, respectively. The variation of codon usage
along the third component is related to the curvature of the mean-field
approximation. First three eigenvalues in codon usage PCA explain 59.1%, 7.8%
and 4.7% of variation. The eubacterial and archaeal genomes codon usage is clearly distributed along
two third order curves with genomic G+C content as a parameter.
A.N. Gorban, O. Radulescu
Dynamic and static limitation in
reaction networks, revisited, http://arxiv.org/abs/physics/0703278
[physics.chem-ph] GorRadLimarXiv0703278v2.pdf
The concept of limiting step gives the limit simplification: the whole network
behaves as a single step. This is the most popular approach for model
simplification in chemical kinetics. However, in its simplest form this idea is
applicable only to the simplest linear cycles in steady states. For such the
simplest cycles the nonstationary behaviour is also limited by a single step,
but not the same step that limits the stationary rate. In this paper, we
develop a general theory of static and dynamic limitation for all linear
multiscale networks, not only for simple cycles. Our main mathematical tools
are auxiliary discrete dynamical systems on finite sets and specially developed
algorithms of ``cycles surgery" for reaction graphs. New estimates of
eigenvectors for diagonally dominant matrices are used.
Multiscale ensembles of reaction networks with well separated
constants are introduced and typical properties of such systems are studied.
For any given ordering of reaction rate constants the explicit approximation of
steady state, relaxation spectrum and related eigenvectors (``modes") is
presented. In particular, we proved that for systems with well separated
constants eigenvalues are real (damped oscillations are improbable). For
systems with modular structure, we propose to select such modules that it is
possible to solve the kinetic equation for every module in the explicit form.
All such ``solvable" networks are described. The obtained multiscale
approximations that we call ``dominant systems" are computationally cheap
and robust. These dominant systems can be used for direct computation of steady
states and relaxation dynamics, especially when kinetic information is
incomplete, for design of experiments and mining of experimental data, and
could serve as a robust first approximation in perturbation theory or for
preconditioning.
R.A. Brownlee, A.N. Gorban, J. Levesley,
Nonequilibrium entropy limiters in
lattice Boltzmann methods, arXiv:0704.0043v1
[cond-mat.stat-mech] BrowGorLevLimitersArXiv.pdf
We construct a
system of nonequilibrium entropy limiters for the lattice Boltzmann methods
(LBM). These limiters erase spurious oscillations without blurring of shocks,
and do not affect smooth solutions. In general, they do the same work for LBM
as flux limiters do for finite differences, finite volumes and finite elements
methods, but for LBM the main idea behind the construction of nonequilibrium
entropy limiter schemes is to transform a field of a scalar quantity -
nonequilibrium entropy. There are two families of limiters: (i) based on
restriction of nonequilibrium entropy (entropy "trimming") and (ii)
based on filtering of nonequilibrium entropy (entropy filtering). The physical
properties of LBM provide some additional benefits: the control of entropy
production and accurate estimate of introduced artificial dissipation are
possible. The constructed limiters are tested on classical numerical examples:
1D athermal shock tubes with an initial density ratio 1:2 and the 2D lid-driven
cavity for Reynolds numbers Re between 2000 and 7500 on a coarse 100*100 grid.
All limiter constructions are applicable for both entropic and non-entropic
quasiequilibria.
R. A. Brownlee, A. N. Gorban, and
J. Levesley,
Stability and stabilization of the
lattice Boltzmann method, Phys.
Rev. E 75, 036711 (2007) (17 pages) BGJPhyRev2007.pdf
We revisit the classical stability versus accuracy dilemma for the
lattice Boltzmann methods (LBM). Our goal is a stable method of
second-order accuracy for fluid dynamics based on the lattice Bhatnager-Gross-Krook method
(LBGK). The LBGK scheme can be recognized as a discrete dynamical
system generated by free flight and entropic involution. In this
framework the stability and accuracy analysis are more natural. We
find the necessary and sufficient conditions for second-order accurate
fluid dynamics modeling. In particular, it is proven
that in order to guarantee second-order accuracy the distribution
should belong to a distinguished surfacethe invariant film (up to
second order in the time step). This surface is the trajectory of
the (quasi)equilibrium distribution surface under free flight. The main
instability mechanisms are identified. The simplest recipes for
stabilization add no artificial dissipation (up to second order) and
provide second-order accuracy of the method. Two other prescriptions
add some artificial dissipation locally and prevent the system from
loss of positivity and local blowup.
Demonstration of the proposed stable LBGK schemes are provided by
the numerical simulation of a one-dimensional (1D) shock tube and
the unsteady 2D flow around a square cylinder up to Reynolds number
Re~20,000.
E. Chiavazzo,
A.N. Gorban, and I.V. Karlin,
Comparison of Invariant Manifolds for
Model Reduction in Chemical Kinetics, Commun. Comput.
Phys. Vol. 2, No. 5 (2007), pp. 964-992 CiCP2007vol2_n5_p964.pdf
A modern approach to model reduction in
chemical kinetics is often based on the notion of slow invariant manifold. The
goal of this paper is to give a comparison of various methods of construction
of slow invariant manifolds using a simple Michaelis-Menten
catalytic reaction. We explore a recently introduced Method of Invariant Grids
(MIG) for iteratively solving the invariance equation. Various initial
approximations for the grid are considered such as Quasi Equilibrium Manifold,
Spectral Quasi Equilibrium Manifold, Intrinsic Low Dimensional Manifold and
Symmetric Entropic Intrinsic Low Dimensional Manifold. Slow invariant manifold
was also computed using the Computational Singular Perturbation (CSP) method. A
comparison between MIG and CSP is also reported.
A.N. Gorban, N.R.
Sumner, and A.Y. Zinovyev,
Topological grammars for data approximation, Applied Mathematics Letters Volume
20, Issue 4 (2007), 382-386 GorSummnZinAML2006.pdf
A method of topological
grammars is proposed for multidimensional data approximation. For data with
complex topology we define a principal cubic complex of low dimension
and given complexity that gives the best approximation for the dataset. This
complex is a generalization of linear and non-linear principal manifolds and
includes them as particular cases. The problem of optimal principal complex
construction is transformed into a series of minimization problems for
quadratic functionals. These quadratic functionals have a physically transparent interpretation in
terms of elastic energy. For the energy computation, the whole complex is
represented as a system of nodes and springs. Topologically, the principal
complex is a product of one-dimensional continuums (represented by graphs), and
the grammars describe how these continuums transform during the process of
optimal complex construction. This factorization of the whole process onto
one-dimensional transformations using minimization of quadratic energy functionals allows us to construct efficient algorithms.
A.N. Gorban,
Orderdisorder
separation: Geometric revision, Physica A Volume
374, Issue 1 , 15 January 2007, Pages 85-102 GorPhysA2006Order.pdf
After Boltzmann and Gibbs, the notion of disorder in statistical physics
relates to ensembles, not to individual states. This disorder is measured by
the logarithm of ensemble volume, the entropy. But recent results about measure
concentration effects in analysis and geometry allow us to return from the
ensemble-based point of view to a state-based one, at least, partially. In this
paper, the orderdisorder problem is represented as a problem of relation
between distance and measure. The effect of strong orderdisorder separation
for multiparticle systems is described: the phase
space could be divided into two subsets, one of them (set of disordered states)
has almost zero diameter, the second one has almost zero measure. The symmetry
with respect to permutations of particles is responsible for this type of
concentration. Dynamics of systems with strong orderdisorder separation has
high average acceleration squared, which can be interpreted as evolution
through a series of collisions (acceleration-dominated dynamics). The time
arrow direction from order to disorder follows from the strong orderdisorder
separation. But, inverse, for systems in space of symmetric configurations with
sticky boundaries the way back from disorder to order is typical (Natural
selection). Recommendations for mining of molecular dynamics results are also
presented.
Ovidiu Radulescu,
Alexander N. Gorban, Sergei Vakulenko,
Andrei Zinovyev
Hierarchies and modules in complex
biological systems, In: Proceedings of European Conference on Complex Systems (paper
ECCS06-114), Oxford, UK, September 2006. http://complexsystems.lri.fr/FinalReview/FILES/PDF/p114.pdf
or OxfordHiModP114.pdf
We review several
mathematical methods allowing to identify modules and hierarchies with several
levels of complexity in biological systems. These methods are based either on
the properties of the input-output characteristic of the modules or on global
properties of the dynamics such as the distribution of timescales or the
stratification of attractors with variable dimension. We also discuss the
consequences of the hierarchical structure on the robustness of biological
processes. Stratified attractors lead to Waddington's type canalization
effects. Successive application of the many to one mapping relating parameters
of different levels in an hierarchy of models (analogue to the renormalization
operation from statistical mechanics) leads to concentration and robustness of
those properties that are common to many levels of complexity. Examples such as
the response of the transcription factor NFB to signalling, and the
segmentation patterns in the development of Drosophila are used as
illustrations of the theoretical ideas.
R. A. Brownlee, A. N. Gorban, and
J. Levesley,
Stabilization of the lattice Boltzmann
method using the Ehrenfests' coarse-graining idea,
Phys. Rev. E 74, 037703 (2006) RobBrowGorbLeveslPRE2006.pdf
The lattice Boltzmann method (LBM) and its variants have emerged as
promising, computationally efficient and increasingly popular numerical
methods for modeling complex fluid flow.
However, it is acknowledged that the method can demonstrate
numerical instabilities, e.g., in the vicinity of shocks. We propose
a simple technique to stabilize the LBM by monitoring the difference
between microscopic and macroscopic entropy. Populations are returned
to their equilibrium states if a threshold value is exceeded. We
coin the name Ehrenfests' steps for
this procedure in homage to the vehicle that we use to introduce
the procedure, namely, the Ehrenfests'
coarse-graining idea.
A.N. Gorban, B.M. Kaganovich,
S.P. Filippov, A.V. Keiko, V.A. Shamansky,
I.A. Shirkalin,
Thermodynamic
Equilibria and Extrema:
Analysis of Attainability Regions and Partial Equilibria,
Springer, Berlin-Heidelberg-New York, 2006.
Model
Reduction and Coarse--Graining Approaches for Multiscale
Phenomena,
Ed. by Alexander N. Gorban, Nikolaos Kazantzis, Ioannis G. Kevrekidis, Hans
Christian Öttinger, Constantinos
Theodoropoulos ,
Springer,
A. Gorban,
Invariant Grids: Method of Complexity Reduction in Reaction Networks, Complexus, V.
2, 110127. ComPlexUs2006.pdf
Complexity in the description of big chemical reaction
networks has both structural (number of species and reactions) and temporal
(very different reaction rates) aspects. A consistent way to make model
reduction is to construct the invariant manifold which describes the asymptotic
system behaviour. In this paper we present a discrete analogue of this object:
an invariant grid. The invariant grid is introduced independently from the
invariant manifold notion and can serve to represent the dynamic system
behaviour as well as to approximate the invariant manifold after refinement.
The method is designed for pure dissipative systems and widely uses their
thermodynamic properties but allows also generalizations for some classes of open
systems. The method is illustrated by two examples: the simplest catalytic
reaction (Michaelis-Menten mechanism) and the
hydrogen oxidation.
A.N. Gorban,
Basic Types of Coarse-Graining, e-print http://arxiv.org/abs/cond-mat/0602024
(local copy CoaGrWorkSpri7.pdf).
42 pgs, 11 figs. A talk given at the
research workshop: "Model Reduction and
Coarse-Graining Approaches for Multiscale Phenomena,"
We consider two
basic types of coarse-graining: the Ehrenfest's coarse-graining and its
extension to a general principle of non-equilibrium thermodynamics, and the coarse-graining
based on uncertainty of dynamical models and $\epsilon$-motions (orbits).
Non-technical discussion of basic notions and main coarse-graining theorems are
presented: the theorem about entropy overproduction for the Ehrenfest's
coarse-graining and its generalizations, both for conservative and for
dissipative systems, and the theorems about stable properties and the Smale
order for $\epsilon$-motions of general dynamical systems including
structurally unstable systems. A brief discussion of two other types,
coarse-graining by rounding and by small noise, is also presented.
Computational kinetic models of macroscopic dynamics are considered. We
construct a theoretical basis for these kinetic models using generalizations of
the Ehrenfest's coarse-graining.
A.N. Gorban, I.V. Karlin,
Quasi-Equilibrium Closure Hierarchies for the Boltzmann Equation, Physica A 360 (2006) 325364 GKQEBoltzPhysA2006.pdf
In this paper, explicit method of
constructing approximations (the
Triangle Entropy Method) is developed for nonequilibrium
problems. This method enables one to
treat any complicated nonlinear functionals that fit
best the physics of a problem (such as, for
example, rates of processes) as new independent variables.
The work of the method was demonstrated on the Boltzmann's - type kinetics.
New macroscopic variables are
introduced (moments of the Boltzmann
collision integral, or scattering rates). They are treated as
independent variables rather than as infinite moment series. This approach gives the complete account
of rates of scattering
processes. Transport equations for scattering rates are obtained
(the second hydrodynamic chain), similar
to the usual moment chain (the first
hydrodynamic chain). Various examples of the closure of the first, of the
second, and of the mixed hydrodynamic chains
are considered for the hard spheres model. It is shown, in particular, that the complete
account of scattering processes leads to a renormalization of transport
coefficients.
The method gives the explicit solution for the closure problem, provides
thermodynamic properties of reduced models, and can be applied to any kinetic
equation with a thermodynamic Lyapunov function
2005
A. Gorban, A. Zinovyev,
Elastic Principal Graphs and Manifolds
and their Practical Applications, Computing 75, 359379 (2005), (DOI) 10.1007/s00607-005-0122-6 , GorbZin2005Computing.pdf
Principal manifolds serve as useful tool for many practical applications. These manifolds are defined as lines or surfaces passing through the middle of data distribution. We propose an algorithm for fast construction of grid approximations of principal manifolds with given topology. It is based on analogy of principal manifold and elastic membrane. First advantage of this method is a form of the functional to be minimized which becomes quadratic at the step of the vertices position refinement. This makes the algorithm very effective, especially for parallel implementations. Another advantage is that the same algorithmic kernel is applied to construct principal manifolds of different dimensions and topologies. We demonstrate how flexibility of the approach allows numerous adaptive strategies like principal graph constructing, etc. The algorithm is implemented as a C++ package elmap and as a part of stand-alone data visualization tool VidaExpert, available on the web. We describe the approach and provide several examples of its application with speed performance characteristics.
A.N. Gorban, I.V. Karlin,
Invariance correction to Grad's equations: Where to go beyond
approximations? Continuum Mechanics and Thermodynamics, 17(4) (2005), 311335, GorKarCMT_05.pdf,
http://arxiv.org/abs/cond-mat/0504221
We review some recent developments of Grad's approach to solving the Boltzmann
equation and creating reduced description. The method of invariant manifold is
put forward as a unified principle to establish corrections to Grad's
equations. A consistent derivation of regularized Grad's equations in the
framework the method of invariant manifold is given. A new class of kinetic
models to lift the finite-moment description to a kinetic theory in the whole
space is established. Relations of Grad's approach to modern mesoscopic integrators such as the entropic lattice
Boltzmann method are also discussed.
A.N. Gorban, T.G.Popova, A.Yu. Zinovyev,
Codon usage trajectories and 7-cluster structure of 143 complete bacterial
genomic sequences Physica A:
Statistical and Theoretical Physics, 353C (2005), 365-387. CodonPhysA2005.pdf (Number 11 in TOP25
articles within the journal: Physica A: Statistical Mechanics and its
Applications, APR - JUN 2005 Top25.pdf)
Three
results are presented. First, we prove the existence of a universal 7-cluster
structure in all 143 completely sequenced bacterial genomes available in Genbank in August 2004, and explained its properties. The
7-cluster structure is responsible for the main part of sequence heterogeneity
in bacterial genomes. In this sense, our 7 clusters is the basic model of
bacterial genome sequence. We demonstrated that there are four basic
``pure" types of this model, observed in nature: ``parallel
triangles", ``perpendicular triangles", degenerated case and the
flower-like type.
Second, we answered the question: how big are the position-specific information
and the contribution connected with correlations between nucleotide. The
accuracy of the mean-field (context-free) approximation is estimated for
bacterial genomes.
We show that codon usage of bacterial genomes is a multi-linear function of
their genomic G+C-content with high accuracy (more precisely, by two similar
functions, one for eubacterial genomes and the other
one for archaea). Description of these two
codon-usage trajectories is the third result.
All 143 cluster animated 3D-scatters are collected in a database and is made
available on our web-site: http://www.ihes.fr/~zinovyev/7clusters
.
A.N. Gorban, T.G.Popova,
A.Yu. Zinovyev,
Four basic symmetry types in the universal 7-cluster structure of microbial
genomic sequences, In Silico Biology, 5 (2005),
0039. Internet site CLUSTER STRUCTURE IN GENOME
with analysis of all bacterial genomes.
Coding information is the main source of heterogeneity (non-randomness) in the
sequences of microbial genomes. The heterogeneity corresponds to a cluster
structure in triplet distributions of relatively short genomic fragments
(200-400bp). We found a universal 7-cluster structure in microbial genomic
sequences and explained its properties. We show that codon usage of bacterial
genomes is a multi-linear function of their genomic G+C-content with high
accuracy. Based on the analysis of 143 completely sequenced bacterial genomes
available in Genbank in August 2004, we show that
there are four "pure" types of the 7-cluster structure observed. All
143 cluster animated 3D-scatters are collected in a database which is made
available on our web-site (http://www.ihes.fr/~zinovyev/7clusters).
The findings can be readily introduced into software for gene prediction,
sequence alignment or microbial genomes classification.
A.N. Gorban, I.V. Karlin,
Invariant Manifolds for Physical and Chemical Kinetics, Lect. Notes
Phys. 660, Springer, Berlin, Heidelberg, 2005 (498 pages). [Preface-Contents-Introduction(pdf)][Reviews(htm)]
A.N. Gorban, T.G. Popova,
A.Yu. Zinovyev,
Four basic symmetry types in the universal 7-cluster structure of 143
complete bacterial genomic sequences E-print: http://arxiv.org/abs/q-bio/0410033
coding information is the main source of heterogeneity (non-randomness) in the
sequences of bacterial genomes. This information can be naturally modeled by analysing cluster structures in the
"in-phase" triplet distributions of relatively short genomic
fragments (200-400bp). We found a universal 7-cluster structure in bacterial
genomic sequences and explained its properties. We show that codon usage of
bacterial genomes is a multi-linear function of their genomic G+C-content with
high accuracy. Based on the analysis of 143 completely sequenced bacterial genomes
available in Genbank in August 2004, we show that
there are four "pure" types of the 7-cluster structure observed. All
143 cluster animated 3D-scatters are collected in a database and is made
available on our web-site: http://www.ihes.fr/~zinovyev/7clusters.
The finding can be readily introduced into any software for gene prediction,
sequence alignment or bacterial genomes classification
Gorban, I.N.;Zinovyev,
A.Y.
Elastic principal manifolds and their practical applications E-print http://arxiv.org/abs/cond-mat/0405648
Principal manifolds defined as lines or surfaces passing through "the
middle" of the data distribution serve as useful objects for many
practical applications. We propose a new algorithm for fast construction of
grid approximations of principal manifolds with given topology. One advantage
of the method is a new form of the functional to be minimized, which becomes
quadratic at the step of the vertexes positions refinement. This makes the
algorithm very effective, especially for parallel implementations. Another
advantage is that the same algorithmic kernel is applied to construct principal
manifolds of different dimensions and topologies. We demonstrate how
flexibility of the approach allows easily numerous adaptive strategies like
principal graph constructing, etc. The algorithm is implemented as a C++
package elmap and as a part of stand-alone data
visualization tool VidaExpert, available on the web.
We describe the approach and provide several examples of its applications with
speed performance characteristics.
Gorban, A.N.
Systems with inheritance: dynamics of distributions with conservation of
support, natural selection and finite-dimensional asymptotics
E-print: http://arxiv.org/abs/cond-mat/0405451
If we find a representation of an infinite-dimensional dynamical system as a
nonlinear kinetic system with {\it conservation of supports} of distributions,
then (after some additional technical steps) we can state that the asymptotics is finite-dimensional. This conservation of
support has a {\it quasi-biological interpretation, inheritance} (if a gene was
not presented initially in a isolated population without mutations, then it
cannot appear at later time). These quasi-biological models can describe
various physical, chemical, and, of course, biological systems. The
finite-dimensional asymptotic demonstrates effects of {\it "natural"
selection}. The estimations of asymptotic dimension are presented. The support
of an individual limit distribution is almost always small. But the union of
such supports can be the whole space even for one solution. Possible are such
situations: a solution is a finite set of narrow peaks getting in time more and
more narrow, moving slower and slower. It is possible that these peaks do not
tend to fixed positions, rather they continue moving, and the path covered
tends to infinity at $t \to \infty$. The {\it drift
equations} for peaks motion are obtained. Various types of stability are
studied. In example, models of cell division self-synchronization are studied.
The appropriate construction of notion of typicalness
in infinite-dimensional spaces is discussed, and the "completely
thin" sets are introduced
Gorban, A.N.
Singularities of transition processes in dynamical systems: Qualitative
theory of critical delays Electron. J. Diff. Eqns. Monograph 5, 2004, 55 p.Slorelax2004EJDE.pdf Online: http://ejde.math.swt.edu/Monographs/05/abstr.html
This monograph presents a systematic analysis of the singularities in the
transition processes for dynamical systems. We study general dynamical systems,
with dependence on a parameter, and construct relaxation times that depend on
three variables: Initial conditions x, parameters k of the system, and accuracy
e of the relaxation. We study the singularities of relaxation times as
functions of (x,k) under fixed e, and then classify
the bifurcations (explosions) of limit sets. We study the relationship between
singularities of relaxation times and bifurcations of limit sets. An analogue
of the Smale order for general dynamical systems
under perturbations is constructed. It is shown that the perturbations simplify
the situation: the interrelations between the singularities of relaxation times
and other peculiarities of dynamics for general dynamical system under small
perturbations are the same as for the Morse-Smale
systems
Gorban, A.N.;Gorban,
P.A.;Karlin, I.V.
Legendre integrators, post-processing and quasiequilibrium
J. Non-Newtonian Fluid Mech. 120 (2004) 149167GoGoKar2004.pdf
Online: http://arxiv.org/abs/cond-mat/0308488
A toolbox for the development and reduction of the dynamical models of nonequilibrium systems is presented. The main components of
this toolbox are: Legendre integrators, dynamical post-processing, and the
thermodynamic projector. The thermodynamic projector is the tool to transform
almost any anzatz to a thermodynamically consistent
model. The post-processing is the cheapestway to
improve the solution obtained by the Legendre integrators. Legendre integrators
give the opportunity to solve linear equations instead of nonlinear ones for quasiequilibrium (maximum entropy, MaxEnt)
approximations. The essentially new element of this toolbox, the method of
thermodynamic projector, is demonstrated on application to the FENE-P model of
polymer kinetic theory. The multi-peak model of polymer dynamics is developed.
Gorban, A.N.;Karlin,
I.V.
Uniqueness of thermodynamic projector and kinetic basis of molecular
individualism Physica A, 336, 2004, 391-432
UniMolIndRepr.pdf Online: http://arxiv.org/abs/cond-mat/0309638
Three results are presented: First, we solve the problem of persistence of
dissipation for reduction of kinetic models. Kinetic equations with
thermodynamic Lyapunov functions are studied.
Uniqueness of the thermodynamic projector is proven: There exists only one
projector which transforms any vector field equipped with the given Lyapunov function into a vector field with the same Lyapunov function for a given anzatz
manifold which is not tangent to the Lyapunov
function levels. Second, we use the thermodynamic projector for developing the
short memory approximation and coarse-graining for general nonlinear dynamic
systems. We prove that in this approximation the entropy production increases.
(The theorem about entropy overproduction.) In example, we apply the thermodynamic
projector to derive the equations of reduced kinetics for the Fokker-Planck
equation. A new class of closures is developed, the kinetic multipeak
polyhedra. Distributions of this type are expected in
kinetic models with multidimensional instability as universally as the Gaussian
distribution appears for stable systems. The number of possible relatively
stable states of a nonequilibrium system grows as
2^m, and the number of macroscopic parameters is in order mn,
where n is the dimension of configuration space, and m is the number of
independent unstable directions in this space. The elaborated class of closures
and equations pretends to describe the effects of molecular individualism. This
is the third result.
Gorban, A.N.;Karlin,
I.V.;Zinovyev, A.Y.
Constructive methods of invariant manifolds for kinetic problems Phys.
Rep., 396, 2004, 197-403 PhysRepCorr.pdf Online: http://arxiv.org/abs/cond-mat/0311017
The concept of the slow invariant manifold is recognized as the central idea
underpinning a transition from micro to macro and model reduction in kinetic
theories. We present the Constructive Methods of Invariant Manifolds for model
reduction in physical and chemical kinetics, developed during last two decades.
The physical problem of reduced description is studied in the most general form
as a problem of constructing the slow invariant manifold. The invariance conditions
are formulated as the differential equation for a manifold immersed in the
phase space (the invariance equation). The equation of motion for immersed
manifolds is obtained (the film extension of the dynamics). Invariant manifolds
are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as
stability.
A collection of methods to derive analytically and to compute numerically the
slow invariant manifolds is presented. Among them, iteration methods based on
incomplete linearization, relaxation method and the method of invariant grids
are developed. The systematic use of thermodynamics structures and of the
quasi-chemical representation allow to construct approximations which are in
concordance with physical restrictions.
The following examples of applications are presented: nonperturbative
derivation of physically consistent hydrodynamics from the Boltzmann equation
and from the reversible dynamics, for Knudsen numbers Kn~1; construction of the
moment equations for nonequilibrium media and their
dynamical correction (instead of extension of list of variables) to gain more
accuracy in description of highly nonequilibrium
flows; determination of molecules dimension (as diameters of equivalent hard
spheres) from experimental viscosity data ; model reduction in chemical
kinetics; derivation and numerical implementation of constitutive equations for
polymeric fluids; the limits of macroscopic description for polymer molecules,
etc.
Gorban, A.N.;Karlin,
I.V.;Zinovyev, A.Y.
Invariant grids for reaction kinetics Physica
A, 333, 2004 106-154 ChemGrPhA2004.pdf Online: http://arxiv.org/abs/cond-mat/0307076
In this paper, we review the construction of low-dimensional manifolds of
reduced description for equations of chemical kinetics from the standpoint of
the method of invariant manifold (MIM). MIM is based on a formulation of the
condition of invariance as an equation, and its solution by
A. Yu. Zinovyev, A. N. Gorban,
T. G. Popova
Self-Organizing Approach for Automated Gene Identification Open Sys. &
Information Dyn., 10, 2003, 321-333 GoZiPo2003final.pdf
Self-training technique for automated gene recognition both in entire genomes
and in unassembled ones is proposed. It is based on a simple measure (namely,
the vector of frequencies of non-overlapping triplets in sliding window), and
needs neither predetermined information, nor preliminary learning. The sliding
window length is the only one tuning parameter. It should be chosen close to
the average exon length typical to the DNA text under investigation. An
essential feature of the technique proposed is preliminary visualization of the
set of vectors in the subspace of the first three principal components. It was
shown, the distribution of DNA sites has the bullet-like structure with one
central cluster (corresponding to non-coding sites) and three or six ank ones (corresponding to protein-coding sites). The
bullet-like structure itself revealed in the distribution seems to be very
interesting illustration of triplet usage in DNA sequence. The method was
examined on several genomes (mitochondrion of P.wickerhamii,
bacteria C.crescentus and primitive eukaryot S.cerevisiae). The
percentage of truly predicted nucleotides exceeds 90%.
In October 2004 this paper was mentioned as one of the five most viewed
paper published in the Journal since 1997 http://www.kluweronline.com/issn/1230-1612
.
A. N. Gorban, A. Yu. Zinovyev,
T. G. Popova
Seven clusters in genomic triplet distributions In Silico Biology,
3, 2003, 471-482 (0039), Online: http://arXiv.org/abs/cond-mat/0305681
Motivation: In several recent papers new algorithms were proposed for detecting
coding regions without requiring learning dataset of already known genes. In
this paper we studied cluster structure of several genomes in the space of
codon usage. This allowed to interpret some of the results obtained in other
studies and propose a simpler method, which is, nevertheless, fully functional.
Results: Several complete genomic sequences were analyzed, using visualization
of tables of triplet counts in a sliding window. The distribution of 64-dimensional
vectors of triplet frequencies displays a well-detectable cluster structure.
The structure was found to consist of seven clusters, corresponding to
protein-coding information in three possible phases in one of the two
complementary strands and in the non-coding regions. Awareness of the existence
of this structure allows development of methods for the segmentation of
sequences into regions with the same coding phase and non-coding regions. This
method may be completely unsupervised or use some external information. Since
the method does not need extraction of ORFs, it can be applied even for
unassembled genomes. Accuracy calculated on the base-pair level (both
sensitivity and specificity) exceeds 90%. This is not worse as compared to such
methods as HMM, however, has the advantage to be much simpler and clear.
Availability: The software and datasets are available at http://www.ihes.fr/~zinovyev/bullet
Gorban, A.N.;Karlin,
I.V.,
Method of invariant manifold for chemical kinetics, Chem.
In this paper, we review the construction of low-dimensional manifolds of
reduced description for equations of chemical kinetics from the standpoint of
the method of invariant manifold (MIM). The MIM is based on a formulation of
the condition of invariance as an equation, and its solution by
A. N. Gorban, A. Y. Zinovyev, D.C. Wunsch
Application of The Method of Elastic Maps In Analysis of Genetic Texts, Proceedings
of IJCNN2003 GZW2003.pdf
Method of elastic maps allows to construct efficiently 1D, 2D and 3D non-linear
approximations to the principal manifolds with different topology (piece of
plane, sphere, torus etc.) and to project data onto it. We describe the idea of
the method and demonstrate its applications in analysis of genetic
sequences.
Gorban A.
N.,
Quasi-Equilibrium Closure Hierarchies for The Boltzmann Equation E-print,
http://arXiv.org/abs/cond-mat/0305599
v1
Explicit method of constructing of approximations (Triangle Entropy Method) is
developed for strongly nonequilibrium problems of
Boltzmann's--type kinetics, i.e. when standard moment variables are
insufficient. This method enables one to treat any complicated nonlinear functionals that fit the physics of a problem (such as, for
example, rates of processes) as new independent variables. The method is
applied to the problem of derivation of hydrodynamics from the Boltzmann
equation. New macroscopic variables are introduced (moments of the Boltzmann
collision integral, or collision moments). They are treated as independent
variables rather than as infinite moment series. This approach gives the
complete account of rates of scattering processes. Transport equations for
scattering rates are obtained (the second hydrodynamic chain), similar to the
usual moment chain (the first hydrodynamic chain). Using the triangle entropy
method, three different types of the macroscopic description are considered.
The first type involves only moments of distribution functions, and results
coincide with those of the Grad method in the Maximum Entropy version. The
second type of description involves only collision moments. Finally, the third
type involves both the moments and the collision moments (the mixed
description). The second and the mixed hydrodynamics are sensitive to the
choice of the collision model. The second hydrodynamics is equivalent to the
first hydrodynamics only for Maxwell molecules, and the mixed hydrodynamics
exists for all types of collision models excluding Maxwell molecules. Various
examples of the closure of the first, of the second, and of the mixed
hydrodynamic chains are considered for the hard spheres model. It is shown, in
particular, that the complete account of scattering processes leads to a
renormalization of transport coefficients.
The paper gives English translation of the first part of the paper: Gorban, A. N., Karlin, I. V.,
Quasi-equilibrium approximation and non-standard expansions in the theory of
the Boltzmann kinetic equation, in: "Mathematical Modelling in Biology and
Chemistry. New Approaches", ed. R. G. Khlebopros,
Nauka,
Gorban A.
N.
Neuroinformatics: What are us, where are we
going, how to measure our way? The lecture was given at the USA-NIS Neurocomputing opportunities workshop,
What is neuroinformatics? We can define it as a
direction of science and information technology, dealing with development and
study of the methods for solution of problems by means of neural networks. A
field of science cannot be determined only by fixing what it is "dealing
with". The main component, actually constituting a scientific direction,
is "THE GREAT PROBLEM", around which the efforts are concentrated.
One may state even categorically: if there is no a great problem, there is no a
field of science, but only more or less skilful imitation. What is "THE
GREAT PROBLEM" for neuroinformatics? The problem
of effective parallelism, the study of brain (solution of mysteries of
thinking), etc are discussed. The neuroinformatics
was considered not only as a science, but as a services sector too. The main
ideas of generalized technology of extraction of explicit knowledge from data
are presented. The mathematical achievements generated by neuroinformatics,
the problem of provability of neurocomputations, and
benefits of neural network realization of solution of a problem are discussed.
Gorban A.
N., Karlin I. V.
Geometry of irreversibility: The film of nonequilibrium
states E-print: http://arxiv.org/abs/cond-mat/0308331
A general geometrical framework of nonequilibrium
thermodynamics is developed. The notion of macroscopically definable ensembles is
developed. The thesis about macroscopically definable ensembles is suggested.
This thesis should play the same role in the nonequilibrium
thermodynamics, as the Church-Turing thesis in the theory of computability. The
primitive macroscopically definable ensembles are described. These are
ensembles with macroscopically prepared initial states. The method for
computing trajectories of primitive macroscopically definable nonequilibrium ensembles is elaborated. These trajectories
are represented as sequences of deformed equilibrium ensembles and simple
quadratic models between them. The primitive macroscopically definable
ensembles form the manifold in the space of ensembles. We call this manifold
the film of nonequilibrium states. The equation for
the film and the equation for the ensemble motion on the film are written down.
The notion of the invariant film of non-equilibrium states, and the method of
its approximate construction transform the the
problem of nonequilibrium kinetics into a series of
problems of equilibrium statistical physics. The developed methods allow us to
solve the problem of macro-kinetics even when there are no autonomous equations
of macro-kinetics
Iliya V. Karlin, Larisa
L. Tatarinova, Alexander N. Gorban,
Hans Christian Ottinger
Irreversibility in the short memory approximation Physica A, 327, 2003, 399-424
Online: http://arXiv.org/abs/cond-mat/0305419 v1 18 May
2003 KTGOe2003LANL.pdf
A recently introduced systematic approach to derivations of the macroscopic
dynamics from the underlying microscopic equations of motions in the
short-memory approximation [Gorban et al, Phys. Rev.
E 63 , 066124 (2001)] is presented in detail. The essence of this method is a
consistent implementation of Ehrenfest's idea of
coarse-graining, realized via a matched expansion of both the microscopic and
the macroscopic motions. Applications of this method to a derivation of the
nonlinear Vlasov-Fokker-Planck equation, diffusion
equation and hydrodynamic equations of the uid with a
long-range mean field interaction are presented in full detail. The advantage
of the method is illustrated by the computation of the post-Navier-Stokes
approximation of the hydrodynamics which is shown to be stable unlike the
Burnett hydrodynamics.
Alexander N. Gorban,
Iliya V. Karlin
Family of additive entropy functions out of thermodynamic limit,
Physical Review E 67, 016104, 2003. Online: http://arXiv.org/abs/cond-mat/0205511
We derive a one-parametric family of entropy functions that respect the additivity condition, and which describe effects of
finiteness of statistical systems, in particular, distribution functions with
long tails. This one-parametric family is different from the Tsallis entropies, and is a convex combination of the
Boltzmann- Gibbs-Shannon entropy and the entropy function proposed by Burg. An
example of how longer tails are described within the present approach is worked
out for the canonical ensemble. We also discuss a possible origin of a hidden
statistical dependence, and give explicit recipes on how to construct
corresponding generalizations of the master equation.
Gorban A. N., Karlin I. V.
Geometry of irreversibility, in: Recent Developments in Mathematical and
Experimental Physics, Volume C: Hydrodynamics and Dynamical Systems, Ed. F.
Uribe (Kluwer,
A general geometrical setting of nonequilibrium
thermodynamics is developed. The approach is based on the notion of the natural
projection which generalizes Ehrenfests'
coarse-graining. It is demonstrated how derivations of irreversible macroscopic
dynamics from the microscopic theories can be addressed through a study of
stability of quasiequilibrium manifolds.
A. Gorban, A. Rossiev, N. Makarenko, Y. Kuandykov, V. Dergachev
Recovering data gaps through neural network methods, International
Journal of Geomagnetism and Aeronomy vol. 3, no. 2,
pages 191-197, December 2002 geomag02.pdf
A new method is presented to recover the lost data in geophysical time series.
It is clear that gaps in data are a substantial problem in obtaining correct
outcomes about phenomenon in time series processing. Moreover, using the data
with irregular coarse steps results in the loss of prime information during
analysis. We suggest an approach to solving these problems, that is based on
the idea of modeling the data with the help of
small-dimension manifolds, and it is implemented with the help of a neural
network. We use this approach on real data and show its proper use for
analyzing time series of cosmogenic isotopes. In
addition, multifractal analysis was applied to the
recovered 14C concentration in the Earth's atmosphere.
Gorban A.N.,
Karlin I.V.
Methods of nonlinear kinetics, Contribution to the "Encyclopedia of Life Support Systems" (EOLSS
Publishers,
Nonlinear kinetic equations are reviewed for a wide audience of specialists and
postgraduate students in physics, mathematical physics, material science,
chemical engineering and interdisciplinary research.
Contents:
1. The Boltzmann equation
2. Phenomenology of the Boltzmann equation
3. Kinetic models
4. Methods of reduced description
4.1. The Hilbert method
4.2. The Chapman-Enskog method
4.3. The Grad moment method
4.4. Special approximations
4.5. The method of invariant manifold
4.6. Quasi-equilibrium approximations
5. Discrete velocity models
6. Direct simulation
7. Lattice Gas and Lattice Boltzmann models
8. Other kinetic equations
8.1. The Enskog equation for hard spheres
8.2. The Vlasov equation
8.3. The Smoluchowski equation
Gorban
A.N., Karlin I.V.
Method of invariant manifold for chemical kinetics Online: http://arXiv.org/abs/cond-mat/0207231
v1
In this paper, we review the construction of low-dimensional manifolds of
reduced description for equations of chemical kinetics from the standpoint of the
method of invariant manifold (MIM). MIM is based on a formulation of the
condition of invariance as an equation, and its solution by
Gorban A.N., Karlin I.V.
Hydrodynamics from Grad's equations: What can we learn from exact solutions?
Annalen der Physics, 2002. Online: http://arXiv.org/abs/cond-mat/0209560 v1 24 Sep 2002. annphys02.pdf
A detailed treatment of the classical Chapman-Enskog
derivation of hydrodynamics is given in the framework of Grad's moment
equations. Grad's systems are considered as the minimal kinetic models where
the Chapman-Enskog method can be studied exactly,
thereby providing the basis to compare various approximations in extending the
hydrodynamic description beyond the Navier-Stokes
approximation. Various techniques, such as the method of partial summation, Pad_e approximants, and invariance principle are compared
both in linear and nonlinear situations.
Karlin I.V., Grmela M., Gorban A.N.
Duality in nonextensive statistical mechanics. Physical
Review E, 2002, Volume 65, 036128. P.1-4. PRE362002.pdf
We revisit recent derivations of kinetic equations based on Tsallis
entropy concept. The method of kinetic functions is introduced as a standard
tool for extensions of classical kinetic equations in the framework of Tsallis statistical mechanics. Our analysis of the
Boltzmann equation demonstrates a remarkable relation between thermodynamics
and kinetics caused by the deformation of macroscopic observables.
Gorban A.N., Karlin
I.V., Ottinger H.C.
The additive generalization of the Boltzmann entropy, Physical Review E, 2003,
Volume 67, 067104,. Online: http://arXiv.org/abs/cond-mat/0209319 v1
There exists only one generalization of the classical Boltzmann-Gibbs-Shannon
entropy functional to a one-parametric family of additive entropy functionals. We find analytical solution to the
corresponding extension of the classical ensembles, and discuss in some detail
the example of the deformation of the uncorrelated state.
Gorban A.N., Karlin I.V.
Macroscopic dynamics through coarse-graining: A solvable example,
Physical Review E, 2002, Volume 65, 026116, P.1-5. PREEhr02.pdf
The recently derived fluctuation-dissipation formula (A. N. Gorban
et al., Phys. Rev. E 63, 066124. 2001) is illustrated by the explicit
computation for McKeans kinetic model (H. P. McKean, J. Math. Phys. 8, 547.
1967). It is demonstrated that the result is identical, on the one hand, to the
sum of the Chapman-Enskog expansion, and, on the
other hand, to the exact solution of the invariance equation. The equality
between all three results holds up to the crossover from the hydrodynamic to
the kinetic domain.
Gorban' A., Braverman M., and Silantyev V.
Modified Kirchhoff flow with a partially penetrable obstacle and its
application to the efficiency of free flow turbines, Mathematical and
Computer Modelling, Volume 35, Issue 13, June 2002, P. 1371-1375. MCM2002-2.pdf
An explicitly solvable analog of the Kirchhoff flow
for the case of a semipenetrable obstacle is
considered. Its application to estimating the efficiency of free flow turbines
is discussed.
Gorban' A., Silantyev V.
Riabouchinsky flow with partially penetrable obstacle,
Mathematical and Computer Modelling, Volume 35, Issue 13, June 2002, P.
1365-1370. MCM2002-1.pdf
An explicitly solvable Riabouchinsky model with a
partially penetrable obstacle is introduced. This model applied to the
estimation of the efficiency of free flow turbines allows us to take into
account the pressure drop past the lamina.
Gorban'
A.N., Gorlov A.N., Silantyev
V.M.
Limits of the Turbine Efficiency for Free Fluid Flow, Journal of Energy
Resources Technology - December 2001 - Volume 123, Issue 4, pp. 311-317. Gorlov2001.pdf
An accurate estimate of the theoretical power limit of turbines in free fluid
flows is important because of growing interest in the development of wind power
and zero-head water power resources. The latter includes the huge kinetic
energy of ocean currents, tidal streams, and rivers without dams. Knowledge of
turbine efficiency limits helps to optimize design of hydro and wind power
farms. An explicitly solvable new mathematical model for estimating the maximum
efficiency of turbines in a free (nonducted) fluid is
presented. This result can be used for hydropower turbines where construction
of dams is impossible (in oceans) or undesirable (in rivers), as well as for
wind power farms. The model deals with a finite two-dimensional, partially
penetrable plate in an incompressible fluid. It is nearly ideal for
two-dimensional propellers and less suitable for three-dimensional cross-flow Darrieus and helical turbines. The most interesting finding
of our analysis is that the maximum efficiency of the plane propeller is about
30 percent for free fluids. This is in a sharp contrast to the 60 percent given
by the Betz limit, commonly used now for decades. It is shown that the Betz
overestimate results from neglecting the curvature of the fluid streams. We
also show that the three-dimensional helical turbine is more efficient than the
two-dimensional propeller, at least in water applications. Moreover,
well-documented tests have shown that the helical turbine has an efficiency of
35 percent, making it preferable for use in free water currents.
Gorban A.N., Zinovyev A.Yu.
Visualization of Data by Method of Elastic Maps and its Applications in
Genomics, Economics and Sociology, Institut des Hautes Etudes Scientifiques
Preprint. IHES M/01/36. Online: http://www.ihes.fr/PREPRINTS/M01/Resu/resu-M01-36.html elmap.pdf
Technology of data visualization and data modeling is
suggested. The basic of the technology is original idea of elastic net and
methods of its construction and application. A short review of relevant methods
has been made. The methods proposed are illustrated by applying them to the
real biological, economical, sociological datasets and to some model data
distributions.
Gorban A.N., Karlin I.V., Ilg P., Ottinger H.C.
Corrections and enhancements of quasi-equilibrium states, J. Non-Newtonian
Fluid Mech. 2001, 96, P. 203-219. NonNew01.pdf
We give a compact non-technical presentation of two basic principles for
reducing the description of nonequilibrium systems
based on the quasi-equilibrium approximation. These two principles are:
construction of invariant manifolds for the dissipative microscopic dynamics,
and coarse-graining for the entropy-conserving microscopic dynamics. Two new
results are presented: first, an application of the invariance principle to
hybridization of micromacro integration schemes is introduced, and is
illustrated with non-linear dumbbell models; second, Ehrenfests
coarse-graining is extended to general quasi-equilibrium approximations, which
gives the simplest way to derive dissipative equations from the Liouville equation in the short memory approximation.
Gorban A.N., Karlin I.V., Ottinger H.C., Tatarinova L.L.
Ehrenfests argument extended to a
formalism of nonequilibrium thermodynamics,
Physical Review E, 2001. Volume 63, 066124, P.1-6. PREEhr01.pdf
A general method of constructing dissipative equations is developed, following Ehrenfests idea of coarse graining. The approach resolves
the major issue of discrete time coarse graining versus continuous time
macroscopic equations. Proof of the H theorem for macroscopic equations is
given, several examples supporting the construction are presented, and
generalizations are suggested.
Gorban A.N., Zinovyev A.Yu., Popova T.G.
Self-organizing approach for automated gene identification in whole genomes,
Institut des Hautes Etudes Scientifiques Preprint. IHES. December 12, 2001, Online: http://arXiv.org/abs/physics/0108016
v1 10 Aug 2001 lanlgpz01.pdf
An approach based on using the idea of distinguished coding phase in explicit
form for identi cation of
protein-coding regions in whole genome has been proposed. For several genomes
an optimal window length for averaging GC-content function and calculating
codon frequencies has been found. Self-training procedure based on clustering
in multidimensional space of triplet frequencies is proposed.
Gorban A.N., Zinovyev A.Yu., Popova T.G.
Statistical approaches to automated gene identification without teacher. Institut des Hautes Etudes Scientifiques Preprint. IHES M/01/34. Online: http://www.ihes.fr/PREPRINTS/M01/Resu/resu-M01-34.html geneid.pdf
Overview of statistical methods of gene identification is made. Particular
attention is given to the methods which need not a training set of already
known genes. After analysis several statistical approaches are proposed for
computational exon identification in whole genomes. For several genomes an
optimal window length for averaging GC-content function and calculating codon
frequencies has been found. Self-training procedure based on clustering in
multidimensional codon frequencies space is proposed.
A. N. Gorban,
K. O. Gorbunova, D. C. Wunsch
II
Liquid Brain: Kinetic Model of Structureless
Parallelism, liquidbrain.pdf
A new formal model of parallel computations, the Kirdin
kinetic machine, is suggested. It is expected that this model will play the
role for parallel computations similar to Markov normal algorithms, Kolmogorov
and Turing machine or Post schemes for sequential computations. The basic ways
in which computations are realized are described; correctness of the elementary
programs for the Kirdin kinetic machine is
investigated. It is proved that the determined Kirdin
kinetic machine is an effective calculator. A simple application of the Kirdin kinetic machine, heap encoding, is suggested.
Subprograms similar to usual programming enlarge the Kirdin
kinetic machine.
Gorban A.N., Karlin I.V., Zmievskii V.B., Dymova S.V.
Reduced description in the reaction kinetics, Physica
A, 2000, 275, P.361-379. GKZD2000.pdf
Models of complex reactions in thermodynamically isolated systems often
demonstrate evolution towards low-dimensional manifolds in the phase space. For
this class of models, we suggest a direct method to construct such manifolds,
and thereby to reduce the effective dimension of the problem. The approach
realizes the invariance principle of the reduced description, it is based on
iterations rather than on a small parameter expansion, it leads to tractable
linear problems, and is consistent with thermodynamic requirements. The
approach is tested with a model of catalytic reaction.
Gorban A.N., Popova T.G., Sadovsky M.G.
Classification Of Symbol Sequences Over Thier
Frequency Dictionaries: Towards The Connection Between Structure And Natural
Taxonomy, Open Sys. & Information Dyn. 7:
1-17, 2000. opsygps00.pdf
The classifications of bacterial 16S RNA sequences developed over the real and
transformed frequency dictionaries have been studied. Two sequences considered
to be close each other, when their frequency dictionaries were close in
Euclidean metrics. A procedure to transform a dictionary is proposed that makes
clear some features of the information pattern of a symbol sequence. A
comparative study of classifications developed over the real frequency
dictionaries vs. the transformed ones has been carried out. A correlation
between an information pattern of nucleotide sequences and taxonomy of the
bearer of the sequence was found. The sites with high information value are
found, that were the main factors of the difference between the classes in a
classification. The classification of nucleotide sequences developed over the
real frequency dictionaries of the thickness 3 reveals the best correlation to
a gender of bacteria. A set of sequences of the same gender is included
entirely into one class, as a rule, and the exclusions occur rarely. A
hierarchical classification yields one or two taxonomy groups on each level of
the classification. An unexpectedly often (in comparison to the expected), or
unexpectedly rare occurrence of some sites within a sequence makes a basic difference
between the structure patterns of the classes yielded; a number of those sites
is not too great. Further investigations are necessary in order to compare the
sites revealed with those determined due to other methodology.
A. N. Gorban, I.V. Karlin, and V.B. Zmievskii
Two-Step Approximation of Space-Independent Relaxation, TRANSPORT THEORY AND STATISTICAL
PHYSICS, 28(3) (1999), 271-296. GorKarZmiTTSP99.pdf Local copy
In this paper we introduce a
new method of constructing approximate trajectories for space independent
kinetic equations confirming to the second law of thermodynamics. Classical
examples are the space independent Boltzmann equation and chemical kinetics
equations for closed homogeneous
systems. This family of kinetic equations is characterized by the following
general properties:
(1). There exists a set of
functions which remain constant on a solution (these are density, momentum and
energy in context of the Boltzmann equation).
(ii). There exists a convex
function which monotonically decreases along any solution from its value in the
initial state to an absolute minima in the final equilibrium state (this is the
H-theorem for the Boltzmann equation) .
Usually we do know only the
initial and the final (equilibrium) states, and the kinetic equation neither
can be solved exactly, nor contains small parameters to develop a reliable
perturbation theory. Still, we would like to get (perhaps a rather rough but a
simple) approximation of the relaxation trajectory.
An
express method to approximate trajectories of space independent kinetic
equations is developed. It involves a two-step treatment of relaxation through
a quasiequilibria located on a line emerging from the
initial state in the direction prescribed by the kinetic equation. A test
for the Boltzmann equation shows the validity of the method.
A.N. Gorban, A.A. Rossiev, D. C. Wunsch II
Neural Network Modeling of Data with Gaps: Method
of Principal Curves, Carleman's Formula, and Other, The
talk was given at the USA-NIS Neurocomputing
opportunities workshop,
Online: http://arXiv.org/abs/cond-mat/0305508
A method of modeling data with gaps by a sequence of
curves has been developed. The new method is a generalization of iterative construction
of singular expansion of matrices with gaps. Under discussion are three
versions of the method featuring clear physical interpretation:
1) linear: modeling the data by a sequence of linear
manifolds of small dimension;
2) quasilinear: constructing "principal
curves": (or "principal surfaces"), univalently
projected on the linear principal components;
3) essentially non-linear, based on constructing "principal curves":
(principal strings and beams) employing the variation principle; the iteration
implementation of this method is close to Kohonen
self-organizing maps.
The derived dependencies are extrapolated by Carlemans
formulas. The method is interpreted as a construction of neural network
conveyor designed to solve the following problems:
1) to fill gaps in data;
2) to repair data, to correct initial data values in such a way as to make the
constructed models work best;
3) to construct a calculator to fill gaps in the data line fed to the input.
Gorban A. N.
Neuroinformatics: What are us, where are we
going, how to measure our way? The lecture was given at the USA-NIS Neurocomputing opportunities workshop ( http://phy025.lubb.ttuhsc.edu/wldb/Witali/WWW/NSF_NOW.html
),
What is neuroinformatics? For me here and now neuroinformatics is a direction of science and information
technology, dealing with development and study of the methods for solution of
problems by means of neural networks. A base example of artificial neural network,
which will be referred to below, is a feed-forward network from standard
neurons.
Alexander N. Gorban,
Eugeniy M. Mirkes and
Victor G. Tsaregorodtsev
Generation of Explicit Knowledge from Empirical Data through Pruning of
Trainable Neural Networks, International Joint Conference on Neural
Networks, Washington, DC July 10-16, 1999. ( http://www.cas.american.edu/~medsker/ijcnn99/ijcnn99.html
). know.pdf E-print: http://arxiv.org/abs/cond-mat/0307083
This paper presents a generalized technology of extraction of explicit
knowledge from data. The main ideas are:
1) maximal reduction of network complexity (not only removal of neurons or
synapses, but removal all the unnecessary elements and signals and reduction of
the complexity of elements),
2) using of adjustable and flexible pruning process (the pruning sequence
shouldn't be predetermined - the user should have a possibility to prune
network on his own way in order to achieve a desired network structure for the
purpose of extraction of rules of desired type and form),
3) extraction of rules not in predetermined but any desired form.
Some considerations and notes about network architecture and training process
and applicability of currently developed pruning techniques and rule extraction
algorithms are discussed. This technology, being developed by us for more than
10 years, allowed us to create dozens of knowledge-based expert systems.
Karlin I.V., Gorban A.N., Dukek G., Nonnenmacher T. F.
Dynamic correction to moment approximations. Physical Review E, February
1998 Volume 57, Number 2, P.1668-1672. KGDN98.pdf
Considering the Grad moment ansatz as a suitable
first approximation to a closed finite-moment dynamics, the correction is
derived from the Boltzmann equation. The correction consists of two parts,
local and nonlocal. Locally corrected thirteen-moment equations are
demonstrated to contain exact transport coefficients. Equations resulting from
the nonlocal correction give a microscopic justification to some
phenomenological theories of extended hydrodynamics.
Gorban A. N.
Approximation of Continuos Functions of Several
Variables by an Arbitrary Nonlinear Continuous Function of One Variable, Linear
Functions, and Their Superpositions, Appl. Math. Lett., Vol. 11, No. 3, pp 45-49,
1998 approx98.pdf
Karlin I.V., Gorban A.N., Succi S., Boffi V.
Maximum Entropy Principle for Lattice Kinetic Equations. Physical Review
Letters Volume 81, Number 1,
The entropy maximum approach to constructing equilibria
in lattice kinetic equations is revisited. For a suitable entropy function, we
derive explicitly the hydrodynamic local equilibrium, prove the H theorem for
lattice Bhatnagar-Gross-Krook
models, and develop a systematic method to account for additional constraints.
Gorban A.N.,
Shokin Yu.I., Verbitskii V.I.
Simultaneously dissipative operators and the infinitesimal
In solving a system of ordinary differential equations by an interval method
the approximate solution at any considered moment of time t represents a set
(called interval) containing the exact solution at the moment t. The intervals
determining the solution of a system are often expanded in the course of time
irrespective of the method and step used.
The phenomenon of interval expansion, called the
M.Yu. Senashova, A.N. Gorban, D. C. Wunsch II
Back-propagation of accuracy, The talk given on ICNN97 (The 1997
IEEE International Conference on Neural Networks, Houston, USA),
Online: http://arXiv.org/abs/cond-mat/0305527
gorsenwu.pdf
In this paper we solve the problem: how to determine maximal allowable errors,
possible for signals and parameters of each element of a network proceeding
from the condition that the vector of output signals of the network should be
calculated with given accuracy? "Back-propagation of accuracy" is
developed to solve this problem.
A. N: Gorban,
Ye. M. Mirkes, D.C. Wunsch
II
High order ortogonal tensor networks: information
capacity and reliability. The talk given on ICNN97 (The 1997 IEEE
International Conference on Neural Networks, Houston, USA), gomirwu1.pdf
Neural networks based on construction of ortogonal
projectors in the tensor power of space of signals are described. A sharp
estimate of their ultimate information capacity is obtained. The numbers of
stored prototype patterns (prototypes) can many times exceed the number of
neurons. A comparison with the error control codes is made.
Gorban A.N.,
Karlin I.V.
Short-Wave Limit of Hydrodynamics: A Soluble Example. Physical Review
Letters, Volume 77, Number 2,
The Chapman-Enskog series for shear stress is summed
up in a closed form for a simple model of Grad moment equations. The resulting
linear hydrodynamics is demonstrated to be stable for all wavelengths, and the
exact asymptotic of the acoustic spectrum in the short-wave domain is obtained.
Gorban A.N., Karlin I.V. Nonnenmacher T. F., Zmievskii V.B.
Relaxation Trajectories: Global approximation. Physica
A, 1996, 231, P.648-672. GKZNPhA96.pdf
Gorban A.
N., Karlin I. V.
Scattering rates versus moments: Alternative Grad equations, Physical
Review E October 1996 Volume 54, Number 4, P. 3109-3112. pR3109_11996.pdf
Scattering rates (moments of collision integral) are treated as independent
variables, and as an alternative to moments of the distribution function, to
describe the rarefied gas near local equilibrium. A version of the entropy
maximum principle is used to derive the Grad-like description in terms of a
finite number of scattering rates. The equations are compared to the Grad
moment system in the heat nonconductive case. Estimations for hard spheres
demonstrate, in particular, some 10% excess of the viscosity coefficient
resulting from the scattering rate description, as compared to the Grad moment
estimation.
Alexander N. Gorban'
, Iliya V. Karlin
General approach to constructing models of the Boltzmann equation, Physica A, 1994, 206, P.401-420. GKPhA94.pdf
The problem of thermodynamic parameterization of an arbitrary approximation of
reduced description is solved. On the base of this solution a new class of
model kinetic equations is constructed that gives a model extension of the
chosen approximation to a kinetic model. Model equations describe two
processes: rapid relaxation to the chosen approximation along the planes of
rapid motions, and the slow motion caused by the chosen approximation. The
H-theorem is proved for these models. It is shown, that the rapid process
always leads to entropy growth, and also a neighborhood
of the approximation is determined inside which the slow process satisfies the
H-theorem. Kinetic models for Grad moment approximations and for the
Tamm-Mott-Smith approximation are constructed explicitly. In particular, the
problem of concordance of the ES-model with the H-theorem is solved.
Alexander N. Gorban'
, Iliya V. Karlin
Thermodynamic parameterization, Physica A,
1992, 190, P.393-404 GKPhA92.pdf
A new method of successive construction of a solution is developed for problems
of strongly nonequilibrium Boltzmann kinetics beyond
normal solutions. Firstly, the method provides dynamic equations for any
manifold of distributions where one looks for an approximate solution.
Secondly, it gives a successive procedure of obtaining corrections to these
approximations. The method requires neither small parameters, nor strong
restrictions upon the initial approximation; it involves solutions of linear
problems. It is concordant with the H-theorem at every step. In particular, for
the Tamm-Mott-Smith approximation, dynamic equations are obtained, an expansion
for the strong shock is introduced, and a linear equation for the first
correction is found.
V. I.
Verbitskii and A. N. Gorban'
Jointly dissipative operators and their
applications, Siberian Mathematical Journal, Volume 33, Number 1
(1992), 19-23, DOI: 10.1007/BF00972932
The jointly dissipative operators were introduced by Verbitskii
and Gorban' (1989). Let E be an n-dimensional real or complex linear space, and let L(E)
be the space of linear operators in E.
Let us introduce a norm |||| on E
and the corresponding norm in L(E). An operator A from L(E) is said to be dissipative if ||exp(tA)||≤1 for
all t≥0. It is roughly dissipative if there is ε
> 0 such that ||exp(tA)||≤exp(-εt) for all t≥0.
For the existence of a norm with respect to which the operator A is be roughly dissipative
it is necessary and sufficient that the system (i) be
asymptotically stable, i.e., that the matrix of A be stable (i.e., that the
spectrum of A Lie in the open left halfplane). A
family of operators is said to be jointly dissipative (resp. jointly
roughly dissipative) if there exists a norm with respect to which all operators
from this family are dissipative (resp., roughly dissipative). The jointly dissipative operators find application
in the analysis of dynamical properties of nonlinear systems of ordinary
differential equations and in some applications (chemical kinetics, numerical
analysis). In the present paper we discuss the properties of jointly
dissipative operators and some of their applications. For example, the
following theorems are proved: (Theorem 1) Suppose the family {A} is compact, generates a solvable
Lie algebra, and all matrices in {A}
are stable. Then {A} is jointly
roughly dissipative. (Theorem 2) Suppose the family {A} is finite, generates a nilpotent Lie algebra, and for each
operator from {A} there exists a
norm with respect to which it is is dissipative. Then the family {A} is jointly dissipative.
Gorban A.N., Bykov V.I.
A model of autooscillations in association
reactions, Chemical Engineering Science. 1987, Vol. 42, No. 5. P.
1249-1251. BG1987.pdf
The aim of this paper is to show that association reactions can result in the
appearance of autooscillations in nonlinear systems.
Gorban A.N., Bykov V.I., Yablonskii G.S.
Thermodynamic function analogue for reactions proceeding without interaction
of various substances, Chemical Engineering Science, 1986. Vol. 41, No. 11.
P. 2739-2745. BGYa1986.pdf
Function similar to Lyapunovs function has been
constructed for reactions with $a_i A_i \to b_j A_j$
stages. This provides for the quasi-thermodynamics of the appropriate kinetic
model, which implies steady-state uniqueness and global stability in the
reaction polyhedron. The kinetic law generalizing the Marcelin-de
Donder kinetics has been written for a separate
stage. Explicit Lyapunov thermodynamic functions have
been written for various conditions of the reaction proceeding in closed
systems. The matrix of linear approximation close to equilibrium is expressed
by means of the introduced scalar product. Particularly, the absence of damped
oscillations as equilibrium is approached as shown.
Gorban A.N., Bykov V.I.
Macroscopic clusters induced by diffusion in a catalytic oxidation
reactions, ChemicaI Engineering Science, 1980.
Vol. 35, P. 2351-2352 BG1980.pdf
Gorban A.N.
Singularities of Transition Processes In Dynamical Systems. http://arXiv.org/abs/chao-dyn/9703010
v1
The paper gives the systematic analysis of singularities of transition
processes in general dynamical systems. Dynamical systems depending on
parameter are studied. A system of relaxation times is constructed. Each
relaxation time depends on three variables: initial conditions, parameters k of
the system and accuracy \epsilon of relaxation. This system of times contains:
the time before the first entering of the motion into \epsilon -neighbourhood
of the limit set, the time of final entering in this neighbourhood and the time
of stay of the motion outside the \epsilon -neighbourhood of the limit set. The
singularities of relaxation times as functions of (x_0; k) under fixed \epsilon
are studied. A classification of different bifurcations (explosions) of limit
sets is performed. The bifurcations fall into those with appearance of new
limit points and bifurcations with appearance of new limit sets at finite
distance from the existing ones. The relations between the singularities of
relaxation times and bifurcations of limit sets are studied. The peculiarities
of dynamics which entail singularities of transition processes without
bifurcations are described as well. The peculiarities of transition processes
under perturbations are studied. It is shown that the perturbations simplify
the situation: the interrelations between the singularities of relaxation times
and other peculiarities of dynamics for general dynamical system under small
perturbations are the same as for smooth two-dimensional structural stable
systems.