Time averages and Boltzmann extremals for Markov chains, discrete Liouville equations, and the Kac circular model

Adzhiev, S. Z.; Vedenyapin, V. V.

doi:10.1134/s0965542511110029

Cited by 21 publications

(15 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The Hamilton-Jacobi equation methods [26][27][28][29][30][31] would be useful here. A very important task is to obtain for the Vlasov-like equations a statement like «Time averages coincide with the Boltzmann extremals» [32][33][34]…”

Section: Resultsmentioning

confidence: 99%

Vlasov-Maxwell-Einstein Equation and Einstein Lambda

Vedenyapin

Pershin²

2019

KIAM Prepr.

View full text Add to dashboard Cite

Москва 2019 УДК 517.958 Веденяпин В.В., Першин И.С. Уравнение Власова-Максвелла-Эйнштейна и лямбда Эйнштейна Выводятся уравнения Власова-Максвелла-Эйнштейна из классического действия Лоренца-Шваршильда-Гильберта-Эйнштейна. При этом требуется и удаётся синхронизовать собственные времена различных частиц. На основе полученных выражений для действия анализируется лямбда Эйнштейна и вклад её в тёмную энергию. Ключевые слова: лямбда Эйнштейна, синхронизация времён, тёмная энергия, уравнение Власова, уравнение Власова-Эйнштейна.

show abstract

Section: Resultsmentioning

confidence: 99%

Vlasov-Maxwell-Einstein Equation and Einstein Lambda

Vedenyapin

Pershin²

2019

KIAM Prepr.

View full text Add to dashboard Cite

show abstract

“…Let us note that the consideration of the H-theorem for nonlinear systems with discrete time, in particular, even for the Becker-Döring system of equations, becomes an extremely important problem as the computer simulation has a significance in the solution of fundamental problem of the creating of new materials. In the linear case, the transition from continuous time to discrete gives the transition from a Markov process to a Markov chain and the H-theorem is valid and studied (see [54] and references in it, [55]). In the nonlinear case, for explicit time discretization it is fulfilled in rare cases [19,20] and for the implicit one, it is investigated in [20,21].…”

Section: Discussion Of the Resultsmentioning

confidence: 99%

Approaches to determining the kinetics for the formation of a nano-dispersed substance from the experimental distribution functions of its nanoparticle properties

Adzhiev¹,

Мелихов²,

Vedenyapin³

2019

Nanosystems: Phys. Chem. Math.

View full text Add to dashboard Cite

In the present paper, we discuss the kinetic equations for the evolution of particles of a nanodispersed substance, distinguishing by properties (sizes, velocities, positions, etc.). The aim of the present investigation is to determine the coefficients for the equations by the distribution functions, which are obtained experimentally. The experiment is characterized by the time interval, which is needed for the measurement of the distribution function. However, the nanodispersed substance is obtained from a highly supersaturated solution or vapor and this time interval is large, thus, one is able to measure distribution functions only when the processes of the integration and the fragmentation of the particles become rather slow. So it is advisable to reconstruct the kinetics for the formation of a nanodispersed substance by the experimental distribution functions measured when the processes are rather slow. The first problem that arises is the obtaining of correct equations, and, hence, the derivation of the equations from each other. From the discrete system of equations for the evolution of discrete distribution functions of particles of a nanodispersed substance, we obtain the continuum equation of the Fokker-Planck type, or of the Einstein-Kolmogorov type, or of the diffuse approximation on the distribution function of nanoparticles distinguishing by the numbers of molecules forming them. We consider the distribution functions, which approximate the experimental data. We determine the coefficients for the equation of the Fokker-Planck type by the stationary and non-stationary distribution functions of a nanodispersed substance. Due to unity of the kinetic approach, the present work may be useful for specialists of various areas, who study the evolution of structures (not only with nanosize) with differing properties.

show abstract

“…In this example, the Boltzmann-Orlov-Moser-Bruno vector is unique up to a factor: α − β = (1, 1, −1, −1). Thus, the stoichiometric coefficients are the numbers of molecules participating in the reaction (4). S i , i = 1, 2, .…”

Section: The Methods and Definitionsmentioning

confidence: 99%

“…Thus, Boltzmann defines the simplest discrete model as what we now term as the Boltzmann extremals [3] (see also [4,5]). In [3][4][5][6][7][8][9][10], it was shown that the stationary solution can be obtained without solving the equation in different cases such as for discrete Boltzmann equations, for general chemical kinetics and for Liouville equations. It is also interesting to note that Boltzmann generalized his H-theorem for chemical kinetics in his work, but modern generalization of chemical (1)…”

Section: Introductionmentioning

confidence: 99%

“…[C(l, 1)N(l, t)N(1, t) − F(l, 1)N(l + 1, t)] , dN(2,t) dt The conditions of the H-theorem for Equation (1) have been investigated in [18,19], and those for the Becker-Döring equations, Equation (2), in [20,21]. It is interesting to rewrite the results of [18][19][20][21] in terms of the Boltzmann extremals [1,4,5], which is the argument of the conditional minimum the H-function, provided that the constants of linear conservation laws are fixed. Namely, according to the H-theorem solutions of the equations converge to the Boltzmann extremals as soon as time tends to infinity.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Kinetic Equations for Particle Clusters Differing in Shape and the H-theorem

Adzhiev

Batishcheva

Мелихов

et al. 2019

Physics

View full text Add to dashboard Cite

The question of constructing models for the evolution of clusters that differ in shape based on the Boltzmann’s H-theorem is investigated. The first, simplest kinetic equations are proposed and their properties are studied: the conditions for fulfilling the H-theorem (the conditions for detailed and semidetailed balance). These equations are to generalize the classical coagulation–fragmentation type equations for cases when not only mass but also particle shape is taken into account. To construct correct (physically grounded) kinetic models, the fulfillment of the condition of detailed balance is shown to be necessary to monitor, since it is proved that for accepted frequency functions, the condition of detailed balance is fulfilled and the H-theorem is valid. It is shown that for particular and very important cases, the H-theorem holds: the fulfillment of the Arrhenius law and the additivity of the activation energy for interacting particles are found to be essential. In addition, based on the connection of the principle of detailed balance with the Boltzmann equation for the probability of state, the expressions for the reaction rate coefficients are obtained.

show abstract

Time averages and Boltzmann extremals for Markov chains, discrete Liouville equations, and the Kac circular model

Cited by 21 publications

References 8 publications

Vlasov-Maxwell-Einstein Equation and Einstein Lambda

Vlasov-Maxwell-Einstein Equation and Einstein Lambda

Approaches to determining the kinetics for the formation of a nano-dispersed substance from the experimental distribution functions of its nanoparticle properties

Kinetic Equations for Particle Clusters Differing in Shape and the H-theorem

Contact Info

Product

Resources

About