Entropy in the sense of Boltzmann and Poincaré

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

doi:10.1070/rm2014v069n06abeh004926

Cited by 24 publications

(14 citation statements)

References 14 publications

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“…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.

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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.

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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.

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“…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%

“…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.Physics 2019, 1 230 kinetic equations was found 100 years later in [11][12][13][14]. The theory was also generalized to the quantum case; see [5,15,16] for review. Therefore, the classical H-theorem not only substantiates the second law of thermodynamics for the described systems but also provides information about the behavior of solutions.…”

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confidence: 99%

“…Hereinafter, we write the rate constants of the processes as coefficients in equations as in Equation (3). The meaning of the indices in the coefficients in Equation (5) are clarified in the next section, where we define the notion of a shape consisting from cubes and write out a more general model via Equations (8)- (10). Note that the systems in Equations (1) and (2) represent the physicochemical kinetics equations with an infinite number of reactions.…”

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confidence: 99%

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

Adzhiev

Batishcheva

Мелихов

et al. 2019

Physics

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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.

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