The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi>H</mml:mi></mml:math>-theorem for the physico-chemical kinetic equations with explicit time discretization

Adzhiev, S. Z.; Мелихов, И. В.; Vedenyapin, V. V.

doi:10.1016/j.physa.2017.03.032

Cited by 7 publications

(6 citation statements)

References 11 publications

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“…In the linear case, when passing from continuous to discrete time, we have a transition from a Markov process to a Markov chain, and the H-theorem in this case is valid, as investigated earlier (see [8] and references therein). In the nonlinear case, it is valid in rare cases for explicit time discretization [8], and for the implicit case as investigated in [9].…”

Section: Discussionmentioning

confidence: 93%

“…Consideration of the H-theorem for nonlinear systems with discrete time, in particular, even for the system of Becker-Döring equations, becomes an extremely important and urgent task, since computer modeling plays a key role in solving the fundamental problem of creating new materials [22,23]. In the linear case, when passing from continuous to discrete time, we have a transition from a Markov process to a Markov chain, and the H-theorem in this case is valid, as investigated earlier (see [8] and references therein). In the nonlinear case, it is valid in rare cases for explicit time discretization [8], and for the implicit case as investigated in [9].…”

Section: Discussionmentioning

confidence: 99%

“…We first consider the first of our models-the system in Equation (15). Reactions (8) can be written in the form of (18):…”

Section: The Linear Conservation Laws and The Convergence To The Equimentioning

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%

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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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Section: Discussionmentioning

confidence: 93%

Section: Discussionmentioning

confidence: 99%

“…We first consider the first of our models-the system in Equation (15). Reactions (8) can be written in the form of (18):…”

Section: The Linear Conservation Laws and The Convergence To The Equimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Adzhiev

Batishcheva

Мелихов

et al. 2019

Physics

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“…If ∆N (n, t) ≡ N (n, t + ∆t)−N (n, t), then we have the explicit time discretization of the system (2.1)-(2.3). For the system of equation (21)-(2.3) the H-theorem is fulfilled, but for the system with the explicit time discretization the H-theorem is not valid: it is proved in [19,20] that the H-theorem is not fulfilled for the case of this system when only single molecules and dimers are considered. Also, it is valid for the implicit time discretization: when ∆N (n, t) ≡ N (n, t) − N (n, t − ∆t) [20,21], and thus, we can't use the explicit time discretization for the computer simulations.…”

Section: The Becker-doring Case and The Continuum Description Of The mentioning

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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Colloidal transport in anisotropic porous media: Kinetic equation and its upscaling

Russell

Bedrikovetsky

2023

Journal of Computational and Applied Mathematics

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The H-theorem for the physico-chemical kinetic equations with explicit time discretization

Cited by 7 publications

References 11 publications

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

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

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

Colloidal transport in anisotropic porous media: Kinetic equation and its upscaling

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