TheH-theorem for the physico-chemical kinetic equations with discrete time and for their generalizations

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

“…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]. The diffuse approximation is widely used for the modeling of crystallization processes of a dispersed substance [5][6][7][8].…”

“…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. Instead the equation (2.1) one can write the conservation law of the number of all molecules of the forming the phase substance in the system N 0 :…”

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

“…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].…”

“…Therefore, the 1-dimensional case of the evolution of clusters of different mass and shape under the Becker-Döring constraint makes sense to consider only when the growth of chains occurs on the surfaces, and then the system of equations describing the evolution has the form of (2). (1,(1,1,1))(n,a) (n+1,a+e i ) N 1 (t)N(n, a, t) − K (n+1,a+e i ) (1,(1,1,1))(n,a) N(n + 1, a + e i , t) , (9) dN(n,a,t) dt = F(n, a, t) ≡ ≡ 3 i=0 K (1,(1,1,1))(n−1,a−e i ) (n,a) N 1 (t)N(n − 1, a − e i , t) − K (1,(1,1,1))(n−1,a−e i ) (n,a) N(n, a, t) − K (1,(1,1,1))(n,a) (n+1,a+e i )…”

“…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)…”