THE EINSTEIN LAW FOR THE SYSTEM ”BROWNIAN PARTICLE IN THERMOSTAT” BASED ON THE PRESENTED PROBABILITY APPROACH

A one-dimensional system ”Brownian particle in thermostat” is considered. The equation — the Einstein law for Brownian motion is derived on the basis of presented probability approach in the paper. The kinetic equation for space part of the distribution function is obtained. AMS Subject Classification: 35Q82


Introduction
The studying of simple models of particle systems such as systems of hard spheres allows us to solve a number of problems of description of dynamics of many-particle systems.The problems are rigorous justification of kinetic equations, justification of approximate methods of description of dynamics etc. [1], [2], [3], [4], [5] and [6].
The system "Brownian particle in thermostat" is an important case of many-particle many kind system, and the studying of this system is of selfdependent interest [7].
In the present paper it is derived the equation -the Einstein law for Brownian motion on the basis of the presented probability approach used to describe the considerable many-particle system.It is obtained the kinetic equation for space part of the distribution function of Brownian particle in thermostat.

Formulation of the Problem
We consider a one-dimensional system "Brownian particle in thermostat" of many particles.All particles of thermostat are identical and have the mass m 0 = 1.The velocity distribution of particles of thermostat is equilibrium, the Maxwellian: The coordinates of the particles are uniformly distributed along a straight line on the admissible configurations.In the thermostat considered there is a massive Brownian particle with mass m ≫ 1.We can neglect the particle sizes.We assume collisions between particles be absolutely hard.
The aim of the present paper is the studying of massive Brownian particle motion, derivation of the equation -the Einstein law for the system "Brownian particle in thermostat", derivation of the kinetic equation for space part of the distribution function of Brownian particle on the basis of the presented probability approach used to describe the considerable many-particle system.

Statistical Characteristics of Brownian Particle Motion
We calculate the mean time and free path (displacement) of Brownian particle as functions of its initial velocity u 0 .
The probability P (u 0 , t) of lacking of collisions of the Brownian particle moving with velocity u 0 at time t is determined by the function [8]: where, taking into account that the velocity distribution function of particles in thermostat is equilibrium, the Maxwellian (1), function f (u 0 ) can be determined in terms of special function erf u 0 √ 2 , which is the erf integral, i.e.
Particles of thermostat, which collide with a massive Brownian particle with mass m ≫ 1, have identical and vice versa, in comparison with Brownian particle, very small mass m 0 = 1.Then velocity of particles of thermostat after collision with Brownian particle will be much bigger than velocity of Brownian particle.If we assume mean modulus of velocity of particles of thermostat after collision with Brownian particle be unity then according to momentum and energy conservation laws we obtain that Brownian particle velocity |u 0 | ≪ 1.By this we conclude that the change of velocity of Brownian particle after collision with the particle of thermostat is a small quantity.Therefore, approximation of the function (2) results to expression where . It is obvious, that time t between collisions of the Brownian particle having velocity u 0 with particles of thermostat (mean free time of Brownian particle) is random variable with distribution function By this function we conclude that mean time between collisions of Brownian particle with particles of thermostat (mean free time of Brownian particle) is equal to (by (3)) Since |u 0 | ≪ 1, then t weakly depends on Brownian particle velocity u 0 .Therefore, we can suppose We calculate relaxation time [8]-to-mean time between collisions of Brownian particle with particles of thermostat ratio (this is mean number of free paths of Brownian particle during relaxation time of its velocity): Mean free path (displacement) of Brownian particle according to (4) and ( 5): Denote λ by λ 0 , i.e.
We calculate mean-square free path (displacement) of Brownian particle according to (4) and ( 5): We calculate mean-square velocity u of Brownian particle: where ϕ(u) is the Maxwell distribution function for Brownian particle.Analogously we denote u 2 by u 2 0 , i.e.

The Kinetic Equation for Space Part of the Distribution Function
Main problem of calculation space part of the distribution function is occurrence of strong correlations between neighboring free displacements of Brownian particle.
We calculate statistical characteristics of Brownian particle motion during relaxation time of its velocity.Introduce the following denotations: λ i is the i-th displacement of Brownian particle between the i-th and the (i + 1)-th collisions; is mean-square displacement of Brownian particle during relaxation time of its velocity, where N = n − 1 ( n is from formula ( 6)).
Averaging we take into account that λ i+1 depends on λ i , i.e. there is correlation.Therefore, it is necessary to obtain conditional distribution functions and their characteristics.
Taking into account that m ≫ 1 and according to formula (6), we have According to equality (5) from [8] we write distribution function of velocity u of Brownian particle after collision under condition, that before collision this particle had velocity u 0 (i.e.conditional distribution function): To calculate the normalization factor C we use the normalization condition: Calculating this integral we obtain: , dz is the function from [8].Here u 0 − u = 2 m+1 z, and for G k (u) from [8]
Here u is Brownian particle velocity after collision, i.e. u is the following after u 0 velocity of Brownian particle.Analogously for λ and λ 0 .Then using formulas (7) and ( 8) we obtain: Here, for the sake of simplicity, we do not use indices 1 for u and λ.
Using expressions (13), ( 11), ( 7) and ( 14), ( 12), ( 8), we find mean characteristics of displacement of Brownian particle: then we obtain (taking into account that m ≫ 1) We calculate mean square of displacement of Brownian particle during relaxation time of its velocity, starting from the last summand since this summand depends on previous ones.Therefore, numbering k (k = 0, 1, 2, . . ., N ) of averaged quantities we perform from the end of the chain.We start, for example, from the k-th (from the end) summand.Denote We average over the last summand (i.e. over the (N −k)-th), using formulas (19), ( 17), (18): where R k+1 is obtained from denotation Note, that in numbering It is obvious (formula (20)), that step-by-step averaging does not change the linear form Thus, we obtain the recurrent relations: Therefore, Altogether, after all averagings for k = N (i.e.before relaxation of velocity of Brownian particle), taking into account that numbering of averaged quantities was conducted from the end, i.e. λ N −k = λ N −N = λ 0 and according to formulas (19), (21) S N = λ 2 0 , R N = λ 2 0 , substituting a N , b N , c N into formula (23), we obtain: Substituting λ 2 0 , N, γ, δ according to formulas (8), ( 9), (10), ( 15), ( 16) and taking into account that m ≫ 1 and L N = L 2 (since L N is calculated mean square of displacement of Brownian particle during relaxation time of its velocity), we obtain since m ≫ 1 and therefore the first and the second summands are much smaller than the third summand.Then Taking out of context πm 16 and performing arithmetic operations we obtain: Taking into account that relaxation time of velocity of Brownian particle [8] T rel.= m √ 2π 8 we have Thus, mean square of displacement L 2 of Brownian particle during large number of collisions of this particle with particles of thermostat is in proportion to time, i.e. we obtain the law, corresponding to diffusion motion.Therefore, the kinetic equation for space part of the distribution function of Brownian particle in thermostat has the form of the diffusion equation: where x is the coordinate of Brownian particle, t is time, D is diffusion coefficient.
For this diffusion equation where τ is observation time interval, sufficiently large for Brownian particle to collide many times with particles of thermostat.The equation (25) represents the Einstein law.Derived equation (24) corresponds to the Einstein law.
Thus, we denote T rel.= τ .Therefore, using formulas (25), (24), we obtain: i.e.D does not depend on mass of Brownian particle.Thus, it is derived the equation -the Einstein law and obtained the kinetic equation for space part of the distribution function of Brownian particle, which has the form of diffusion equation, for the system "Brownian particle in thermostat".