# Basis of Local Approach in Classical Statistical Mechanics

## Abstract

**:**

## Introduction.

## Mechanics and Ensembles.

- q
_{α}is 3n of generalized coordinates, - p
_{α}is 3n of generalized momenta - H = H(q
_{1},…q_{3N},p_{1},…p_{3N},t) is the Hamiltonian of the whole system, which depends on 3N coordinates and momenta of all N particles and, probably, on time t.

^{0}is not equal to the corresponding phase volume calculated at the time point t ≠ t

^{0}. To do this, it would seem possible to use a well-known technique of change from integration with respect to one set of variables to integration with respect to another one using the Jacobian

^{0}.

_{0}= 1, i.e. formula (2) holds for the case of subsystems for infinitesimal transformations. Let us show that dJ/dt is not equal to zero identically. To make it obvious, first we consider the simplest case where the initial system involves only two particles executing one-dimensional motion. And we take one particle as the subsystem.

_{2}and p

_{2}, i. e. with respect to coordinates and momenta of surrounding particles. When substituting (4) into (3), either of determinants is divided into the sum of four determinants, with determinants free of partial derivatives with respect to q

_{2}and p

_{2}being equal to zero (similarly to the case of Gibbs's ensemble [1]), and combining the retaining terms we obtain:

_{2}and p

_{2}and the variables q

_{1}and p

_{p}are lost. It is seen that in general case

_{α}and p

_{α}of coordinates and momenta of the subsystem particles, which are caused by interactions of the subsystem particles with environmental particles. It is obvious that in the limiting case of absence of subsystems interaction with the environment, we obtain the Gibbs's ensemble and the phase volume conservation. One can note that it is seen from equations (5) and (6) that a phase volume change for subsystems is not directly related to velocity divergence (see in more detail hereinafter). Moreover, from the above one can easily obtain that for the subsystems

_{n}for the ensemble of subsystems of n particles similarly to the case of Gibbs's statistics. This function defining probability of values of coordinates q

_{α}and momenta p

_{α}of n subsystem particles can also depend on time t and parametrically on coordinates q

_{κ}and momenta p

_{κ}of particles surrounding the subsystem. Therefore the total derivative of the distribution function ρ

_{n}with respect to time should be written in the form:

_{α}, for the velocity ${\dot{p}}_{\alpha}$ in it represents a force acting on the particle. In principle, a situation is possible when at an infinitely small difference in the positions of two points in the 6n-space, forces acting from the environment differ by a finite quantity due to a finite difference in the positions of surrounding particles. This case, however, can be easily avoided by means of proper choice of subsystems. For this purpose, there must be no discontinuity in the hypersurface intersecting Gibbs's ensemble so as to form an ensemble of subsystems, i. e. the hypersurface must be smooth. This condition is not essentially burdensome and we can use the continuity equation in the 6n-space as well, where it takes the form

_{n}depends, generally speaking, on 6N variables and time, 6N equations will be equations of characteristics for it, from them 6n equations being of the form

_{0}= 0 we see that the length of any segment X-direction is not constant in time. That is in the one-dimensional ensemble under consideration, the volume is not conserved though the velocity divergence is equal to 0.

_{1}(q

_{α}, p

_{α}) is energy of the subsystem under consideration without regard for the environment; H

_{w}(q

_{α}, q

_{κ}) is energy of interaction with the environment of the subsystem; H

_{2}(q

_{κ}, p

_{κ}) is energy of the environment. For simplicity, we have supposed here H

_{w}as independent of momenta, which is not a principal restriction. Let us denote by E the expression of the form

_{n}satisfying equations (9) and (10) as well as the condition $\frac{\partial {\rho}_{n}}{\partial t}=0$

_{n}is the distribution function of subsystems in the phase space, then it is subject to normalization to the unit.

_{κ}, but not on q

_{α}and p

_{α}. So equations (9) and (10) are satisfied by the solution (15). This solution is a local, more exactly quasilocal distribution in the assigned field of temperatures, that depends on the environment. The environmental dependence enters through the interaction energy H

_{w}which provides fluctuations in an equilibrium case and evolution in a non-equilibrium case.

_{w}→ 0

## Entropy

_{κ}as parameters, i.e. it is a conditional distribution function. But the values of q

_{κ}are unknown for us. Nevertheless, we can construct an ensemble for a neighboring subsystem and define probability of values of all the coordinates q

_{κ}or a part thereof. It is coordinates q

_{α}of the first subsystem that will now enter into the distribution function as parameters. We can not generally establish such boundary conditions for a subsystem that they should be independent of the subsystem itself. It is not a drawback of this method but presents a more general case of describing nature. Indeed, when we establish boundary conditions "strictly", we introduce certain idealization neglecting a reverse action of the system under consideration on the surroundings. It is clear that such idealization is not always permissible. Since (15) involves the interaction energy of subsystems, which reflects the fact that the subsystems are not statistically independent, then the distribution function of the whole system can not be obtained by factorization of distribution functions of the subsystems. However, with weak coupling being present, such an approximation is possible. Since in this case subsystem temperatures can differ, the total distribution function will not be symmetrical in respect to a rearrangement of coordinates of particles belonging to two different subsystems. Nevertheless, in rearranging particle coordinates within a subsystem, the distribution function will not change, i.e. a local symmetry takes place.

_{1}… dq

_{3n}dp

_{1}… dp

_{3n}and take a variation from both sides of (16) in changing the coordinates q

_{κ}and parameter Θ .

_{α}and momenta p

_{α}.

_{n}= [(F − E)/Θ]

_{n}〉 can be interpreted as local entropy in appropriate units, Θ is the subsystem temperature. The second term of expression (17) is related to the expansion work −PδV where P is pressure, δV is the volume variation. A detailed analysis of the expression will allow one to understand under what conditions the approximation of local thermodynamics for the expansion work is valid. Turning back to the definition of local entropy we see that it is the same in form as the Gibbs's one but has somewhat another content. Now entropy is a dynamical variable changing in accordance with a change of q

_{κ}and it is not a strictly additive value. In integral consideration of the whole nonequilibrium system, its entropy can increase. This problem requires further investigation. One circumstance, however, deserves mentioning. In the literature (e.g. [6]) one can find a statement that in the local-equilibrium distribution the entropy can not increase. In our opinion, this statement is not true and is based on the fact that energy of interaction with the environment of the considered part of the system has not been properly taken into account. Moreover, from the above consideration it should be clear that for the mechanical substantiation of the second law of thermodynamics, there is no need in a non- physical assumption on discrepancy of calculated and measured trajectories, which was made in a widely cited paper by N. S. Krylov [7].

_{α},v

_{α},t) is one-particle distribution function, x

_{α}are the particle coordinates, v

_{α}are velocities, F

_{α}are components of a volume force affecting the particle. J

_{c}is the collision integral, the particular form of which is not very important for us. Now essential is just the fact that in this term, and only in it, all the interactions of the particle under consideration with the environment are taken into account.

## Stochasticity

## To problem of irreversibility.

## References

- The collected works of J. Willard Gibbs. In two volume. (N. Y. etc.: Longmans, Green and Co 1928.)
- J.B.Zeldovich, A.D.Myshkis. Elementy matematicheskoi phiziki.(M.Nauka, 1973.)
- P. Resibois, M. De Leener. Classical kinetic theory of fluids.(Hiley-Interscience, New-York, 1977.)
- H. G. Schuster. Deterministic chaos.(Weinheim, 1984.)
- S. R. De Groot, P. Mazur. Nonequilibrium thermodynamics.(North Holland, Amsterdam, 1962.)
- D. N. Zubarev. Neravnovesnaya statisticheskaya termodinamika.(M. Nauka, 1971.)
- N. S. Krylov., Works on the foundation of statistical physics, Princeton (U.Press, Princeton, NJ, 1979.)
- I. Prigogine. From being to becoming.(W. H. Freeman and company. San Francisco, 1980.)

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**MDPI and ACS Style**

Sharov, S.R.
Basis of Local Approach in Classical Statistical Mechanics. *Entropy* **2005**, *7*, 122-133.
https://doi.org/10.3390/e7020122

**AMA Style**

Sharov SR.
Basis of Local Approach in Classical Statistical Mechanics. *Entropy*. 2005; 7(2):122-133.
https://doi.org/10.3390/e7020122

**Chicago/Turabian Style**

Sharov, Sergey R.
2005. "Basis of Local Approach in Classical Statistical Mechanics" *Entropy* 7, no. 2: 122-133.
https://doi.org/10.3390/e7020122