By Berezin F.
''I learn this lecture direction in 1966–67 on the division of Mechanics and arithmetic of the Moscow country college. presently statistical physics is greater than ninety% heuristic technology. which means the evidence verified during this technological know-how are usually not proved, normally, within the mathematical which means of this be aware even supposing the arguments resulting in them are relatively convincing.'' - F. Berezin, 1972
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Sample text
DS = βdQ is the total differential of the function S(β, λ1 , . . , λS ) = N (ln z + βE) + const. 5) § 5. Relation to thermodynamics 33 Proof reduces to the computation of partial derivatives of the function S: 1 N 1 N ∂S ∂ ln z ∂E ∂E ∂E = +E +β = −E + E + β =β , ∂β ∂β ∂β ∂β ∂β ∂S ∂ ln z ∂E ∂E = +β = −βλi + β . ∂λi ∂λi ∂λi ∂λi Now, let us compute dS: dS =N β ∂E dβ + ∂β N β(dE − β ∂E − pi dλi = ∂λi pi dλi ) = βdQ. 6) Note that the entropy is only determined up to an additive constant. the following question naturally arises: Question.
2) We can consider the gases A and B before the partition between them was removed and they had been intermixed as a united system whose phase space L1 is the product of phase spaces LA and LB of gases A and B. Obviously, L1 is a part of the phase space L of the system obtained from gases A and B as a result of removing the partition and intermixing: L = {(p(α) , q (α) ) | q (α) ∈ K}, L1 = {(p(α) , q (α) ) | q (α) ∈ KA or q (α) ∈ KB }, K = K A ∪ KB . The group S1 = SA × SB , consisting of the permutations of coordinates of particles of gases A and B separately, acts in the space L1 .
Real gases We may assume that the interaction between the subsystems Cα1 is small. Since, due to the fact that the potential v(q) has a compact support, the energy of interaction of subsystems vα,β is of the same order of magnitude as the area of the surface that separates the volumes Ωα , whereas the potential energy of the system is proportional to the volume of Ωα and is therefore much larger. Obviously, from the physical point of view, the replacement of the initial system C by an auxiliary system C 1 is unsatisfactory: the presence of impenetrable walls changes the properties of the gas confined in Ω: For example, it makes the large-scale (as compared with the size of Ωα ) movements of molecules (such as wind) impossible.