By Berezin, Felix; Leites, Dimitry

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Suppose we are given two systems (1) and (2) at distinct temperatures T2 > T1 . Let us mix these systems and wait till the equilibrium is established. Then: 1) t h e t e m p e r a t u r e T o f t h e o b t a i n e d s y s t e m C i s s t r i c t l y b e t w e e n T1 a n d T2 , 2) t h e m e a n e n e r g y o f e a c h s u b s y s t e m t h a t e a r l i e r e n tered the system (1) has increased whereas the mean energy of each subsystem which earlier entered the system (2) has diminished. These two statements constitute the second law of thermodynamics.

The assumption that v(q) = ∞ for |q| ≤ a physically means that the molecules cannot be at distances smaller than a from each other; that is each of them is confined in the center of an impenetrable spherical envelope of radius a. It is a also possible to assume that every molecule is a hard ball of radius centered a 2 at qi . The number is sometimes called the radius of the hard core. 2 The assumption that v(q) = 0 for |q| > b means that the molecules do not interact if the distance between them is > b.

V = v= V VA + VB = , and therefore NA + NB NA + NB lim N −→∞ NA +NB NB , . VB VA = , we have NA NB Veff. = 1. V1 eff. The result obtained apparently demonstrates that although the part of L complementary to L1 might be huge, it is“a hick town”, hardly essential for the statistical properties of the system. Therefore, the systems before and after intermixing are close and nothing essential happens after partition is removed. The equilibrium takes place practically instantly. 6. 1. The maximum principle.

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