Statistical analogues of the entropy and Helmholtz free energy

Statistical analogues of the entropy and Helmholtz free energy 

In equation (11 14) the P/s are all positive fractions and therefore S is always positive. Suppose that the system could exist in only a single quantum state, say state 0. Then for this state we should have f0 l and ail other P s would be zero. Such a distribution would correspond to no randomness, and (11 14) shows that it would give rise also to zero S. On the other hand, a distribution of the ensemble over a large number of states, for each of which P is very small, will clearly give rise to a large positive value of S The similarity of (11 14) to the thermodynamic expr ions for the entropy increase of mixiug in perfect gases and ideal solutions may also be noted. These various points provide some preliminary justification for the idea that 8 is related to entropy.f 

Consider a virtual variation in the ensemble which involves the j and h states only, their probabilities changing by dP and dP respectively. Since SP =0 it follows that df = — dl and the last equation 

It follows that dS /dPf is zero whenever the conditions of the system are such that P and Pj are equal. The same applies to a virtual change affecting any other pair of accessible quantum states.  

The latter equality follows since 2 =0. For the same reason this quantity 

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