Temperature and Entropy in Quantum Statistics

TEe proof of tEe fact tEat tEe quantity occurring in statistics is inversely proportional to tbe absolute temperature can be presented in tbe same form for all tbree types of statistics, tbe Boltzmann (B.), tEe Bose-Einstein (B.E.), and tbe Eermi-Dirac (F.D.). TEe fimdamental formulae for tbe probability of a state for all three statistics, can be unified by introducing tbe symbol 

The second term of the last expression is equal to zero for y = 0, and equal to 1 for y = +1 we can therefore replace it simply Iby y  

We now assume that the system (a gas) is enclosed in a vessel of variable volume this volume may be defined, say, by the position coordinate a of a piston. Thus the energy values are functions of a. If changes in a are made extremely slowly, no quantum jumps are excited by these changes the numbers % for the quantum states are therefore not changed. Such processes are called adiabatic (a better word would be quasistatic ). The work done in a small change is 

however, the change of a takes place rapidly, uncontrollable quantum jumps occur, and therefore alterations in the values Wg. The corresponding change in the energy is called heat supphed 

Here it is to be noted that not only but also a and are to be regarded as functions of a. Now we have 

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