Thermodynamics, Statistical Physics, and Quantum Mechanics

214 SOLUTIONS

The converse is also true: If the variables and are statis-

tically independent. Now for a closed system we can write

where is the total entropy of the system and is the entropy change
due to the fluctuation. On the other hand,

where is the minimum work to change reversibly the thermodynamic

variables of a small part of a system (the rest of the system works as a heat

bath), and is the average temperature of the system (and therefore the

temperature of the heat bath). Hence,

However,

where and are changes of a small part of a system due to

fluctuations and P are the average temperature and pressure. So,

Expanding (for small fluctuations) gives

Substituting (S.4.93.16) into (S.4.93.14), we obtain