Isolated systems and the Boltzmann definition of entropy

The definition of the Gibbs free energy G = E — TS + PV may be combined with Equation (1.1) to show the following. 

In the following sections we will see how temperature, entropy, and free energy are statistical properties that emerge in systems composed of large numbers of particles. In Chapter 12, the appendix to this book, we dig more deeply into statistical thermodynamics, derive a set of statistical laws that are used in this chapter, and show how Equation (1.1) - the fundamental equation of macroscopic thermodynamics - is in fact a statistical consequence of more fundamental principles operating at the microscopic level. 

An isolated system is defined to be a system that does not exchange material or energy with its environment. Thus the extensive thermodynamic variables N, V, and E are held fixed. Boltzmann s formula for the entropy of such a system is 

Similarly, if one is interested in a macroscopic thermodynamic state (i.e., a subset of microstates that corresponds to a macroscopically observable system with bxed mass, volume, and energy), then the corresponding entropy for the thermodynamic state is computed from the number of microstates compatible with the particular macrostate. All of the basic formulae of macroscopic thermodynamics can be obtained from Boltzmann s definition of entropy and a few basic postulates regarding the statistical behavior of ensembles of large numbers of particles. Most notably for our purposes, it is postulated that the probability of a thermodynamic state of a closed isolated system is proportional to 2, the number of associated microstates. As a consequence, closed isolated systems move naturally from thermodynamic states of lower 2 to higher 2. In fact for systems composed of many particles, the likelihood of 2 ever decreasing with time is vanishingly small and the second law of thermodynamics is immediately apparent. 

Combining Equations (1.1) and (1.4), we can develop a statistical interpretation of the thermodynamic quantity temperature, 

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