Maxima and minima at equilibrium

Using this equation, the average number of atoms in the system is determined to be 

Similar to the NVT ensemble, fluctuations in the energy vanish at the thermodynamic limit, as do particle number fluctuations. Indeed, 

In the /VPT or isothermal-isobaric ensemble, the pressure and the temperature are kept constant. The NPT ensemble is useful for determining the thermodynamics of systems that are open to the atmosphere, e.g., for biological systems. Following a similar derivation as before, the W FT probability density is 

Equation 6.38 indicates that a system at constant temperature and pressure is at equilibrium when its Gibbs free energy is minimum. We have seen that for isolated systems, it is the entropy that is maximum at equilibrium, and that for NVT systems the Helmholtz free energy is minimum at equilibrium. 

Consider, for example, two distinct microscopic states of two separate, isolated systems with N ideal gas particles, energy E and volume V depicted in Fig. 6.4. In the first state, particles are uniformly distributed across the system volume. In the second state, particles are confined in one half of the available system volume. Both of these microscopic states correspond to the same NVE macroscopic state. Both of these microscopic states are available and actually have the same probability of occurring, which is equal to 1/Q(N, V, E). 

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Maxima and minima at equilibrium The partition function is then... 

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