Chapter 18 - Microcanonical and grand canonical ensembles

Abstract

First in this chapter, we introduced what it means by microcanonical ensemble, and wrote probability distribution of the canonical ensemble in term of the microcanonical partition function. The appropriate constraint on mass was discussed, and we show that the mean entropy is directly correlated to the probability of thermodynamic distribution. All members of the microcanonical ensemble have equal probability. Mean entropy, mean pressure, mean enthalpy, mean Helmholtz energy, and mean Gibbs energy were found out in terms of microcanonical partition function. Then, we define the main differences between grand canonical and canonical ensembles. Appropriate constraints, expressions for the equilibrium distribution and partition function of the grand canonical ensemble were identified. Suitable fundamental equations of the mean thermodynamic quantities, and Lagrange multipliers, were figured out. We indicate that the chemical potential, volume, and temperature are the natural independent thermodynamics variables for the grand canonical ensemble. The grand canonical partition function for independent indistinguishable identical particles, mean internal energy, and the mean entropy of ideal gases, and crystalline solid were expressed in term of molecular partition function.