6. Thermodynamics in different ensembles:

6.1 Microcanonical ensemble (E, N,V) : In this ensemble, the macrostate is defined by the total energy E, the number of particles N and the volume V. However, for calculation purpose, a small range of energy E to (with ) is considered instead of a sharply defined energy value E. The systems of the ensemble may be in any one of a large number of microstates between E and . In the phase space, the representative points will lie within a hypershell defined by the condition

At statistical equilibrium, all representative points are uniformly distributed and between E and otherwise zero. As per equal a priori probability, any accessible state is equally probable and the number of accessible microstates Ω is proportional to the phase space volume enclosed within the hypershell and it is given by

for a system of N particles and of total energy E. If the particles are indistinguishable, the number of microstates Ω should be divided by as the Gibb's correction. However, if the energy states are discrete, the particles are distributed among the different energy levels as, particles in the energy level and satisfies the following conditions

The total number of possible distributions or microstates of N such particles is then given by

The thermodynamic properties can be obtained by associating entropy S of the system to the number of accessible microstates Ω. The statistical definition of entropy by Boltzmann is given by

where is the Boltzmann constant, JK . In a natural process the equilibrium corresponds to maximum Ω or equivalently maximum entropy S as is stated in the second law of thermodynamics. It is to be noted that, as , the system is going to be in its ground state and the value of Ω is going to be 1. Consequently, the entropy which is the third law of thermodynamics. If a thermodynamic potential like entropy S is known in terms of the number of microstates, the thermodynamic properties of the system can be obtained by taking suitable derivative of S with respect to the relevant parameters as given below.