The partition function for Cannonical ensemble is given by

Equation 1 can be written as

where g{np} is the proper statistical weight factor for the distribution set np, with g{np} = 1 in the case of Bose and Fermi, and

for Boltzmann gas. The grand partition function for BE and FD is

This double summation is equivalent to summing each np independently i.e

where ∑ extends over the values n=0,1,2,.........for Bosons and n=0 or 1 for fermions.

which is a geometric occupation number <np> are given by

For fermions

We have grand partition function

For fermions n=0,1 then

Now the average occupation number <np> is

This is the expression for average occupation number.

For Maxwell Boltzmann gas

The partition function for MB case,

For Maxwell Boltzmann case , statistical weight

The grand partition function is given by

The double sum is equivalent to sum of each np independently

Here for Maxwell Boltzmann case,

Here, the average occupation number for MB case we get

The use of the grand partition function in statistical mechanics provides several advantages for analyzing systems described by Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics. Here are some advantages:

Comprehensive Description: The grand partition function allows for a unified and comprehensive description of statistical systems, encompassing all possible energy levels and incorporating the effects of temperature, chemical potential, and quantum statistics. It provides a single mathematical framework that can be applied to different statistical distributions.
Statistical Quantities: The grand partition function allows for the calculation of various statistical quantities, including the average occupation number, energy, entropy, and fluctuations. These quantities provide valuable insights into the behavior of the system and can be derived from the grand partition function through appropriate differentiations and manipulations.
Thermodynamic Relations: The grand partition function facilitates the derivation of thermodynamic relations and equations of state. By taking appropriate derivatives of the grand partition function with respect to temperature, volume, and chemical potential, one can obtain relations such as the equation of state, heat capacity, compressibility, and chemical potential.
Flexibility for Different Statistics: The grand partition function accommodates different statistical distributions, enabling the analysis of systems composed of distinguishable particles (Maxwell-Boltzmann), indistinguishable bosons (Bose-Einstein), and indistinguishable fermions (Fermi-Dirac). The statistical behavior of each system can be captured by choosing the appropriate form of the grand partition function and applying the corresponding statistical principles.
Connection to Quantum Mechanics: The grand partition function connects statistical mechanics to quantum mechanics. By considering quantum statistics and incorporating the effects of quantum indistinguishability, the grand partition function allows for the description of systems at the quantum level, capturing phenomena such as Bose-Einstein condensation and Fermi-Dirac statistics for fermionic systems.

Overall, the use of the grand partition function in Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics provides a powerful and versatile framework for analyzing and understanding the statistical behavior of diverse physical systems, incorporating both classical and quantum effects.

This note is taken from Quantum Statistical Mechanics, Msc physics, Nepal.

This note is a part of the Physics Repository.

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