Question

Describe the different models used to model the distribution of particles in statistical mechanics, including Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac statistics. In each case, describe the counting techniques used in the model.

Answer 1

in order to understand the different models, first, we need to understand the microstates and macrostates.

macrostates give the quantities which sums/averages over the whole particles. like volume, pressure, etc. microstates gives the properties of each of the system like momentum, velocity, etc.

now there are three main models

Maxwell Boltzmann distribution

Bose-Einstein distribution

Fermi- Dirac distribution

other models are toy models like  Bethe ansatz, hard hexagon model or algorithms like Monte Carlo algorithm.

Maxwell Boltzmann: they are indistinguishable, non-integral spin particle where

Bose-Einstein: they are indistinguishable particles which have full integral spin(0,1,2,....)

where

Fermi-Dirac: they are indistinguishable particles which have half odd integral spin(1/2,3/2) where

Bethe Ansatz: exact solution of 1D quantum particles are found. It uses Pauli's exclusion principle and by solving the boundary conditions, we can find the exact solution. From this model, few models like the Heisenberg model, the Hubbard model have been found.

Hard hexagon model: it is a 2D gas model which allows the particle to be on a triangular system with particles on the vertices but no two particles are adjacent to each other. the count is determined by using the grand partition function.

Monte Carlo method: this method is computational. That is a few codes are written using the physics formula and derivation. these are then fed into the system and the computer gives a random solution using the samples. here it takes the empirical mean of other independent variables and uses it on the problem given.