Thermodynamics And Statistical Mechanics

For a 1-D simple harmonic oscillator, the phase space area is given by:

For independent spin-1/2 particles, each spin can be in two possible states. The number of accessible microstates is:
The entropy is given by Boltzmann’s relation:

An adiabatic process is one in which there is no heat exchange, meaning entropy remains constant in an ideal reversible case.

A microcanonical ensemble describes a system with fixed energy, volume, and number of particles, meaning it is isolated from external influences.

For fermions, the number of ways to distribute 3 indistinguishable fermions in 4 states obeying the Pauli exclusion principle is given by the combinatorial formula:
The entropy is calculated using:
Substituting :

The specific heat is given by:
For a photon gas, the internal energy scales as , leading to: