In quantum statistical mechanics, the thermodynamics of an ideal Bose gas is described using quantum principles and statistical methods. An ideal Bose gas consists of a collection of identical bosonic particles (particles with integer spin) that do not interact with each other except through the fundamental principles of quantum mechanics. Examples of bosonic particles include photons, mesons, and composite particles like helium-4 atoms.

Here are some key concepts related to the thermodynamics of an ideal Bose gas in quantum statistical mechanics:

Quantum State and Energy Levels: Quantum mechanics allows particles to exist in discrete energy levels, rather than in a continuous range of energies. The energy levels in the system are quantized, and each energy level can accommodate multiple particles due to the bosonic nature of the particles.
Bose-Einstein Distribution Function: The Bose-Einstein distribution function describes the probability of finding a boson in a particular quantum state at a given temperature. It is given by:
n(E) = 1 / [exp((E - μ) / (k_B * T)) - 1]
where:
- n(E) is the average number of particles in the quantum state with energy E.
- μ is the chemical potential, which determines the average number of particles in the system.
- k_B is Boltzmann's constant.
- T is the temperature.
Bose-Einstein Condensation: As the temperature is lowered, the Bose-Einstein distribution function approaches a macroscopic number of particles occupying the lowest energy state (the ground state). This phenomenon is known as Bose-Einstein condensation. At a critical temperature T_c, a macroscopic fraction of the particles condenses into the ground state, forming a coherent quantum state called a Bose-Einstein condensate (BEC).
Thermodynamic Quantities: Using the Bose-Einstein distribution function, various thermodynamic quantities can be calculated for the ideal Bose gas. These include the internal energy, entropy, free energy, and specific heat. At low temperatures, the specific heat exhibits a power-law behaviour with temperature, as mentioned earlier.
Quantum Statistics: Quantum statistics, particularly Bose-Einstein statistics, is a crucial aspect of the thermodynamics of an ideal Bose gas. Bose statistics allow multiple bosons to occupy the same quantum state, leading to the possibility of Bose-Einstein condensation.
Critical Temperature and BEC Transition: The critical temperature T_c, below which Bose-Einstein condensation occurs, is a fundamental property of the ideal Bose gas. It depends on the particle density and the mass of the bosons. For an ideal Bose gas, T_c is inversely proportional to the particle density.

Grand partition function for Boson gas is

Taking log on both sides, we get

For an ideal Bose gas, the summands in equation 3 diverges as z -> 1 because the single term corresponding to p= 0 diverges. Thus the single term at p=0 may be an important as the entire sum. So we split off the term corresponding to p= 0 and replace rest of the term by integral.

The average occupation number is given by

We know from Bose Einstein distribution law,

for a single state particle with momentum p =0

where <no> is the number of particles occupying the single state with p=0

Using 7 in 6, we get

for 0⩽Ƶ ⩽ 1, the function g3/2(Ƶ) is bounded, positive monotonically increasing function of Ƶ. It has largest value at Ƶ = 1.

Equation 8 can be rewritten as

This expression shows that at low temperatures, the thermal de Broglie wavelength cubed (𝜆³) divided by the volume (v) of the system is greater than or equal to the Bose function g3/2(1). The inequality is a characteristic feature of the quantum statistics of bosons, and it plays a crucial role in understanding Bose-Einstein condensation and the behaviour of an ideal Bose gas at low temperatures. This indicates that a limited fraction of the particle is in the state where p=0 is present. Bose Einstein condensation is the name given to the event in which a macroscopically small portion of the particle (making up the B.E assembly) accumulates in a single quantum state corresponding to p = 0.

for 𝜆³/v < 2.612, Ƶ is given by the solution of 𝜆³/v constant and g3/2 (Ƶ) = 𝜆³/v > 2.612, Ƶ = 1. So equation 11 defines the transition region for Bose Einstein condensation. The transition temperature for a fixed v and transition volume for fixed T can be obtained from equation

𝜆³/v ⩾ g3/2(1) ....12

In this region the system can be considered to be a mixture of two thermodynamic phases. One phase being composed of particles with p -0 and the other with p ≠ 0.

For a fixed volume, transition temperature To is

𝜆³/v = g3/2(1)

Also

transition condensation temperature

Similarly

Transition volume Vo for a fixed temperature T is

Fraction of particles in the zero momentum space i.e p= 0 is obtained as

This is the result that the pressure of a boson gas independent of number of particles.

The study of the thermodynamics of an ideal Bose gas in quantum statistical mechanics is essential for understanding various phenomena in condensed matter physics, quantum optics, and cosmology. Bose-Einstein condensation, in particular, is a fascinating quantum phenomenon that has been experimentally observed and studied in ultracold atomic systems, leading to exciting new applications in areas such as quantum computing and quantum simulation.

This note is a part of the Physics Repository.

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