Unit 3: Identical Particles and Bose-Einstein Statistics

1. Entropy of Mixing and Gibb's Paradox

The entropy of mixing refers to the increase in the total entropy when two or more different gases are mixed at constant temperature and pressure. According to classical thermodynamics, if two different gases occupy volumes V1 and V2 and are allowed to mix, the entropy increase is proportional to the number of particles and the logs of the volume ratios.

Gibb's Paradox arises when one considers the mixing of two volumes of the same gas. Classical calculations predict a non-zero entropy increase even for identical gases, which contradicts experimental observations and the principle that the state of the system should not change by mixing identical parts.

2. Resolution of Gibb's Paradox

The resolution to Gibb's paradox lies in the indistinguishability of identical particles. In quantum mechanics, we cannot track individual identical particles as we do in classical mechanics.

To resolve this classically, the partition function must be divided by N! (N-factorial) to account for the fact that permutations of identical particles do not result in a new physical state. This adjustment ensures that entropy is an extensive property, meaning mixing identical gases results in zero entropy change.

3. Concept of Identical Particles

Identical particles are particles that cannot be distinguished from one another by any experimental means. In quantum mechanics, because of the wave nature of matter and the uncertainty principle, their paths cannot be uniquely determined, making them truly indistinguishable.

Key Concept: If two identical particles are swapped, the physical state (the probability density) must remain the same.

4. Limitations of Classical Statistics

Classical (Maxwell-Boltzmann) statistics assume particles are distinguishable and can occupy any energy state without restriction. These assumptions fail at:

• Low Temperatures: Where the de Broglie wavelength becomes comparable to the inter-particle distance.
• High Densities: Where quantum effects dominate.

Classical statistics cannot explain phenomena like the specific heat of solids at low temperatures or the blackbody radiation spectrum.

5. Fermions and Bosons

Identical particles are divided into two categories based on their spin and the symmetry of their wavefunctions:

6. Bose-Einstein Distribution Function

The Bose-Einstein (BE) distribution applies to identical, indistinguishable particles with integral spin (Bosons). The number of particles in a state with energy E is given by:

Where mu is the chemical potential. For bosons, mu must always be less than or equal to the lowest energy state to keep the occupancy positive.

7. Bose-Einstein Condensation (BEC)

Bose-Einstein Condensation is a phenomenon that occurs in a boson gas at extremely low temperatures.

As the temperature drops below a Critical Temperature (Tc), a macroscopic fraction of the particles "condenses" into the lowest available energy state (the ground state). This results in a new state of matter where quantum effects become visible on a macroscopic scale, such as in superfluidity.

8. Thermodynamical Quantities of Photon Gas

A photon gas is a collection of photons, which are massless bosons with spin 1. Blackbody radiation is essentially a photon gas in thermal equilibrium with its container.

Key Properties:

• Chemical Potential: For a photon gas, mu = 0 because photons can be created or destroyed easily.
• Internal Energy (U): Proportional to T^4 (Stefan-Boltzmann Law).
• Pressure (P): P = U / 3V.
• Entropy (S): S = 4U / 3T.