Statistical Mechanics Notes

Fundamental Concepts

• Study of microscopic behavior to explain macroscopic properties.
• Bridges microscopic and macroscopic physics.
• Based on probability theory and quantum mechanics.
• Deals with systems of many particles (typically > 10²³).

Statistical Ensembles

1. Microcanonical Ensemble

• Isolated system with fixed energy.
• All microstates equally probable.
• Boltzmann's entropy formula: S = k ln(Ω)

2. Canonical Ensemble

• System in thermal contact with heat bath.
• Fixed temperature.
• Probability distribution: P = e^(-E/kT)/Z (Z = partition function)

3. Grand Canonical Ensemble

• System can exchange particles and energy.
• Fixed chemical potential.
• Used for open systems.

Partition Functions

• Partition function: Z = Σ e^(-E/kT)
• Contains all thermodynamic information.
• Used to derive:
  • Free energy
  • Internal energy
  • Entropy
  • Pressure
  • Heat capacity

Maxwell-Boltzmann Distribution

• Describes velocity distribution of gas particles.
• Formula: f(v) = (m/2πkT)^(3/2) * e^(-mv²/2kT)
• Foundation for kinetic theory.

Quantum Statistics

1. Bose-Einstein Statistics

• For bosons (integer spin).
• Multiple particles can occupy the same state.
• Leads to Bose-Einstein condensation.

2. Fermi-Dirac Statistics

• For fermions (half-integer spin).
• Pauli exclusion principle applies.
• Important for electron behavior in metals.

Phase Transitions

• Critical phenomena.
• Order parameters.
• Critical exponents.
• Universality classes.
• Mean field theory.

Applications

• Ideal gases.
• Quantum gases.
• Magnetism.
• Heat capacities.
• Blackbody radiation.
• Chemical reactions.
• Semiconductor physics.

Key Equations

• F = -kT ln(Z)
• S = -k Σ Pi ln(Pi)
• U = -∂/∂β ln(Z)
• P = kT ∂/∂V ln(Z)

Historical Development

• Boltzmann's H-theorem.
• Maxwell's demon.
• Gibbs paradox.
• Einstein's contributions.
• Development of quantum statistics.

These principles form the foundation for understanding bulk matter behavior and are crucial in modern physics and chemistry.