Thermodynamics and Statistical Physics
Thermodynamics and Statistical Physics in GATE PH tests the laws of thermodynamics, thermodynamic potentials, Maxwell relations, ensembles, partition functions, quantum statistics, and phase transitions. Questions often ask for a derivative, a heat capacity, or the classical limit of a quantum distribution.
Core rules and concepts
- First law: dU = δQ - δW, with δW = P dV for reversible pressure-volume work. The sign convention matters.
- Thermodynamic potentials: F = U - TS, H = U + PV, G = U - TS + PV. Maxwell relations follow from equality of mixed second derivatives.
- Canonical partition function Z = Σ exp(-βE_i), with β = 1/(k_BT). Then U = -∂lnZ/∂β and C_V = (1/k_BT²)∂²lnZ/∂β².
- Fermi-Dirac and Bose-Einstein distributions reduce to Maxwell-Boltzmann at high temperature and low density.
- Indistinguishability introduces a 1/N! factor in the classical partition function for identical particles.
Worked example
A two-level system has energies 0 and ε. Its partition function is Z = 1 + exp(-βε). The average energy is U = ε exp(-βε)/(1 + exp(-βε)). The heat capacity is C = dU/dT = (ε²/(k_BT²)) exp(-βε)/(1 + exp(-βε))². As T → 0, C → 0; as T → ∞, C → 0, showing a Schottky peak at intermediate temperature.
Traps that cost marks
- Using the wrong sign for work done by or on the system.
- Taking a partial derivative while holding the wrong variable constant.
- Forgetting the 1/N! factor for indistinguishable particles.
- Applying quantum distributions in a regime where the classical limit is valid.