academic-thermodynamics-statistical-physics

# Thermodynamics and Statistical Mechanics (Pathria & Callen) This skill establishes the rigorous microscopic and macroscopic bridge between the phenomenological laws of thermodynamics, the statistical mechanics of ensembles, and the degenerate quantum gases of fermions and bosons. --- ## 🔥 1. Laws of Thermodynamics and Thermodynamic Potentials ### 1.1 Fundamental Laws - **First Law**: $dU = \delta Q - \delta W + \mu dN$. - **Second Law**: $dS \ge \frac{\delta Q}{T}$ (for reversible processes, $dS = \frac{\delta Q_{rev}}{T}$). - Carnot Cycle Efficiency: $\eta = 1 - \frac{T_C}{T_H}$. - **Third Law (Nernst Theorem)**: $S \to 0$ as $T \to 0\text{ K}$ for pure crystalline systems. ### 1.2 Thermodynamic Potentials and Maxwell Relations | Potential | Legendre Transform | Fundamental Differential | Associated Maxwell Relation | | :--- | :--- | :--- | :--- | | **Internal Energy ($U$)** | $U(S, V, N)$ | $dU = T dS - P dV + \mu dN$ | $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$ | | **Enthalpy ($H$)** | $H = U + PV$ | $dH = T dS + V dP + \mu dN$ | $\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P$ | | **Helmholtz Free Energy ($F$)** | $F = U - TS$ | $dF = -S dT - P dV + \mu dN$ | $\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$ | | **Gibbs Free Energy ($G$)** | $G = H - TS$ | $dG = -S dT + V dP + \mu dN$ | $\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P$ | --- ## 📊 2. Statistical Mechanics Ensembles (Pathria) ```mermaid graph TD subgraph Ensembles EM["1. Microcanonical Ensemble<br/>(E, V, N isolated)"] -->|Boltzmann Entropy| S["S = kB ln Ω(E)"] EC["2. Canonical Ensemble<br/>(T, V, N thermal bath)"] -->|Partition Function Z| F["F = -kB T ln Z"] EGC["3. Grand Canonical Ensemble<br/>(T, V, μ open particles)"] -->|Grand Partition Function 𝒵| PHI["Φ = -kB T ln 𝒵 = -PV"] end ``` ### 2.1 Canonical Ensemble - **Canonical Partition Function**: $Z = \sum_{i} e^{-\beta E_i}$ where $\beta = \frac{1}{k_B T}$. - **Average Internal Energy**: $U = \langle E \rangle = -\frac{\partial \ln Z}{\partial \beta} = k_B T^2 \frac{\partial \ln Z}{\partial T}$. - **Energy Fluctuations and Heat Capacity**: $\sigma_E^2 = \langle E^2 \rangle - \langle E \rangle^2 = k_B T^2 C_v$. --- ## ⚛️ 3. Quantum Statistics and Degenerate Gases The average occupation number of a single-particle quantum state of energy $\epsilon_i$: $$\langle n_i \rangle = \frac{1}{e^{\beta(\epsilon_i - \mu)} + a}$$ - $a = 0$: **Classical Maxwell-Boltzmann Statistics** (dilute limit $n \lambda_{th}^3 \ll 1$). - $a = +1$: **Fermi-Dirac Quantum Statistics** (half-integer spin fermions with the Pauli Exclusion Principle). - $a = -1$: **Bose-Einstein Quantum Statistics** (integer spin bosons with condensation). ### 3.1 Degenerate Electron Gas (Fermi-Dirac) - **Fermi Energy ($T = 0\text{ K}$)**: $$E_F = \frac{\hbar^2}{2m} (3\pi^2 n)^{2/3}, \quad k_F = (3\pi^2 n)^{1/3}$$ - **Sommerfeld Electronic Heat Capacity**: $C_{v,el} = \frac{\pi^2}{2} N k_B \left( \frac{T}{T_F} \right) = \gamma T$ (linear dependence on temperature). ### 3.2 Bose-Einstein Condensation (BEC) and Black-Body Radiation - **Critical Condensation Temperature ($T_c$)**: Below $T_c$, a macroscopic fraction of bosons occupies the zero-energy ground state: $$T_c = \frac{2\pi \hbar^2}{m k_B} \left( \frac{n}{\zeta(3/2)} \right)^{2/3} \approx 3.31 \frac{\hbar^2 n^{2/3}}{m k_B}$$ - **Planck's Law for Photon Radiation ($\mu = 0$)**: $$u(\nu, T) = \frac{8\pi h \nu^3}{c^3} \frac{1}{e^{h\nu/k_B T} - 1}$$