Part IV: Statistical Thermodynamics
Statistical thermodynamics connects the microscopic world of atoms and molecules to macroscopic thermodynamic observables. By counting microstates and weighting them with Boltzmann factors, we derive all of classical thermodynamics from first principles — and predict fluctuations that classical theory cannot.
Part Overview
Starting from the canonical partition function, we build a complete statistical mechanical framework. Molecular partition functions are factored into translational, rotational, vibrational, and electronic contributions. Ensemble theory generalizes from the canonical to grand canonical and microcanonical descriptions. The fluctuation-dissipation theorem connects spontaneous fluctuations to the system's linear response, bridging equilibrium and non-equilibrium physics.
Key Equations
Partition Function: $Z = \sum_i e^{-E_i / (k_B T)}$
Boltzmann Distribution: $P_i = \frac{e^{-E_i / (k_B T)}}{Z}$
Helmholtz Free Energy: $A = -k_B T \ln Z$
Fluctuation-Dissipation: $\langle (\Delta E)^2 \rangle = k_B T^2 C_V$
4 chapters | Partition functions to fluctuations | Micro to macro
Chapters
Chapter 1: Partition Functions
The canonical partition function $Z = \sum_i e^{-E_i/(k_BT)}$ as the central object of statistical mechanics. Factorization into molecular partition functions:$q = q_{\text{trans}} \cdot q_{\text{rot}} \cdot q_{\text{vib}} \cdot q_{\text{elec}}$. Translational partition function from the particle-in-a-box. Rotational and vibrational contributions for diatomic and polyatomic molecules. Symmetry numbers. The classical limit and the thermal de Broglie wavelength.
Chapter 2: Molecular Thermodynamics
Deriving all thermodynamic quantities from $Z$: internal energy$U = -\frac{\partial \ln Z}{\partial \beta}$, entropy$S = k_B \ln Z + k_B T \frac{\partial \ln Z}{\partial T}$, and pressure$P = k_B T \frac{\partial \ln Z}{\partial V}$. Heat capacities of ideal gases: translational, rotational, and vibrational contributions as a function of temperature. Chemical equilibrium constants from partition functions. The Sackur-Tetrode equation for ideal gas entropy.
Chapter 3: Ensemble Theory
The microcanonical ensemble (fixed $N, V, E$) and the fundamental postulate of equal a priori probabilities. The canonical ensemble (fixed $N, V, T$) and the Boltzmann distribution. The grand canonical ensemble (fixed $\mu, V, T$) with the grand partition function $\Xi = \sum_N e^{\beta \mu N} Z_N$. Equivalence of ensembles in the thermodynamic limit. Gibbs entropy formula $S = -k_B \sum_i P_i \ln P_i$.
Chapter 4: Fluctuations & Response
Energy fluctuations in the canonical ensemble: $\langle (\Delta E)^2 \rangle = k_B T^2 C_V$. Particle number fluctuations in the grand canonical ensemble:$\langle (\Delta N)^2 \rangle = k_B T \left(\frac{\partial \langle N \rangle}{\partial \mu}\right)_{T,V}$. Relative fluctuations scale as $1/\sqrt{N}$, justifying the thermodynamic limit. The fluctuation-dissipation theorem: spontaneous fluctuations determine the system's response to external perturbations. Connection to linear response theory and the Onsager regression hypothesis. Applications to specific heat, compressibility, and magnetic susceptibility.