Non-Equilibrium Statistical Mechanics
Non-Equilibrium Statistical Mechanics
Real systems are rarely in equilibrium. Transport phenomena — heat conduction, diffusion, electrical conductivity — require non-equilibrium statistical mechanics. The Boltzmann transport equation provides the microscopic foundation.
Definition
Boltzmann equation: \(\partial_t f + \mathbf{v}\cdot\nabla f + \mathbf{F}/m\cdot\nabla_v f = (\partial f/\partial t)_{coll}\). In the relaxation-time approximation: \((\partial f/\partial t)_{coll} = -(f-f_0)/\tau\).
Key Result
Electrical conductivity from Boltzmann: \(\sigma = ne^2\tau/m\) (Drude formula). Thermal conductivity: \(\kappa = \frac{1}{3}C_V v_{th}\ell\) where \(\ell = v_{th}\tau\) is the mean free path.
Example 1
Brownian motion: a particle with friction coefficient \(\gamma\) obeys \(m\dot{v} = -\gamma v + F_{noise}(t)\). The power spectral density of the noise force equals \(2\gamma k_BT\) (fluctuation-dissipation).
Example 2
Entropy production: for irreversible processes, \(dS/dt \geq 0\) strictly. Near equilibrium, entropy production rate \(\dot{S} = \sum_i J_i X_i\) (fluxes × forces) — Onsager's theory of irreversible thermodynamics.
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Practice
- Derive Fick's law of diffusion from the Boltzmann equation.
- What is the H-theorem, and what does it prove?
- Explain why the Boltzmann equation captures irreversibility despite Newton's laws being reversible.
- What is the Green-Kubo relation for transport coefficients?
Show Answer Key
1. Boltzmann equation: $\frac{\partial f}{\partial t} + \mathbf{v}\cdot\nabla f + \frac{\mathbf{F}}{m}\cdot\nabla_v f = \left(\frac{\partial f}{\partial t}\right)_{\text{coll}}$. Near equilibrium, $f = f_0 + \delta f$. The collision integral linearized gives $\delta f \propto -\tau\mathbf{v}\cdot\nabla n$. The particle flux $\mathbf{J} = \int \mathbf{v}\,\delta f\,d^3v = -D\nabla n$: Fick's law with $D = \frac{1}{3}v_{\text{th}}\ell$ (mean free path $\ell = v_{\text{th}}\tau$).
2. Boltzmann's H-theorem: define $H = \int f\ln f\,d^3v$. For the Boltzmann equation with molecular chaos (Stosszahlansatz): $dH/dt \leq 0$, with equality iff $f$ is the Maxwell-Boltzmann distribution. Since $S = -k_B H$ (up to constants), entropy increases monotonically until equilibrium. This provides a statistical-mechanical basis for the second law.
3. Newton's laws are time-reversible, but the Boltzmann equation is not — due to the molecular chaos assumption (Stosszahlansatz): velocities of colliding particles are assumed uncorrelated before collision. This introduces a time asymmetry (an arrow of time). The assumption is valid for dilute gases where recollisions are rare, but breaks down for dense systems or short times.
4. Green-Kubo: transport coefficient $L = \int_0^\infty \langle J(0)J(t)\rangle\,dt$ where $J$ is the corresponding current (heat current for thermal conductivity, stress for viscosity, particle current for diffusion). This relates equilibrium time-correlation functions to non-equilibrium transport coefficients — a cornerstone of linear response theory.