Nonequilibrium Statistical Mechanics §

Step 1: The master equation as a probability flow.

The master equation says: the rate of change of probability in state $x$ equals the total probability flowing into $x$ minus the total flowing out. For a two-state system with states 0 and 1 and transition rates $W_{01}$ (0→1) and $W_{10}$ (1→0):

Step 2: Solve the two-state example.

With $P_0 + P_1 = 1$, the solution is

where $P_0^{\text{eq}} = W_{10}/(W_{01}+W_{10})$. The relaxation time is $\tau = 1/(W_{01}+W_{10})$. At equilibrium, detailed balance holds: $W_{01}\,P_0^{\text{eq}} = W_{10}\,P_1^{\text{eq}}$.

Step 3: From discrete to continuous — the Fokker–Planck equation.

When the state space is continuous (e.g., position $x$), the Kramers–Moyal expansion of the master equation truncates at second order if the steps are small. The result is the Fokker–Planck equation with drift $A(x)$ (deterministic force) and diffusion $D(x)$ (noise strength). For the Langevin equation $m\ddot{x} = -\gamma\dot{x} + F + \xi(t)$ with $\langle\xi(t)\xi(t')\rangle = 2\gamma k_B T\,\delta(t-t')$, the overdamped limit ($m\to 0$) gives $A(x) = F/\gamma$ and $D = k_B T/\gamma$.

Step 4: Detailed balance and nonequilibrium currents.

The condition $W(x\to x')P_{\text{eq}}(x) = W(x'\to x)P_{\text{eq}}(x')$ ensures zero net probability current at equilibrium. When violated (e.g., by a non-conservative driving force), a sustained current $J(x) = A(x)P - \partial_x[D(x)P] \neq 0$ persists, and the system produces entropy. See CanonicalEnsemble for the equilibrium fixed point $P_{\text{eq}} \propto e^{-\beta E}$.

Step 5: Interactive simulation.

The simulation below shows a Brownian particle under the Langevin equation. You can adjust the friction $\gamma$ and temperature $T$ to observe the transition from ballistic (inertial) to diffusive (overdamped) motion, and verify the Einstein relation $D = k_B T/\gamma$ from the mean-squared displacement.

Authoritative sources

• van Kampen, Stochastic Processes in Physics and Chemistry, Ch. 3
• Risken, The Fokker–Planck Equation, 2nd ed.

Step 1: What the FDT says.

The FDT states that the linear response of a system at equilibrium to a small perturbation is completely determined by the equilibrium fluctuations of the same observable. In the classical limit, the susceptibility (response) is proportional to the symmetrized correlation function:

This is the nonequilibrium generalization of the equilibrium fluctuation relations: $\operatorname{Var}(E) = k_B T^2 C_V$ (canonical) and $\operatorname{Var}(N) = k_B T\, \partial_\mu \langle N\rangle$ (grand canonical). See Statistical Foundations of Thermodynamics.

Step 2: Worked example — magnetization in a field.

Consider $N$ non-interacting spins in a magnetic field $B(t)$. The observable is the magnetization $M = \sum_i \mu \sigma_i$. The perturbation is $h(t) = B(t)$.

• Equilibrium: $\langle M \rangle_{\text{eq}} = N\mu\tanh(\beta\mu B_0)$.
• Susceptibility: $\chi_{MM} = \partial\langle M\rangle/\partial B = N\mu^2\beta\,\text{sech}^2(\beta\mu B_0)$.
• Equilibrium correlation: $S_{MM}(\omega=0) = \langle M^2\rangle - \langle M\rangle^2 = N\mu^2\,\text{sech}^2(\beta\mu B_0)$.
• FDT check: $\chi_{MM} = S_{MM}(0)/(k_B T)$ ✓.

Step 3: Quantum generalization.

In the quantum case, the FDT takes the form

where $S_{AB}(\omega)$ is the symmetrized spectral density. In the classical limit $\hbar\omega \ll k_B T$, $\tanh(\beta\hbar\omega/2) \approx \beta\hbar\omega/2$, giving $\chi_{AB}(\omega) \approx \omega/(2k_B T)\, S_{AB}(\omega)$, which is consistent with the classical FDT up to the conventional factor of $\omega$ relating the one-sided and two-sided spectra.

Step 4: When the FDT breaks down.

The FDT holds only for linear response (small $h$). Far from equilibrium, the response is nonlinear: higher-order response functions $\chi^{(2)}, \chi^{(3)}, \ldots$ appear, and the simple proportionality between response and correlation no longer holds. The diagram above illustrates this: for $h < h^*$ the response is linear (FDT valid); for $h > h^*$ it saturates or becomes nonlinear (FDT invalid).

Authoritative sources

• Kubo, JPSJ 12, 570 (1957)
• Callen & Welton, Phys. Rev. 83, 34 (1951)

Step 1: Derivation from time-reversal symmetry.

The Onsager relations follow from the microscopic reversibility of the equations of motion. If the dynamics is invariant under $t \to -t$ combined with $\mathbf{p} \to -\mathbf{p}$, then the cross-correlation of two fluxes satisfies $L_{ij} = L_{ji}$ (or $L_{ij} = -L_{ji}$ if one flux is odd under time reversal, e.g., magnetization).

Step 2: Concrete example — thermoelectric coupling.

In a conductor with both a temperature gradient and an electric field, the coupled transport equations are

The Onsager symmetry requires $S = \Pi/\sigma$ (Seebeck coefficient equals Peltier coefficient divided by conductivity, up to $T$). This is the Thomson relation.

Step 3: Entropy production and the second law.

The local entropy production rate is $\sigma_s = \sum_i J_i X_i \geq 0$. For a symmetric positive-semidefinite Onsager matrix, this is automatically satisfied. Far from equilibrium, the linear relation $J = LX$ breaks down and nonlinear transport (turbulence, pattern formation) emerges. See PhaseTransitions for how nonequilibrium driving can induce phase transitions.

Authoritative sources

• Onsager, Phys. Rev. 37, 405 (1931)
• de Groot & Mazur, Non-Equilibrium Thermodynamics

Step 1: Local entropy balance.

For a continuum, the entropy density $s(\mathbf{r},t)$ obeys

where $\mathbf{J}_s$ is the entropy flux and $\sigma_s \geq 0$ is the local entropy production rate. In the linear regime, $\sigma_s = \sum_i J_i X_i$ with fluxes and forces as in the Onsager section. The second law ($\sigma_s \geq 0$) is a local statement: entropy is produced at every point, not just on average.

Step 2: Minimum entropy production (Prigogine, 1945).

For a system near equilibrium with fixed boundary conditions (e.g., fixed temperature difference across a slab), the steady state minimizes the total entropy production. Example: in a 1D slab with fixed $T_1, T_2$, the steady-state temperature profile is linear, and any deviation increases $\int \sigma_s\, dV$. This principle does not hold far from equilibrium.

Step 3: Second law from the Jarzynski equality.

The Jarzynski equality $\langle e^{-\beta W_d}\rangle = e^{-\beta\Delta F}$ is an exact identity. Applying Jensen’s inequality to the convex function $e^{-x}$:

This recovers the second law $\langle W_d\rangle \geq \Delta F$ as a consequence of an exact equality. The equality holds only for reversible (quasistatic) processes; any irreversibility makes the inequality strict. This connects the macroscopic second law to the microscopic fluctuation theorems below.

Authoritative sources

• Prigogine, Introduction to Thermodynamics of Irreversible Processes
• Jarzynski, PRL 78, 2690 (1997)

Step 1: The Jarzynski equality as an exact identity.

The Jarzynski equality is not an approximation: it holds for arbitrary driving protocols, arbitrary initial equilibrium states, and arbitrary system size. It reduces to the second law by Jensen’s inequality (see the Entropy production section above). The equality encodes the full statistics of work, not just the mean.

Step 2: The Crooks theorem and time-reversal.

The Crooks fluctuation theorem relates the probability $P_F(W)$ of observing work $W$ in a forward protocol to the probability $P_R(-W)$ of observing work $-W$ in the time-reversed protocol. The ratio is fixed by the free-energy difference. In the limit of large systems or long times, the ratio becomes sharply peaked and the second law emerges as a statement about typical trajectories.

Step 3: Experimental verification.

These theorems have been verified in single-molecule pulling experiments (RNA unfolding, DNA stretching) and in colloidal particles in optical traps. In the RNA experiments, the work distribution for unfolding and refolding shows the predicted exponential ratio, and the Jarzynski equality correctly extracts the free-energy difference even for fast, far-from-equilibrium pulling.

Step 4: Connection to equilibrium.

In the quasistatic limit (infinitely slow driving), $W_d \to \Delta F$ with zero variance, and both the Jarzynski equality and Crooks theorem reduce to the equilibrium statement $\Delta F = W_{\text{rev}}$. See MicrocanonicalEnsemble for the isolated-system limit where work is the only energy exchange.

Authoritative sources

• Jarzynski, PRL 78, 2690 (1997)
• Crooks, PRE 60, 2721 (1999)
• Collin, Ritter, Berry, Levental, Austin, Liphardt, Crooks, Nature 437, 23 (2005)

Start from the Langevin equation:

Step 1: Overdamped limit.

In the limit $m \to 0$ (or equivalently $t \gg m/\gamma$), the inertial term vanishes and the equation becomes

where $\eta(t)$ is a white noise with $\langle\eta(t)\eta(t')\rangle = 2D\,\delta(t-t')$ and $D = k_B T/\gamma$.

Step 2: Mean velocity.

Taking the ensemble average: $\langle\dot{x}\rangle = F/\gamma + \langle\eta\rangle = F/\gamma$. The particle drifts at the force divided by friction.

Step 3: Diffusion coefficient (Einstein relation).

The mean-squared displacement is

For $F = 0$ (pure diffusion): $\langle\Delta x^2\rangle = 2Dt$, giving $D = k_B T/\gamma$.

Step 4: Connection to the FDT.

The Einstein relation $D = \mu_{\text{mob}}\, k_B T$ (where $\mu_{\text{mob}} = 1/\gamma$ is the mobility) is a direct consequence of the fluctuation–dissipation theorem: the same friction $\gamma$ that causes dissipation (the drift term) also sets the noise amplitude (the diffusion term). This is the simplest physical manifestation of the FDT.