Thermodynamics · Advanced Topics

Nonequilibrium Statistical Mechanics

Most of nature operates far from equilibrium — living cells, driven chemical systems, turbulent flows, and active matter. Nonequilibrium statistical mechanics goes beyond Boltzmann to characterize driven, dissipative, and fluctuating systems.

PrerequisitesStatisticalmechanics(Ch.S)⋅Irreversibleprocesses(Ch.IR)⋅Probability(Ch.PR)⋅DStatistical mechanics (Ch. S) \cdot Irreversible processes (Ch. IR) \cdot Probability (Ch. PR) \cdot Differential equations (Ch. DE)
Learning Goals
  • Write the Boltzmann transport equation and explain the H-theorem as a derivation of the second law from molecular dynamics.
  • Derive the Smoluchowski Fokker-PlanckequationforaBrownianparticleinapotentialandidentifytheEinsteinrelationD=nck equation for a Brownian particle in a potential and identify the Einstein relation D =
  • Apply Kramers' escape-ratetheoryandexplaintheexponentialsensitivityofreactionratestobarrierheight\DeltaVrate theory and explain the exponential sensitivity of reaction rates to barrier height \DeltaV
  • StatetheJarzynskiequality⟨e−W/kBT⟩=e−\DeltaF/kBTanddescribehowitisusedtoeState the Jarzynski equality ⟨e^{-W/k_{BT}}⟩ = e^{-\DeltaF/k_{BT}} and describe how it is used to extract equilibrium free energies from irreversible pulling experiments.
  • Write and solve the master equation for a two-state Markov chain and relate it to ion-channel gating kinetics.

NE.1 The Boltzmann Equation

The Boltzmann equation describes the time evolution of the one-particle distribution function f(r, v, t) — the probability density of finding a particle at position r with velocity v at time t:

\partialf/\partialt+v⋅∇rf+(F/m)⋅∇vf=(\partialf/\partialt)coll(Boltzmanntransportequation)\partialf/\partialt + v\cdot\nabla_r f + (F/m)\cdot\nabla_v f = (\partialf/\partialt)_coll \qquad (Boltzmann transport equation)(NE.1)

The left side is the Liouville streaming term (free evolution). The right side is the collision term — the rate of change of f due to two-particle collisions. For elastic hard-sphere collisions:

(\partialf/\partialt)coll=∫d3v1dΩσ(Ω)∣v−v1∣[f(v′)f(v1′)−f(v)f(v1)](\partialf/\partialt)_coll = \int d^{3}v_{1} d\Omega \sigma(\Omega)|v-v_{1}| [f(v')f(v_{1}') - f(v)f(v_{1})](NE.2)

where v' and v₁' are post-collision velocities. The Boltzmann equation assumesmolecular chaos (Stosszahlansatz): pre-collision velocities are uncorrelated.

Theorem NE.1 — Boltzmann's H-Theorem
DefineH=∫flnfd3rd3v.ThendH/dt≤0—Halwaysdecreases(orstaysconstantinequDefine H = \int f ln f d^{3}r d^{3}v. Then dH/dt \le 0 — H always decreases (or stays constant in equilibrium).SincethermodynamicentropyS=−kBH,thisgivesdS/dt≥0—thesecondlawoilibrium). Since thermodynamic entropy S = -k_{B} H, this gives dS/dt \ge 0 — the second law of thermodynamics, derived from molecular dynamics!Equilibriumdistribution:f=0in(\partialf/\partialt)coll→Maxwell−Boltzmannfeq∝e(−mv2/(2kBTEquilibrium distribution: f = 0 in (\partialf/\partialt)_coll \to Maxwell-Boltzmann f_{eq} \propto e^(-mv^{2}/(2k_BT). The H-theorem proves that any initial distribution relaxes to Maxwell-Boltzmann. Controversy: it assumes time-irreversibility (molecular chaos), which is not present in the time-symmetric underlying mechanics — Loschmidt's paradox.

NE.2 The Fokker-Planck Equation

For a Brownian particle in a potential V(x), driven by noise, the probability distribution P(x, t) evolves by the Fokker-Planck equation:

\partialP/\partialt=∂/\partialx[P/(mγ)×dV/dx]+(kBT/(mγ))∂2P/\partialx2(Smoluchowski,overdamped)\partialP/\partialt = \partial/\partialx [P/(m\gamma) \times dV/dx] + (k_{BT}/(m\gamma)) \partial^{2}P/\partialx^{2} \qquad (Smoluchowski, overdamped)(NE.3)

where γ is the friction coefficient. The first term is drift (down the potential gradient); the second is diffusion (D = k_BT/(mγ) — Einstein relation). Steady-state solution: P_eq(x) ∝ e^(−V(x)/(k_BT)) (Boltzmann distribution).

For a particle in a periodic potential tilted by a constant force F (modeling ion channels, molecular motors):

V(x)=V0sin(2\pix/a)−Fx(tiltedwashboardpotential)V(x) = V_{0} sin(2\pix/a) - Fx \qquad (tilted washboard potential)(NE.4)

Below a critical force F_c = πV₀/a: particle is trapped (locked). Above F_c: particle drifts (running). The transition is a saddle-node bifurcation. Thermal fluctuations allow escape (Kramers theory) at rate Γ = ω₀ ω_b/(2πγ) × e^(−ΔV/k_BT).

Example NE.1 — Escape Rate from a Potential Well — Kramers Theory

Estimatethethermalescaperateofamoleculefromapotentialwellofdepth\DeltaV=1eVatEstimate the thermal escape rate of a molecule from a potential well of depth \DeltaV = 1 eV at T=300KT = 300 K

Kramers rate:Γ=(ωminωmax)/(2πγ)×e(−\DeltaV/(kBT))foroverdampedcase.HereωminandωmaxaretheΓ = (\omega_min \omega_max)/(2\pi\gamma) \times e^(-\DeltaV/(k_{BT})) for overdamped case. Here \omega_min and \omega_max are the frequencies at the minimum and saddle point.
Exponent:\DeltaV/(kBT)=1.6×10−19/(1.38×10−23×300)=1.6×10−19/4.14×10−21=38.6.Thisisahugenumbe\DeltaV/(k_{BT}) = 1.6\times10^{-19}/(1.38\times10^{-23}\times300) = 1.6\times10^{-19}/4.14\times10^{-21} = 38.6. This is a huge number.
Rate:e−38.6≈2×10−17.Foramolecularoscillationfrequencyω0∼1013s−1(infrared):Γ≈1e^{-38.6} \approx 2\times10^{-17}. For a molecular oscillation frequency \omega_{0} ∼ 10^{13} s^{-1} (infrared): Γ \approx 1013×2×10−17=2×10−4s−1.Escapetimeτ≈5000s≈1.4hours0^{13} \times 2\times10^{-17} = 2\times10^{-4} s^{-1}. Escape time \tau \approx 5000 s \approx 1.4 hours.
Sensitivity:\DeltaV=0.5eV:e−19.3≈4×10−9,Γ≈4×104s−1,τ≈25\mus.Factor2inbarrier→1020chan\DeltaV = 0.5 eV: e^{-19.3} \approx 4\times10^{-9}, Γ \approx 4\times10^{4} s^{-1}, \tau \approx 25 \mus. Factor 2 in barrier \to 10^{20} change in rate! Chemical kinetics (Arrhenius) and protein folding both governed by this exponential sensitivity.

NE.3 Fluctuation Theorems

For systems driven out of equilibrium, exact relations hold between fluctuations of the entropy production σ over a time interval τ:

P(στ=A)/P(στ=−A)=eA(Crooksfluctuationtheorem)P(\sigma_\tau = A) / P(\sigma_\tau = -A) = e^A \qquad (Crooks fluctuation theorem)(NE.5)

Entropy-decreasing trajectories (σ < 0) exist — they're just exponentially rare. The second law follows on average: ⟨σ⟩ ≥ 0.

The Jarzynski equality (1997) relates the free energy difference ΔF between equilibrium states to the work W done in a non-equilibrium process:

⟨e(−W/(kBT))⟩=e(−\DeltaF/(kBT))(Jarzynskiequality)⟨e^(-W/(k_{BT}))⟩ = e^(-\DeltaF/(k_{BT})) \qquad (Jarzynski equality)(NE.6)

This is remarkable: ΔF (an equilibrium quantity) can be extracted from non-equilibrium work measurements — even if the process is irreversible. Verified by single-molecule RNA unfolding experiments (Liphardt et al., 2002, Science). Jensen's inequality W ≥ ΔF follows immediately (second law).

NE.4 Stochastic Thermodynamics

Stochastic thermodynamics extends thermodynamic concepts to individual stochastic trajectories. For a single molecule following path x(t) in time [0, τ]:

Trajectory entropy: s(t) = −k_B ln P(x(t), t).Medium entropy: δs_med = −δQ/T (heat exchanged with bath).Total entropy production: Δs_tot = Δs + Δs_med ≥ 0 (along any trajectory).

Active matter: driven systems with local energy input — bacteria, bird flocks, cytoskeleton. Self-propelled particles break detailed balance at the microscopic level. Flocking (Vicsek model): polar order emerges above a density threshold — a non-equilibrium phase transition with no equilibrium analog.

Definition NE.2 — Common Traps
  • The Boltzmann equation assumes molecular chaos: irreversibility enters through the collision assumption, not Newton's laws alone.
  • Fokker-Planck describes distributions: it is not one particle trajectory but the evolution of probability density.
  • Rare trajectories matter in Jarzynski averages: the exponential average is dominated by unusually low-work samples.
  • Detailed balance is an equilibrium condition: active matter and driven steady states usually break it.
Exercises — NE.1–NE.4 Nonequilibrium Stat Mech
1.Show how the Boltzmann equation yields Navier-Stokes equations in the hydrodynamic limit. What determines the shear viscosity of a dilute gas?
Straightforward
2.
SolvetheLangevinequationforaBrownianparticleandfind⟨x2(t)⟩.ShowthecrossoverfSolve the Langevin equation for a Brownian particle and find ⟨x^{2}(t)⟩. Show the crossover fromballistictodiffusivebehavior.CalculateD(m2/s)fora1\mumpollengraininwaterarom ballistic to diffusive behavior. Calculate D (m^{2}/s) for a 1 \mum pollen grain in water at 300 K.
m²/s
Intermediate
3.Describe the Jarzynski equality. How was it applied to single−moleculeRNAunfoldingexperiments?WhatarethepracticalchallengesinextractingΔngle-molecule RNA unfolding experiments? What are the practical challenges in extracting \Delta
Intermediate
4.Write and solve the master equation for a two-state Markov chain. Show the relaxation to steady state. How does this describe ion channel gating?
Challenging
Key Takeaways
  • Boltzmannequation:\partialf/\partialt+v⋅\nablaf+(F/m)⋅∇vf=collisionterm.H−theorem:dH/dt≤0→2nBoltzmann equation: \partialf/\partialt + v\cdot\nablaf + (F/m)\cdot\nabla_v f = collision term. H-theorem: dH/dt \le 0 \to 2nd law.
  • Fokker−Planck:\partialP/\partialt=−∂(drift\cdotP)/\partialx+D∂2P/\partialx2.Steadystate:BoltzmanndistributionFokker-Planck: \partialP/\partialt = -\partial(drift\cdotP)/\partialx + D \partial^{2}P/\partialx^{2}. Steady state: Boltzmann distribution.
  • Kramersrate:Γ∝e−\DeltaV/kBT.Exponentialsensitivitytobarrierheight—controlschemiKramers rate: Γ \propto e^{-\DeltaV/k_{BT}}. Exponential sensitivity to barrier height — controls chemistry.
  • Jarzynski:⟨e−W/kBT⟩=e−\DeltaF/kBT.Extractequilibrium\DeltaFfromirreversibleworkJarzynski: ⟨e^{-W/k_{BT}}⟩ = e^{-\DeltaF/k_{BT}}. Extract equilibrium \DeltaF from irreversible work.
  • Crookstheorem:P(σ)/P(−σ)=eσ.Secondlaw(⟨σ⟩≥0)isastatementaboutfluctuationasymCrooks theorem: P(\sigma)/P(-\sigma) = e^\sigma. Second law (⟨\sigma⟩\ge0) is a statement about fluctuation asymmetry.
  • Master equations describe stochastic kinetics: protein folding, ion channels, gene expression.