Statistical mechanics away from Boltzmann equilibrium: linear-response, stochastic thermodynamics, fluctuation theorems, large-deviation theory, transport. Unifies hydrodynamic and kinetic descriptions and furnishes rigorous second-law…
non-equilibrium
Linear response theory
Small perturbation H' = –A(t)f(t) gives ⟨B(t)⟩ = ∫χ_BA(t–t') f(t') dt' with response function χ_BA computable from equilibrium correlations…
Green–Kubo formula
Transport coefficients equal integrals of equilibrium autocorrelations: η = (V/k_BT)∫₀^∞⟨σ_xy(t)σ_xy(0)⟩dt, similar for κ, σ, D. Derives…
Jarzynski equality
Over arbitrary out-of-equilibrium driving protocols, ⟨e^{–βW}⟩ = e^{–βΔF}, where the average is over realisations. Gives equilibrium…
Crooks fluctuation theorem
Forward vs time-reversed work distributions satisfy P_F(W)/P_R(–W) = e^{β(W–ΔF)}. Jarzynski is an immediate corollary; the theorem…
Gallavotti–Cohen fluctuation theorem
Asymptotic large-deviation symmetry for entropy production Σ over long time τ: P(Σ=+σ)/P(Σ=–σ) ≈ e^{στ}. Experimentally verified in driven…
Stochastic thermodynamics
Thermodynamic quantities (heat Q, work W, entropy S) defined at the level of individual trajectories of an overdamped Langevin or jump…
Onsager reciprocal relations
For conjugate flux–force pairs J_i = Σ_j L_{ij} X_j near equilibrium, L_{ij} = L_{ji} under time-reversal symmetry. Basis of…
Large-deviation principle
For non-equilibrium observables A_τ averaged over time τ, P(A_τ = a) ≍ e^{–τ I(a)} with rate function I(a) ≥ 0 and I(a*)=0 at the typical…
Hydrodynamic limit
Macroscopic PDE emerges from microscopic stochastic dynamics on coarse-grained space-time scales ε∼x/L, τ∼t/L². Euler and Navier-Stokes…
Thermodynamic uncertainty relation (TUR)
For any current J in a stationary non-equilibrium process, Var(J)/⟨J⟩² ≥ 2k_B/⟨σ⟩, tying precision directly to entropy production.…
Cramér–Chernoff large-deviation rate function (non-equilibrium)
Non-equilibrium statistical-mechanics application of L0 large-deviations theory and moment-generating-functions. Cramér's theorem: for iid…
Integral fluctuation theorem as a martingale identity (non-equilibrium)
Non-equilibrium statistical-mechanics application of L0 martingale theory and optional-stopping. Entropy production Σ_t along a stochastic…
Onsager reciprocal (1931)
L Onsager 1931 (Nobel 1968) reciprocal-relations L_ij = L_ji in linear-response; foundation of irreversible thermodynamics.
Jarzynski (1997)
C Jarzynski 1997 + Crooks 1999 fluctuation theorems; <exp(-W/kT)> = exp(-Delta-A/kT) far-from-equilibrium; modern stochastic-thermodynamics…
KPZ (Kardar-Parisi-Zhang 1986)
Kardar-Parisi-Zhang 1986 stochastic-PDE for surface-growth; 1+1D exact solution Tracy-Widom + KPZ universality class; modern integrability.
SOC (BTW 1987)
Bak-Tang-Wiesenfeld 1987 sandpile self-organized-criticality; power-law avalanche-distribution; modern earthquake + neuronal-avalanche…
Active matter (Cates 2010s)
M Cates 2010s motility-induced-phase-separation; Vicsek 1995 self-propelled particles; modern bacterial-swarming + cytoskeleton-actomyosin.
NESS (Zia-Schmittmann 2007)
Zia-Schmittmann 2007 driven-diffusive-systems review; current-carrying steady-states; modern ABC model + KMP-stochastic-energy.
Master equation (Pauli 1928)
W Pauli 1928 master-equation; modern Gillespie 1976 stochastic-simulation algorithm; basis of chemical-kinetics + epidemiology.
Langevin (1908)
P Langevin 1908 stochastic-differential-equation Brownian motion; modern numerical-integration Euler-Maruyama + Milstein schemes.
Fokker-Planck (1914-1917)
Fokker-Planck-Smoluchowski 1914-1917 density-evolution PDE; modern dual to Langevin SDE.
FDT (Callen-Welton 1951)
Callen-Welton 1951 fluctuation-dissipation theorem; modern Kubo 1957 generalized; basis of linear-response.
Entropy production (de Groot-Mazur 1962)
S de Groot-P Mazur 1962 'Non-Equilibrium Thermodynamics'; modern Esposito-Van den Broeck 2010 stochastic-thermo extension.
Active matter (Vicsek 1995)
Vicsek 1995 self-propelled particles; modern Cates-Tailleur 2015 motility-induced-phase-separation review.