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…