[Paper Review] On the Violations of Local Equilibrium and Linear Response

Kenichiro Aoki, Dimitri Kusnezov|ArXiv.org|May 29, 2001

Thermal properties of materials1 references3 citations

TL;DR

This paper quantifies the breakdown of local equilibrium and linear response in classical lattice models under thermal gradients, showing both violations scale quadratically with ∇T/T and occur simultaneously without threshold. It reveals non-local corrections to the equation of state and demonstrates that coarse graining cannot restore local equilibrium, with observable deviations exhibiting non-trivial size dependence.

ABSTRACT

We study how local equilibrium, and linear response predictions of transport coefficients are violated as systems move far from equilibrium. This is done by studying heat flow in classical lattice models with and without bulk transport behavior, in 1--3 dimensions. We see that linear response and local equilibrium assumptions break down at the same rate. The equation of state is also found to develop non-local corrections in the steady state. We quantify the breakdown through the analysis of both microscopic and macroscopic observables, which are found to display non-trivial size dependence.

Motivation & Objective

• To investigate how local equilibrium and linear response break down in non-equilibrium steady states of classical lattice systems.
• To quantify the rate and functional form of deviations in transport coefficients and thermodynamic observables from equilibrium predictions.
• To examine the role of system size and dimensionality in the emergence of non-local effects in the equation of state.
• To assess the validity of coarse graining in recovering local equilibrium when these assumptions are violated.
• To establish a connection between momentum cumulants and the breakdown of local equilibrium.

Proposed method

• Numerical simulations of 1–3D classical lattice models, including the FPU-β and φ⁴ models, with thermostatted boundaries to impose thermal gradients.
• Use of Fourier's law and temperature profiles derived from power-law thermal conductivity κ(T) = cT⁻ᵞ to compute heat flux and non-equilibrium gradients.
• Expansion of observables (e.g., pressure, energy density, momentum cumulants) in powers of (∇T/T)² to quantify deviations from local equilibrium and linear response.
• Fitting of non-equilibrium spatial distributions of observables to analytical forms derived from the gradient expansion, including size-dependent coefficients.
• Analysis of momentum cumulants ⟨pⁿ⟩/Tⁿ/² as a robust, temperature-redefinition-invariant criterion for local equilibrium violation.
• Comparison of results across models with different transport behaviors (e.g., L-dependent κ in FPU-β vs. bulk limit in φ⁴) to test generality of findings.

Experimental results

Research questions

• RQ1 How do local equilibrium and linear response break down quantitatively as a function of thermal gradient ∇T/T?
• RQ2 To what extent do transport coefficients and thermodynamic observables like pressure and energy density develop non-local, size-dependent corrections in non-equilibrium steady states?
• RQ3 Is there a quantitative link between the breakdown of local equilibrium and the breakdown of linear response?
• RQ4 Can coarse graining over length scales ℓ restore local equilibrium when these assumptions are violated?
• RQ5 How do non-equilibrium definitions of temperature affect the observed violations, and are key observables invariant under redefinition?

Key findings

• The violations of local equilibrium and linear response occur at the same rate, with δLE ∼ δLR, and scale quadratically with (∇T/T)², showing no threshold for breakdown.
• The equation of state acquires non-local corrections, with pressure and energy density deviating as P(T,∇T,L) = Peq(T)[1 + CP(∇T/T)²], where CP exhibits non-trivial size dependence (e.g., CP^φ⁴ ∝ L⁰.⁹ at T=1).
• The coefficient CP for the φ⁴ model is 1.5(1.2)L⁰.⁹(2) at T=1, and for the FPU-β model it is 4.1(6)L⁰.³⁰(4) at T=8.8, indicating strong system-size dependence.
• Momentum cumulants ⟨pⁿ⟩/Tⁿ/² provide a robust, redefinition-invariant criterion for detecting local equilibrium violation, with deviations observed already at small gradients.
• Coarse graining over length scales ℓ > λ (mean free path) fails to restore local equilibrium, as the functional form of deviations is positive definite and non-local.
• The same quadratic dependence is observed in both the φ⁴ model and FPU-β model across d=1–3, even though the FPU-β model lacks a bulk thermal conductivity in d=1, indicating the breakdown is not tied to integrability.