Rigorous weak-coupling expansion and zero-noise limit of the Green-Kubo conductivity for weakly coupled systems with energy-conserving noise

The Green-Kubo (GK) conductivity $\kappa(\varepsilon,\varsigma)$ of an infinite chain of weakly coupled Hamiltonian cells (interaction strength $\varepsilon$), each cell's dynamics perturbed by an energy-conserving noise of strength $\varsigma$, exists and is finite for every $\varsigma>0$. Bernardin, Huveneers, Lebowitz, Liverani and Olla (2015) formally expanded $\kappa(\varepsilon,\varsigma)=\varepsilon^2\sum_{n\ge2}\varepsilon^{n-2}\kappa_n(\varsigma)$ and identified the lowest coefficient $\kappa_2(\varsigma)$ as the diffusivity of a limiting Ginzburg-Landau energy dynamics (an autonomous diffusion of the cell energies); but the identification $\lim_{\varepsilon\to0}\varepsilon^{-2}\kappa(\varepsilon,\varsigma)=\kappa_2(\varsigma)$ is conjectured in all cases, the coefficients $\kappa_n(\varsigma)$ for $n\ge3$ are not known to be well defined, and the zero-noise behavior of $\kappa_2(\varsigma)$ as $\varsigma\to0$ is undetermined — including whether it vanishes or stays positive for the rotor chain, where resonances make the energy-exchange coefficient blow up like $1/\varsigma$. The open problem is to make this program rigorous: prove the $\varepsilon\to0$ identification and the well-definedness of the full expansion, and determine the existence and value of $\lim_{\varsigma\to0}\kappa_2(\varsigma)$, conjecturally the purely Hamiltonian weak-coupling conductivity. Work published in 2026 by two of the source authors still describes the Green-Kubo relation for such models as obtainable 'only formally'.

Background

Deriving macroscopic heat transport from microscopic dynamics is a central open problem of nonequilibrium statistical mechanics. The Green-Kubo (GK) formula expresses the thermal conductivity of an infinite system as a space-time integral of the equilibrium autocorrelation of the energy current; it is the standard microscopic definition of conductivity, but proving that the resulting quantity is finite and well behaved for genuinely interacting Hamiltonian systems is notoriously difficult.

Bernardin, Huveneers, Lebowitz, Liverani and Olla (2015) study an infinite chain of identical finite Hamiltonian systems ('cells') with energy $H=p^2/2+W(q)$, where the position $q$ takes values in a $d$-dimensional manifold, coupled to nearest neighbors by a smooth potential $\varepsilon V$ with $\varepsilon\ll1$. Within each cell the Hamiltonian flow is perturbed by a stochastic noise of strength $\varsigma$ acting on the momentum and conserving each cell's kinetic energy — hence the total energy. With this noise the GK conductivity $\kappa(\varepsilon,\varsigma)$ exists and is finite for every $\varsigma>0$; the noise is a technical regularization whose eventual removal is part of the program's goal. The equilibrium reference state is the Gibbs measure at inverse temperature $\beta$.

Expanding the resolvent that appears in the GK expression around $\varepsilon=0$ — the generator of the full dynamics is $L_\varepsilon=L_0+\varepsilon G$, with $L_0$ the cell dynamics and $G$ the interaction generator — the authors obtain the formal series $\kappa(\varepsilon,\varsigma)=\varepsilon^2\sum_{n\ge2}\varepsilon^{n-2}\kappa_n(\varsigma)$. In the unpinned case ($W=0$) the lowest coefficient is a mathematically well-defined object: $\beta^{-2}\kappa_2(\varsigma)=\langle\gamma_{0,1}^2\rangle_\beta-\langle\langle\alpha_{0,1},(-L)^{-1}\alpha_{0,1}\rangle\rangle_\beta$, where $\langle\cdot\rangle_\beta$ denotes the Gibbs average, $\gamma_{0,1}$ and $\alpha_{0,1}$ are the symmetric and antisymmetric parts of the intercell energy-exchange coefficients, and $L$ is the generator of an autonomous diffusion of the cell energies of Ginzburg-Landau type. This $\kappa_2(\varsigma)$ coincides with the conductivity obtained in the weak-coupling (van Hove) limit — in which the interacting chain is replaced, on the diffusive time scale $\varepsilon^{-2}t$, by an energy diffusion — wherever that limit has been established rigorously: for a noisy anharmonic crystal (Liverani and Olla 2012) and for cells given by fast-slow Hamiltonian systems — systems with two well-separated time scales — such as geodesic flows on negatively curved manifolds, the Hamiltonian flow of free particle motion on a negatively curved manifold and the canonical example of a strongly chaotic dynamics (Dolgopyat and Liverani 2011). The source also formally identifies $\kappa_2(\varsigma)$ with the conductivity of the finite open chain placed between Langevin reservoirs at temperatures differing by $\delta T$, in the limit $\delta T\to0$, $N\to\infty$, $\varepsilon\to0$.

The zero-noise behavior as $\varsigma\to0$ is the subtle part. Two-sided bounds $\langle\gamma_{0,1}^{-2}\rangle_\beta^{-1}\le\beta^{-2}\kappa_2(\varsigma)\le\langle\gamma_{0,1}^2\rangle_\beta$ are proved, and three one-dimensional examples are analyzed: (i) a chain of pinned strongly anharmonic oscillators (on-site potential $W(q)\sim q^r$ with $r>2$), where $\limsup_{\varsigma\to0}\kappa_2(\varsigma)<\infty$ but no positive lower bound is known (for a pinning potential $W(q)=q^2+a\cos q$ the upper-bound quantity even diverges logarithmically, an effect expected to be compensated by the antisymmetric term); (ii) the rotor chain (each cell is a free rotor: an angle variable on a circle with kinetic energy only), where $\kappa_2(\varsigma)>0$ for every $\varsigma>0$ but the authors state they have 'unfortunately not been able to decide' whether $\kappa_2(\varsigma)\to0$ as $\varsigma\to0$ — the exchange coefficient $\gamma^2$ blows up like $1/\varsigma$ near resonances $|e_x-e_{x+1}|\le\varsigma$ between neighboring cell energies — with the deterministic localization results of De Roeck and Huveneers suggesting vanishing; and (iii) the disordered harmonic chain (harmonic oscillators with random pinning frequencies), where $\lim_{\varsigma\to0}\kappa_2(\varsigma)=0$ is proven. The authors conjecture that in general $\lim_{\varsigma\to0}\kappa_2(\varsigma)$ exists and is closely related to the weak-coupling conductivity of the purely Hamiltonian ($\varsigma=0$) system — strictly positive in dimension $d\ge3$ for geodesic-flow cells — and note that a rigorous treatment 'would require the extension to random perturbations of the theory developed for deterministic perturbations', that is, the spectral perturbation theory of hyperbolic (Anosov, strongly chaotic) flows developed by Butterley and Liverani.

What remains formal in the source: the identification $\lim_{\varepsilon\to0}\varepsilon^{-2}\kappa(\varepsilon,\varsigma)=\kappa_2(\varsigma)$ is believed but not proven in every case (in the unpinned case only the well-definedness of the coefficient and the replacement of the $\varepsilon$-dependent Gibbs ensemble by the $\varepsilon=0$ ensemble inside its formula are proved); the higher-order coefficients are conjectured to be well defined; the smoothness of $\gamma^2$ and the closure of the limiting generator $L$ are proved only in special examples; and the existence and uniqueness of the nonequilibrium stationary measure of the open system — needed for the reservoir identification — 'still remains an open problem for most of the dynamics that appear in this work'.

Problem Statement

Make rigorous the formal weak-coupling expansion and the zero-noise analysis of the Green-Kubo conductivity for the class of weakly coupled systems with energy-conserving noise defined above. For self-containedness, the setting is: an infinite chain of identical finite Hamiltonian cells with energy $H=p^2/2+W(q)$, coupled to nearest neighbors by a smooth potential $\varepsilon V$ with $\varepsilon\ll1$, each cell's dynamics perturbed by a stochastic noise of strength $\varsigma>0$ acting on the momentum and conserving each cell's kinetic energy (hence the total energy); $\kappa(\varepsilon,\varsigma)$ is the Green-Kubo conductivity of this dynamics, $\kappa(\varepsilon,\varsigma)=\varepsilon^2\sum_{n\ge2}\varepsilon^{n-2}\kappa_n(\varsigma)$ its formal expansion in powers of $\varepsilon$, and $\kappa_2(\varsigma)$ the lowest coefficient, identified in the source with the diffusivity of a limiting Ginzburg-Landau energy dynamics — in the unpinned case $W=0$ given by $\beta^{-2}\kappa_2(\varsigma)=\langle\gamma_{0,1}^2\rangle_\beta-\langle\langle\alpha_{0,1},(-L)^{-1}\alpha_{0,1}\rangle\rangle_\beta$, where $\langle\cdot\rangle_\beta$ is the Gibbs average at inverse temperature $\beta$, $\gamma_{0,1}$ and $\alpha_{0,1}$ are the symmetric and antisymmetric parts of the intercell energy-exchange coefficient, and $L$ is the generator of the limiting energy diffusion. Concretely, the open problem consists of the following linked components.

(1) $\varepsilon$-identification. Prove that, for fixed $\varsigma>0$, $\lim_{\varepsilon\to0}\varepsilon^{-2}\kappa(\varepsilon,\varsigma)$ exists and equals $\kappa_2(\varsigma)$, the coefficient given by the Ginzburg-Landau diffusivity formula of the source. This is conjectured there in every case, including the unpinned case $W=0$ where only the coefficient's well-definedness is known. In particular the identification should be established for the three one-dimensional classes treated in the source: the pinned strongly anharmonic chain, the rotor chain, and the disordered harmonic chain. The required target level of this component is the general statement for the whole class of cells defined above; the three one-dimensional classes are its hard special cases, which must be settled as well, so establishing the identification only for the enumerated classes (or any other strict subset of the class) is partial progress rather than a complete resolution of this component.

(2) Higher-order terms. Prove that the coefficients $\kappa_n(\varsigma)$, $n\ge3$, of the formal expansion are well defined — the current-correlation expectations entering them exist — or establish the expansion to all orders as an asymptotic series. The source records that further progress on these terms requires inverting the cell generator $L_0$ (the generator of the uncoupled cell dynamics in $L_\varepsilon=L_0+\varepsilon G$) on the class of functions with zero microcanonical average — the average with respect to the measure concentrated on a fixed value of the cell energy — beyond the restricted class treated there.

(3) Zero-noise limit. Prove that $\lim_{\varsigma\to0}\kappa_2(\varsigma)$ exists and identify its value — conjecturally closely related to the weak-coupling conductivity of the purely Hamiltonian ($\varsigma=0$) system, strictly positive in dimension $d\ge3$ for cells given by geodesic flows on negatively curved manifolds. The source notes that a rigorous treatment of this limit would require extending the spectral perturbation theory of hyperbolic and fast-slow systems to the random perturbations considered here. Concrete instances: decide whether $\kappa_2(\varsigma)\to0$ or remains bounded away from zero for the rotor chain (the resonance mechanism $\gamma^2\sim1/\varsigma$ versus the deterministic localization expectation), and whether $\lim_{\varsigma\to0}\kappa_2(\varsigma)$ is strictly positive for pinned strongly anharmonic chains, where only a finite upper bound is known. The required target level of this component is again the general statement for the whole class of cells defined above — existence and identification of the limit — while the two instances just listed are its concrete special cases, which must be settled as well.

A complete solution consists of components (1)–(3). The enabling well-posedness issues recorded in the source for the same program are not separate deliverables: where a proof of (1), (2) or (3) invokes one of them — the regularity of the energy-exchange coefficient $\gamma^2(e_0,e_1)$, the closure/self-adjointness of the limiting generator $L$ beyond the special examples, or the existence, uniqueness and smoothness of the nonequilibrium stationary measure for the open systems underlying the formal identification of $\kappa_2(\varsigma)$ with the reservoir conductivity — the solution must name that issue explicitly and establish it in the generality it uses, by proving it or, where it is itself a conjecture, disproving it. Rigorous results that establish only a subset of the components (1)–(3) (for example, only the rotor-chain decision, or only the $\varepsilon$-identification for one class of cells) constitute partial progress toward the same objective.

References

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