Stochastic stability and the fluctuation-dissipation-ratio/overlap-distribution identity in finite-dimensional aging systems

Franz, Mézard, Parisi and Peliti proposed in 1998 that, for a finite-dimensional system with short-range interactions in the aging regime, the fluctuation-dissipation ratio (FDR) $X(q)$ — the slope of the parametric plot of the integrated response versus the autocorrelation — coincides with the cumulative equilibrium overlap distribution, $X(q) = \int_0^q P(q')dq'$, making the equilibrium order-parameter function of a glass (in particular a spin glass) measurable from thermoremanent-magnetization and noise measurements. The derivation requires two unproven properties: stochastic stability (continuity of the overlap distribution under infinitesimal random perturbations of the Hamiltonian) and commutation of the perturbation-removal limit $\epsilon \to 0$ with the long-time limit. The authors themselves left open which systems are stochastically stable (mean-field spin glasses and Ising ferromagnets in $d \ge 2$ were asserted to belong, but whether short-range spin glasses do was unknown), under what conditions the limits commute (verified only for the Sherrington-Kirkpatrick model, violated in $p$-spin spherical models), the time scales on which the added thermodynamic perturbations equilibrate, and how to realize the measurement experimentally. Nearly three decades later the identity remains a numerically supported conjecture: rigorous stochastic-stability results exist only for mean-field models or bond-overlap analogues, the most accurate three-dimensional Edwards-Anderson simulations confirm the identity only within error bars and finite-time corrections, and no experiment has extracted a full equilibrium overlap distribution from aging data. The problem is to establish the identity — or its exact boundary of validity — for finite-dimensional short-range systems, especially the Edwards-Anderson spin glass, and to make it experimentally testable.

Background

Glassy systems — spin glasses, structural glasses, polymer glasses, dense colloids — do not reach thermal equilibrium at low temperature on any accessible time scale. After a quench, one-time observables such as the energy density settle close to stationary values, while two-time quantities keep depending on the time $t'$ elapsed since the quench: the system ages. In equilibrium, the fluctuation-dissipation theorem (FDT) fixes the linear response $R(t,t')$ of a spin system to a small perturbing field in terms of its autocorrelation $C(t,t')$. Out of equilibrium this link is broken, but Cugliandolo and Kurchan discovered, in exactly solvable mean-field spin-glass models, that the aging regime still obeys a generalized relation: the integrated response $\chi(t,t')$ becomes a function of the correlation alone, and the slope of the parametric $\chi$–$C$ curve defines the fluctuation-dissipation ratio (FDR) $X(q)$, where $q$ denotes the value of $C$. In the short-time stationary regime $X=1$ (ordinary FDT), while for correlations below the Edwards-Anderson value $q_{EA}$ (the long-time equilibrium autocorrelation within one pure thermodynamic state) the ratio can drop below 1, so that $T/X$ behaves as an effective temperature.

For an equilibrium spin glass — say the Edwards-Anderson model, Ising spins with random nearest-neighbor couplings on a $d$-dimensional lattice — the central order parameter is the overlap distribution $P(q)$: the probability distribution of $q = N^{-1}\sum_x S_x S'_x$ between two configurations $S, S'$ drawn independently from the Gibbs measure of the same sample. In the replica-symmetry-breaking (RSB) picture inherited from mean-field theory, $P(q)$ is nontrivial; in the droplet picture of finite-dimensional spin glasses only two symmetry-related pure states exist and the symmetry-reduced overlap distribution is trivial. Numerical simulations support a nontrivial $P(q)$ in three and four dimensions, but $P(q)$ seemed experimentally inaccessible, since measuring it requires comparing two replicas of the same sample at equilibrium.

Franz, Mézard, Parisi and Peliti argued in 1998 that $P(q)$ is nevertheless measurable, because it coincides with a dynamical quantity. Their construction perturbs the Hamiltonian by a weak random thermodynamic term $\epsilon H_p$ (for example $H_2 = \sum_x h_x S_x S_{T(x)}$, coupling each spin to a far-away spin through independent Gaussian variables $h_x$; extensive, but local in a duplicated system), and shows, for short-range systems, that (a) the long-time dynamical expectation $\langle H_p(t)\rangle$ reaches its equilibrium value (via a standard free-energy bubble argument, assuming local equilibration of the free-energy density), and (b) the moments of the perturbed overlap distribution $P_\epsilon(q)$ coincide with those of $dX_\epsilon/dq$, where $X_\epsilon$ is the FDR of the perturbed system. Consequently, if the system is stochastically stable — meaning the overlap distribution is continuous as the random perturbation is removed, $\lim_{\epsilon \to 0} P_\epsilon = \tilde P = \hat P$, with $\hat P$ the symmetry-reduced overlap distribution of the unperturbed system — and if the $\epsilon \to 0$ limit commutes with the long-time limit so that $\tilde X(q) = X(q)$, then the aging FDR of the unperturbed system satisfies

$$X(q) = \int_0^q P(q')dq'.$$

Mean-field spin glasses and Ising ferromagnets in $d \ge 2$ were asserted to be stochastically stable, and the commutation was known to hold for the Sherrington-Kirkpatrick fully connected model but to fail in $p$-spin spherical models (a solvable one-step RSB model whose aging dynamics is dominated by infinitely long-lived metastable threshold states rather than equilibrium states). If both properties hold, the equilibrium order-parameter function of a spin glass becomes measurable in an aging experiment from the thermoremanent magnetization (the integrated response to a small cut-off field) together with the spontaneous magnetization noise (the correlation); moreover, the shape of $X(q)$ would directly discriminate slow domain growth, for which $X(q) = \theta(q - q_{EA})$, from RSB-like behavior with $X(q) \ne 0$ for $q < q_{EA}$.

What the authors could not establish, and explicitly left open: which systems are stochastically stable (the class was not characterized, and it was not known whether short-range spin glasses — the most interesting case — belong to it); under what conditions the two limits commute; on what time scales $\langle H_p(t)\rangle$ equilibrates (noted to be possibly very long, and hopeless on laboratory time scales in structural glasses); and how the required measurements can be realized and controlled experimentally.

Problem Statement

For finite-dimensional systems with short-range interactions — with the $d$-dimensional Edwards-Anderson Ising spin glass below its glassy transition as the central case — establish, or determine the exact boundary of validity of, the identity

$$X(q) = \int_0^q P(q')dq'$$

between the fluctuation-dissipation ratio of the aging, unperturbed system and the cumulative equilibrium overlap distribution, together with the conditions that enter it. The identity breaks into the following linked sub-questions:

1. (Statics.) Prove or refute stochastic stability — continuity of the symmetry-reduced overlap distribution under infinitesimal random perturbations of the Hamiltonian, $\lim_{\epsilon \to 0} P_\epsilon = \hat P$ — for the Edwards-Anderson model in $d = 3$, and, as a general goal, characterize the class of finite-dimensional short-range systems possessing this property. Known rigorous results cover only mean-field models in terms of the spin overlap, and the Edwards-Anderson model in terms of the bond overlap (an overlap between the bond variables of two replicas of the same sample, rather than between their spins) in beta-average (the identities are proven only after an additional average over the inverse temperature, not at fixed temperature).

2. (Dynamics.) Determine under what conditions on the perturbations and the dynamics the $\epsilon \to 0$ limit is uniform in time, so that the $\epsilon \to 0$ and $t \to \infty$ limits commute and the perturbed FDR converges to the unperturbed one; in particular, prove or disprove this commutation for the three-dimensional Edwards-Anderson model with local dynamics (e.g. Glauber single-spin-flip dynamics, or Langevin dynamics for a soft-spin version of the model), where it is numerically supported. Known counterexample classes are $p$-spin spherical models and coarsening/domain-growth systems.

3. (Time scales.) Quantify the time scales on which the dynamical expectation values of the thermodynamic perturbations, $\langle H_p(t) \rangle$, and the energy density reach their equilibrium values — their dependence on temperature, system size, and the age of the system — determining for which systems and protocols the required equilibration holds within experimental or simulation time scales.

4. (Experiment.) Identify and validate experimental protocols — simultaneous measurements of the aging response (thermoremanent magnetization or an analogous susceptibility) and of the spontaneous-fluctuation correlation (noise) in the same aging sample — that allow $X(q)$ to be extracted over the full aging range $q < q_{EA}$ with controlled separation of stationary and aging contributions, and compared quantitatively with the equilibrium overlap distribution, in spin glasses or comparable glass formers.

The identity to be established, or whose exact boundary of validity is to be determined, is the asymptotic identity for finite-dimensional short-range systems, not its mean-field realization; each sub-question above stands as a research target in its own right.

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