Correlation thresholds in the steady states of particle systems and spin glasses

TL;DR

This work quantitatively connects Boltzmann-like order in steady states to a local-dynamical property called rattling by introducing a local-global correlation between the steady-state potential and exit rates. It develops a general Markov-jump framework and derives explicit thresholds for the correlation in two models: a two-particle ring that maps to a harmonic-trap Glauber dynamics, and high-dimensional SK spin-glass dynamics with Glauber transitions. The main finding is a parameter-driven threshold in the correlation ρ, arising from transitions in how the global and local parts of the effective potential relate, with ρ switching sign as dynamics parameters cross critical values. These results illuminate when nonequilibrium steady states admit Boltzmann-like descriptions and offer practical methods to estimate ρ via easily accessible exit-rate statistics.

Abstract

A growing body of theoretical and empirical evidence shows that the global steady-state distributions of many equilibrium and nonequilibrium systems approximately satisfy an analogue of the Boltzmann distribution, with a local dynamical property of states playing the role of energy. The correlation between the effective potential of the steady-state distribution and the logarithm of the exit rates determines the quality of this approximation. We demonstrate and explain this phenomenon in a simple one-dimensional particle system and in random dynamics of the Sherrington-Kirkpatrick spin glass by providing the first explicit estimates of this correlation. We find that, as parameters of the dynamics vary, each system exhibits a threshold above and below which the correlation dramatically differs. We explain how these thresholds arise from underlying transitions in the relationship between the local and global "parts" of the effective potential.

Figures (5)

• Figure 1: Local--global correlations. (a) The Boltzmann distribution entails perfect correlation $\rho$ between the effective potential of an equilibrium steady state and the energy of a uniformly random state. (Each mark corresponds to one state.) (b) For many equilibrium and nonequilibrium steady states, a different local property of a state, called rattling, is highly correlated with the effective potential.
• Figure 2: Contour plot of \ref{['eq:corr']}. Contours and shading indicate the value of $\rho$ for a given pair $(r,\widehat{\rho}\,) \in [0,\infty) \times [-1,1]$.
• Figure 3: Local--global correlations of particles on a ring. (a) The effective potential scattered against the log exit rate of each state for the dynamics in \ref{['eq:asep_rates']} with $L = 100$ and $\alpha = 1.1$. (b) The correlation as a function of $L$ and $\alpha$ for $L = 25k$, $k \in \{2,\dots,6\}$. Dashed lines indicate $\rho = \pm \sqrt{15}/4$.
• Figure 4: Comparison of $\log q (x)$ (blue points) and $\log \widetilde{q} (x)$ (black points), scaled by $1/L \log \alpha$, for (a) $\alpha = 1.1$ and (b) $\alpha = 0.9$. The blue scatters correspond to $L \in \{25,50,75\}$ (darker blue indicates higher $L$); the black scatter has $L = 100$. Note the points in (b) indicated by red arrows, which all blue scatters share.
• Figure 5: Local--global correlations of SK model dynamics. The effective potential scattered against the log exit rate of each state for (a) $\lambda = 0.50$ and (b) $\lambda = 0.05$. (c) The correlation as a function of $\lambda$ for $N = 4$ (green curve) and $N=10$ (blue curve). The curves show the mean correlation over $50$ and $10$ independent trials, respectively, with corresponding error bars of $\pm 1$ standard deviation. Dashed lines linearly interpolate the formula in \ref{['eq:sk']}. (d) Plot of $(r,\widehat{\rho})$ pairs for $\lambda \in [0,1]$ with $N=10$ fixed. Each mark represents the average over $10$ independent trials of $\widehat{\rho}$ and $r$ for a particular $\lambda$. Shading and contours indicate the corresponding value of $\rho$. The arrow indicates the direction of increasing $\lambda$. In all cases, $\beta = 1$.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Lemma A.1
• proof
• proof : Proof of \ref{['thm:main result asep']}
• Lemma C.1
• proof
• Lemma C.2
• proof
• Lemma C.3