Anomalous diffusion in convergence to effective ergodicity

TL;DR

The paper tackles how a functional of a many-body trajectory converges to ergodicity in a strongly correlated system, extending diffusion analysis beyond particle paths. It introduces functional-diffusion based on the Thirumalai-Mountain metric applied to the total magnetization of a finite Ising chain, analyzed under Metropolis and Glauber dynamics, and characterizes two power-law forms: $K_G(t)\to C t^{\alpha}$ and $P(\Gamma_G(t))\to \Gamma_G(t)^{-\alpha}$, with benchmarking against the exact $M_E$ and finite-size scaling to test universality. The main contributions are the first quantitative demonstration of anomalous (sub- and super-diffusive) convergence to ergodicity in a lattice model, supported by robust uncertainty quantification and universal behavior for $N=1024$ and $N=1536$. This framework provides a rigorous, observable-centric approach to ergodicity in strongly correlated systems and offers a versatile test-bed for exploring functional diffusion beyond conventional particle trajectories.

Abstract

The nature of diffusion is usually studied for particles or dynamics generating trajectories over time. Similar in principle, these studies can be executed in tracking how a given function of observable properties evolve over time akin to a system's particle motion, so-called {\it functional-diffusion}. This is not the same as systems' own trajectories but can be considered as a meta-trajectory. Following this idea, we measure how an approach to ergodicity evolves over time for the observed magnetization of a full Ising model with an external field. We compute the functional's diffusive behavior depending on a range of temperatures via Metropolis and Glauber single-spin-flip dynamics. System's ensemble-averaged dynamics are computed using expressions from the exact solution. Power-laws on the approach to ergodicity provide the classification of anomalies in the {\it functional-diffusion}, demonstrating non-linear anomalous behavior over different temperature and field ranges.

Figures (5)

• Figure 1: Diagnostic plots of $40$ different runs for $N=1024$ with Glauber dynamics for the evolution of $K(t)$ log-log regression (a) A field at $H=1.0$ and inverse temperature $\beta=1.16$ (b) A field at $H=1.4$ and inverse temperature $\beta=1.0$. We identify optimal time starting fit with minimal Kolmogorov-Simirnov statistic on a grid-search.
• Figure 2: Uncertainty quantified two power-laws's $\alpha$ exponents over temperature ranges (a) the time-evolution of $K(t)$, via log-log regressions. (b) the distribution of $\Gamma(t)$ via analytical expressions from Newman et. al.. Here we plot absolute value as $\alpha$. In the fit, functional form is negative $\alpha$.
• Figure 3: Uncertainty quantified two power-laws's $\alpha$ exponents over field ranges at $\beta=1.0$ (a) the time-evolution of $K(t)$, via log-log regressions. (b) the distribution of $\Gamma(t)$ via analytical expressions from Newman et. al.. Here we plot absolute value as $\alpha$. In the fit, functional form is negative $\alpha$.
• Figure 4: Uncertainty quantified power-laws of the time-evolution of $K(t)$, via log-log regressions, diffusion coefficients (a) Over temperature ranges. (b) Over field ranges.
• Figure 5: Uncertainty quantified autocorrelation times over $200$ runs. (a) A diagnostic plot for identification of autocorrelation time. (b) Autocorrelation times over temperature ranges and three different sizes.