Exact anomalous current fluctuations in a deterministic interacting model

Abstract

We analytically compute the full counting statistics of charge transfer in a classical automaton of interacting charged particles. Deriving a closed-form expression for the moment generating function with respect to a stationary equilibrium state, we employ asymptotic analysis to infer the structure of charge current fluctuations for a continuous range of timescales. The solution exhibits several unorthodox features. Most prominently, on the timescale of typical fluctuations the probability distribution of the integrated charge current in a stationary ensemble without bias is distinctly non-Gaussian despite diffusive behavior of dynamical charge susceptibility. While inducing a charge imbalance is enough to recover Gaussian fluctuations, we find that higher cumulants grow indefinitely in time with different exponents, implying singular scaled cumulants. We associate this phenomenon with the lack of a regularity condition on moment generating functions and the onset of a dynamical critical point. In effect, the scaled cumulant generating function does not, irrespectively of charge bias, represent a faithful generating function of the scaled cumulants, yet the associated Legendre dual yields the correct large-deviation rate function. Our findings hint at novel types of dynamical universality classes in deterministic many-body systems.

Figures (8)

• Figure 1: Coordinate frame (time vertical, space horizontal) of a deterministic charged hard-core lattice gas (red: $+$ particles, blue: $-$ particles, while thin black lines indicate vacancies). Example of a lightcone (pyramid) section of a typical trajectory, for which initial data on a saw of $4t$ subsequent links uniquely determine the transport through the mid-point (dashed line) for all times times from $0$ to $2t$.
• Figure 2: Rescaled current distribution for unbiased $b=0$, half-filled $\rho=0.5$ charged hard-core lattice gas in normal/log scale (lower/upper data). Colored dashed lines show convergence of exact distributions $\mathcal{P}^{[0]}_{1/4}(\mathcal{J}|t)$ to the asymptotic form (\ref{['eqn:typical_nonGaussian']}) (solid black line). Estimated current distribution (dots), agrees with exact solution within statistical errors (${\rm N}_{\rm sample} = 10^9$).
• Figure 3: Coordinate frame (time vertical, space horizontal) of a deterministic charged hardcore lattice gas (red: $+$ particles, blue: $-$ particles, while thin black lines indicate vacancies). Example of a pyramid section of a typical trajectory, for which initial data on a saw of $4t$ subsequent links uniquely determine the transport through the mid-point (dashed line) for all times from $0$ to $2t$.
• Figure 4: Distinct particle worldlines within a typical many-body trajectory, depicted with two different colors. Lines crossing the origin are thickened. In this specific example $\Lambda_- = \{-9,-8,-5,-3,-1\}$, $\Lambda_+=\emptyset$, i.e. there are five worldlines which in total duration $2t=26$ cross the boundary -- indicated by the dashed vertical line -- from left to right sublattice.
• Figure 5: Ratio $\mathcal{R}^{[0]}_n$ (\ref{['ratio_def']}) (circles/stars/triangles) of numerically estimated cumulants $\tilde{c}_n(t)$ divided by leading order asymptotics $c_{n|0}^{[0]} t^{n/4}$ as a function of time $t$ for $\rho=0.5$ ($\varDelta=0.25$) and $b=0$. Dotted lines show the asymptotic results including an additional sub-leading order, $1 + t^{-1/2} c_{n|1}^{[0]}/{c_{n|0}^{[0]}}$. Number of samples for each time ${\rm N}_{\rm sample} = 10^{8}$.