Anomalous current fluctuations from Euler hydrodynamics

TL;DR

The paper addresses anomalous current fluctuations in a family of stochastic charged cellular automata by formulating ballistic macroscopic fluctuation theory (BMFT) and applying it to both large and typical fluctuations. It demonstrates that large fluctuations are fully fixed by Euler-propagated initial fluctuations and derives an exact BMFT SCGF F(λ) = (1/2) log[1 + Δ^2(μ_b + μ_b^{-1} − 2)] with Δ^2 = ρ(1 − ρ) and μ_b = cosh λ + |b| sinh|λ|, aligning with microscopic results. For deterministic single-file dynamics (Γ = 0), typical fluctuations are likewise determined by Euler propagation of initial fluctuations, while for stochastic dynamics an additional contribution arises, leading to a generalized typical-distribution form P_typ^{[Γ]}(j) that fits simulations across Γ ≥ 0. The results reveal a hydrodynamic origin of anomalous fluctuations and lay groundwork for extending the framework to other integrable-looking systems, such as Dirac fluids and Heisenberg spin chains.

Abstract

We consider the hydrodynamic origin of anomalous current fluctuations in a family of stochastic charged cellular automata. Using ballistic macroscopic fluctuation theory, we study both typical and large fluctuations of the charge current and reproduce microscopic results which are available for the deterministic single-file limit of the models. Our results indicate that in general initial fluctuations propagated by Euler equations fully characterize both scales of anomalous fluctuations. For stochastic dynamics, we find an additional contribution to typical fluctuations and conjecture the functional form of the typical probability distribution, which we confirm by numerical simulations.

Figures (3)

• Figure 1: (left) Stochastic particle scattering in the two-particle sector of the local two-body map $\Phi$. Particles with positive/negative charge (red/blue squares) either cross (top left) with probability $\Gamma$ or are elastically reflected (bottom left) with probability $\overline \Gamma$. Particle worldlines (colored lines) shown for clarity. (right) Many-body dynamics of charged particles and vacancies (black circles) in discrete space-time. Particles move along diagonals except when they encounter another particle when they scatter stochastically.
• Figure 2: Finite-time typical distributions of the integrated charge current $t^{1/4} \mathcal{P}^{[\Gamma]}(Jt^{-1/4}|t)$ (colored lines) in linear and logarithmic scales at different $\Gamma$ compared against distributions $\mathcal{P}^{[\Gamma]}_{\rm typ}$\ref{['generalized_dist']} (black lines) with $\omega$ and $\sigma$ as fitting parameters. Simulation parameters: $\rho=1/2$, $b=0$, $L = 2^{20}$, $t_{\rm max}=2^{16}$, $5\times 10^3$ samples.
• Figure 3: Initial fluctuations that influence the charge current fluctuations. (a): when $b\neq0$, the particle fluctuations $\delta\varrho_\pm(x,0)$ that cross $x=0$ between $t=0$ and $t=\tau$ give rise to the current fluctuations. (b): when $b=0$ the charge fluctuations $\delta\varrho_c(x,0)$ that cross $x=0$ between $t=0$ and $t=\tau$ contribute to the current fluctuations. Note that while the trajectory is now nonlinear, the slope cannot exceed $1$ because $|{\rm d} X_d(t)/{\rm d} t|<1$. Here we assumed $X_d(t)>0$ for $t\in[0,\tau]$.