Upper Bounded Current Fluctuations in One-Dimensional Driven Transport Systems

TL;DR

The paper addresses how interactions in one-dimensional driven transport affect current fluctuations and proposes an universal upper bound on the Fano factor, $\frac{2D}{J} \le \coth\left(\frac{A}{2}\right)$, with $A=\beta(\mu_{\rm L}-\mu_{\rm R})$. It derives the bound from a simple coarse-grained two-reservoir model where $J=W_+-W_-$, $D=(W_++W_-)/2$, and $A=\ln(W_+/W_-)$, proving $\frac{2D}{J}=\coth\left(\frac{A}{2}\right)$ exactly in that limit. The authors then provide a rigorous proof for quantum ballistic transport using the Landauer framework, showing $\frac{2D}{J}\le \coth\left(\frac{A}{2}\right)$ with the last term equaling $\coth\left(\frac{A}{2}\right)$, indicating the bound is transmission-function independent and tied to Pauli exclusion. They further validate the bound in two diffusive models—the symmetric simple exclusion process and charged-particle transport—where equality is approached in noninteracting or low-density limits and current fluctuations are suppressed by interactions. Overall, the work presents a physically transparent constraint linking fluctuations to driving force in 1D nonequilibrium systems, with potential as a diagnostic for interaction strength and non-Markovian transport behavior.

Abstract

We conjecture that the current fluctuations in one-dimensional driven transport systems obey an upper bound determined by the mean current and the driving force. This inequality originates from repulsive interactions between transporting particles, and the bound is approached both in near-equilibrium systems and in far-from-equilibrium systems with weak interactions. We first propose a coarse-grained model describing random particle exchanges between two reservoirs with constant rates, from which the upper bound emerges. We then rigorously prove the inequality in quantum ballistic transport systems. Finally, we demonstrate its validity in two specific diffusive systems: the exclusion process, for which the inequality can be proven, and charged-particle transport, for which numerical evidence supports the inequality.

Figures (2)

• Figure 1: Schematic representations of driven diffusive systems. Top panel: The symmetric simple exclusion process. The black (white) circles denote occupied (vacant) sites. Bottom panel: Charged particles transporting in a conductive channel.
• Figure 2: (Color online) Graphical representations of the bounded current fluctuations. Left panel: The behavior of $\chi\equiv 2DA/J$ as a function of $A$ with different particle densities (controlled by a parameter $x\equiv \gamma/\alpha$ with a small value for high density and a large value for low density) for SSEP. The parameter values are $\gamma=\beta=1$ and $L=10$. Moreover, the condition $y={\rm e}^Ax$ with $y\equiv\beta/\delta$ is also imposed. The black solid line is the affinity-dependent upper bound $A\coth(A/2)$, and the gray solid line is the lower bound $2$ from the TUR. The dashed lines are depicted from Eqs. (\ref{['eq_SSEP_J']})-(\ref{['eq_SSEP_A']}). Right panel: The comparison between numerical affinities (\ref{['eq_An']}) and theoretical affinities (\ref{['eq_At']}) for the transport of charged particles. It is set that $\bar{N}_{\rm L}=\bar{N}_{\rm R}=N_-=n_-\Omega$, and takes different values in three cases, as shown in the legend. The asterisks are numerical points with dashed lines joining them. The dot-dash line indicates the equality between both kinds of affinities. The parameter values used in simulation are $\beta=e=1.0$, $D=\epsilon=0.01$, $\Omega=10000$, $\Delta x=0.1$, $L=10$. The numerical affinities are computed with the mean current $J$ and its diffusivity $D$ evaluated over time interval $[0,\,5000]$ with $20000$ data.