Anomalous current fluctuations and mobility-driven clustering

TL;DR

This work analyzes steady-state current fluctuations in hardcore lattice gases with extended-range hopping on a ring, revealing a nonequilibrium condensation (clustering) transition for infinite-range hopping below a critical density $\rho_c$ where the mobility diverges as $\chi(\rho)\sim(\rho-\rho_c)^{-1}$ while the diffusion coefficient remains finite. A closure-based microscopic theory decomposes the current into diffusive and fluctuating parts, yielding closed equations for density-current correlations and exact expressions for time-integrated current fluctuations, linking them to the mobility via $\frac{1}{LT}\langle Q_{tot}^2\rangle_c=2\chi(\rho)$. At criticality, the variance scales as $\langle Q^2(L,T)\rangle_c=L^{4/3}{\cal W}_c(T/L^{2})$, with a subdiffusive $T^{2/3}$ growth at intermediate times and a system-size dependent linear growth at long times, and finite-size analysis shows $\chi(\rho_c,L)\sim L^{1/3}$. Finite-range hopping ($l_0=2$) yields finite, explicit $D_2(\rho)$ and $\chi_2(\rho)$, while IRH provides exact forms $D_{\infty}(\rho)=1/2$ and $\chi_{\infty}(\rho)$ with a pole at $\rho_c=1/\sqrt{2}$. The results offer a dynamical signature of nonequilibrium condensation and are supported by extensive Monte Carlo simulations.

Abstract

We study steady-state current fluctuations in hardcore lattice gases on a ring of $L$ sites, where $N$ particles perform symmetric, {\it extended-ranged} hopping. The hop length is a random variable depending on a length scale $l_0$ (hopping range) and the inter-particle gap. The systems have mass-conserving dynamics with global density $ρ= N/L$ fixed, but violate detailed balance. We consider two analytically tractable cases: (i) $l_0 = 2$ (finite-ranged) and (ii) $l_0 \to \infty$ (infinite-ranged); in the latter, the system undergoes a clustering or condensation transition below a critical density $ρ_c$. In the steady state, we compute, exactly within a closure scheme, the variance $\langle Q^2(T) \rangle_c = \langle Q^2(T) \rangle - \langle Q(T) \rangle^2$ of the cumulative (time-integrated) current $Q(T)$ across a bond $(i,i+1)$ over a time interval $[0, T]$. We show that for $l_0 \to \infty$, the scaled variance of the time-integrated bond current, or equivalently, the mobility diverges at $ρ_c$. That is, near criticality, the mobility $χ(ρ) = \lim_{L \to \infty} [\lim_{T \to \infty} L \langle Q^2(T, L) \rangle_c / 2T] \sim (ρ- ρ_c)^{-1}$ has a simple-pole singularity, thus providing a dynamical characterization of the condensation transition, previously observed in a related mass aggregation model by Majumdar et al.\ [{\it Phys.\ Rev.\ Lett.\ {\bf 81}, 3691 (1998)}]. At the critical point $ρ= ρ_c$, the variance has a scaling form $\langle Q^2(T, L) \rangle_c = L^γ {\cal W}(T/L^{z})$ with $γ= 4/3$ and the dynamical exponent $z = 2$. Thus, near criticality, the mobility {\it diverges} while the diffusion coefficient remains {\it finite}, {\it unlike} in equilibrium systems with short-ranged hopping, where diffusion coefficient usually {\it vanishes} and mobility remains finite.

Figures (6)

• Figure 1: Finite-ranged hopping with $l_0=2$. Left panel: We plot the steady-state variance of time-integrated bond-current $\langle Q^{2}(L, T) \rangle_c = \langle Q^{2}(L, T) \rangle$ as a function of time $T$, where the data are obtained from simulations (points) for $\rho=0.5,$$0.7$, and $0.9$ at $L=1000$. We then compare the simulation results with the analytical ones as in Eq. \ref{['ch4-bond-current-fluc']} (line). The variance of time-integrated bond current exhibits diffusive growth at early times, subdiffusive growth at the intermediate times, and finally diffusive (linear) growth at large times, as shown by the dotted lines, which are consistent with Eq. \ref{['ch4-cf_limit']}. Right panel: The scaled bond-current fluctuation $D_2\langle Q^{2}(L, T) \rangle/2\chi_2 L$ is plotted against the rescaled hydrodynamic time $y=D_2(\rho)T/L^{2}$ for the above-mentioned combination of densities and system size. We also compare the scaling collapse obtained from simulations with the analytic solution of the scaling function $\mathcal{W}\left(y\right)$ as in Eq. \ref{['ch4-cf_scaling_function']} (red line).
• Figure 2: Verification of Eq. \ref{['ch4-I_fluctuation1']} for $l_0=2$: We plot the numerically obtained scaled variance $\langle Q^{2}_{tot}(L,T)\rangle_c/LT = \langle Q^{2}_{tot}(L,T)\rangle / LT$ of the space-time-integrated current across the entire system as a function of $\rho$ and compare the numerical data with the theoretically calculated mobility $\chi_{2}(\rho)$ (line), as in Eq. \ref{['ch4-chi_l2_final']}.
• Figure 3: The steady-state variance $\langle Q^{2}(L, T) \rangle_c = \langle Q^{2}(L, T) \rangle$ of time-integrated bond current is plotted against the observation time $T$ for various hopping range $l_0=5$, $10$, and $20$ at density $\rho=0.5$ and system size $L=1000$. The dotted lines represent our analytical predictions for the growth law in three time regimes as shown in Eq. \ref{['ch4-cf_limit']}.
• Figure 4: Current fluctuations for infinite-ranged hopping. Left: We plot the steady-state variance $\langle Q^{2}(L, T) \rangle_c = \langle Q^{2}(L, T) \rangle$ of time-integrated bond-current as a function of time $T$, obtained from simulations (points) for $l_0 \rightarrow \infty$ in disordered phase ($\rho > \rho_c$) with densities $\rho=0.75,$$0.8$, $0.85$ and $0.9$. We also compare simulations with the theoretically obtained solution Eq. \ref{['ch4-bond-current-fluc']} (line). The variance $\langle Q^{2}(L, T) \rangle$ exhibits diffusive growth at early times, subdiffusive growth at the intermediate times, and a diffusive (linear) growth at large times, as shown by the dotted lines. Right: The scaled bond-current fluctuation $D_{\infty}\langle Q^{2}(L, T) \rangle/2\chi_{\infty} L$ is plotted against the rescaled time $y=D_{\infty}(\rho)T/L^{2}$ for the above-mentioned densities. We also compare the numerically obtained scaling collapse with the analytic scaling function $\mathcal{W}\left(y\right)$ as in Eq. \ref{['ch4-cf_scaling_function']} (red line).
• Figure 5: Verification of Eq. \ref{['ch4-I_fluctuation1']} for $l_0 \rightarrow \infty$ in disordered phase (i.e., $\rho > \rho_c$): We plot the numerically obtained scaled fluctuation of the space-time integrated current as a function of $\rho$ in the disordered phase and compare the numerical data with the theoretically calculated $\chi_{\infty}(\rho)$ (solid line), as derived in Eq. \ref{['ch4-chi-inf-final']}.