Time-dependent properties of run-and-tumble particles. II. Current fluctuations

We investigate steady-state current fluctuations in two models of hardcore run-and-tumble particles (RTPs) on a periodic one-dimensional lattice of $L$ sites, for arbitrary tumbling rate $\gamma ={\tau }_p^{-1}$ and density $\rho$; model I consists of standard hardcore RTPs, while model II is an analytically tractable variant of model I, called a long-ranged lattice gas (LLG). We show that, in the limit of $L$ large, the fluctuation of cumulative current $Q_i(T,L)$ across the $i\mathrm{th}$ bond in a time interval $T\gg 1/D$ grows first subdiffusively and then diffusively (linearly) with $T$: $\langle Q_i^2\rangle \sim T^{\alpha }$ with $\alpha =1/2$ for $1/D\ll T\ll L^2/D$ and $\alpha =1$ for $T\gg L^2/D$, where $D(\rho ,\gamma )$ is the collective- or bulk-diffusion coefficient; at small times $T\ll 1/D$, exponent $\alpha$ depends on the details. Remarkably, regardless of the model details, the scaled bond-current fluctuations $D\langle Q_i^2(T,L)\rangle /2\chi L\equiv \mathcal{W}\left(y\right)$ as a function of scaled variable $y=DT/L^2$ collapse onto a universal scaling curve $\mathcal{W}\left(y\right)$, where $\chi (\rho ,\gamma )$ is the collective particle mobility. In the limit of small density and tumbling rate, $\rho ,\gamma \to 0$, with $\psi =\rho /\gamma$ fixed, there exists a scaling law: The scaled mobility ${\gamma }^a\chi (\rho ,\gamma )/{\chi }^{\left(0\right)}\equiv \mathcal{H}\left(\psi \right)$ as a function of $\psi$ collapses onto a scaling curve $\mathcal{H}\left(\psi \right)$, where $a=1$ and 2 in models I and II, respectively, and ${\chi }^{\left(0\right)}$ is the mobility in the limiting case of a symmetric simple exclusion process; notably, the scaling function $\mathcal{H}\left(\psi \right)$ is model dependent. For model II (LLG), we calculate exactly, within a truncation scheme, both the scaling functions, $\mathcal{W}\left(y\right)$ and $\mathcal{H}\left(\psi \right)$. We also calculate spatial correlation functions for the current and compare our theory with simulation results of model I; for both models, the correlation functions decay exponentially, with correlation length $\xi \sim {\tau }_p^{1/2}$ diverging with persistence time ${\tau }_p\gg 1$. Overall, our theory is in excellent agreement with simulations and complements the prior findings [T. Chakraborty and P. Pradhan, Phys. Rev. E 109, 024124 (2024)].

Figure 1

Schematic diagram of models I and II. Top: Typical microscopic dynamics in model I, which is composed of standard hardcore RTPs (red circles) on a one-dimensional lattice (green rectangles). RTPs move along the associated spin indicated by the arrows above them. Bottom: We demonstrate microscopic dynamics of hardcore particles (red circle) in model II (i.e., LLG) on a one-dimensional lattice (green rectangle). Particles hop symmetrically by length $l$, which is drawn from an exponential distribution $\varphi \left(l\right)\sim e^{-l/l_p}$ with integer $l=0,1,2,\cdots$. This particular diagram has been taken from our previous work [14].

Figure 2

Verification of Eqs. (56), (57), and (58). We plot the scaled two-point spatial correlation of the instantaneous current ${\mathcal{C}}_r^{JJ}/{\mathcal{C}}_0^{JJ}$ for model II (LLG, left-hand panel) and model I (standard RTPs, middle panel), obtained from simulations (points), at a fixed $\rho =0.5$ and various $\gamma =0.05$ (triangle), 0.02 (inverted triangle), and 0.01 (diamond). We also compare the simulation data in both models with the strong persistence analytical solution (dotted line) given by Eqs. (56), (57), and (58). In the right-hand panel, we plot the correlation length $\xi$, as a function of $\gamma$, at $\rho =0.5$ for model II (LLG, closed points) and model I (RTPs, open points) and compare them with the strong persistence analytical solution (line) provided by Eqs. (57) and (58).

Figure 3

Verification of Eq. (61). We plot the negative scaled equal-space unequal-time instantaneous current correlation function, i.e., $-{\mathcal{C}}_0^{JJ}\left(t\right)/\chi D$, as a function of $Dt$, for model II (LLG, closed symbols) and model I (standard RTPs, open symbols), at a fixed $\gamma =0.1$ and different $\rho =0.3$ (blue circle), 0.5 (red square), and 0.7 (magenta triangle). We also compare the scaled simulation data points with the theoretical prediction (black dotted line), as shown in Eq. (61).

Figure 4

[(a) and (b)] Scaled space-time integrated current fluctuation $\gamma \langle Q_{\mathrm{tot}}^2(L,T)\rangle /2LT$ for the LLG, obtained from simulation (points), as a function of $\rho$ [at different $\gamma =0.001$ (blue square), 0.005 (red circle), 0.01 (black triangle), 0.05 (magenta inverted triangle), and 0.1 (green diamond)] and $\gamma$ [at various $\rho =0.01$ (blue square), 0.05 (red circle), 0.1 (black triangle), and 0.5 (magenta inverted triangle)], respectively. Corresponding dotted lines are $\gamma \chi (\rho ,\gamma )$ calculated by using the numerically obtained $P\left(g\right)$ in Eq. (48). The excellent match between these two quantities verifies Eq. (74). [(c) and (d)] Plots of $\langle Q_{\mathrm{tot}}^2(L,T)\rangle /2LT$ for model I (standard RTPs), obtained from numerical simulation (line point), as a function of $\rho$ and $\gamma$, respectively, for the aforementioned parameters.

Figure 5

Verification of Eqs. (77) and (79). We plot the ratio ${\gamma }^a\chi (\rho ,\gamma )/{\chi }^{\left(0\right)}$ for model II (LLG, top panel, $a=2$) and model I (RTPs, bottom panel, $a=1$) as a function of scaling variable $\psi =\rho /\gamma$ in the parameter ranges $0.01\le \rho \le 0.5$ and $0.001\le \gamma \le 0.1$. For LLG, we compare the collapsed simulation data points with the analytic scaling function ${\mathcal{H}}_{\mathrm{LLG}}\left(\psi \right)$ (solid black line) shown in Eq. (79). For both models, the collapsed data points exhibit ${\psi }^{-3/2}$ decay in the asymptotic limit, which is shown here by the red dotted line.

Figure 6

We plot the time-integrated bond-current fluctuation $\langle Q_i^2\left(T\right)\rangle$, as a function of time $T$, obtained from simulations (points) for model II (LLG, top panel) and model I (standard RTPs, bottom panel) at $\rho =0.3,0.7$ and $\gamma =0.1,0.01$. In the case of model II, we also compare the simulation data points with the analytical solution shown in Eq. (81) (line). For both these models, $\langle Q_i^2\left(T\right)\rangle$ exhibits subdiffusive growth at the intermediate-time regime, followed by a diffusive (linear) growth at very long times (shown by the dotted lines).

Figure 7

Verification of Eqs. (85) and (86). We plot the scaled bond-current fluctuation $D\langle Q_i^2\left(T\right)\rangle /2\chi L$ for model II (LLG, left-hand panel) and model I (standard RTPs, middle panel), obtained from simulations (points) at various $\rho$ and $\gamma$, as a function of the rescaled hydrodynamic time $y=D(\rho ,\gamma )T/L^2$. For LLG, we compare the scaled data points with the analytic scaling function $\mathcal{W}\left(y\right)$ shown in Eq. (86) (black line). In the right-hand panel, we check the universality of $\mathcal{W}\left(y\right)$ by plotting these numerically obtained scaled current fluctuation $D\langle Q_i^2\left(T\right)\rangle /2\chi L$ for model II (LLG, closed points) and model I (standard RTPs, open points) together and compare them with the analytically obtained $\mathcal{W}\left(y\right)$. In all three panels, the red dotted guiding lines reflect the early time subdiffusive growth $(\sim \sqrt{y})$, followed by the diffusive growth of $\mathcal{W}\left(y\right)\sim y$ as derived in Eq. (88).