Fluctuations of entropy production of a run-and-tumble particle

Out-of-equilibrium systems continuously generate entropy, with its rate of production being a fingerprint of nonequilibrium conditions. In small-scale dissipative systems subject to thermal noise, fluctuations of entropy production are significant. Hitherto, mean and variance have been abundantly studied, even if higher moments might be important to fully characterize the system of interest. Here, we introduce a graphical method to compute any moment of entropy production for a generic discrete-state system. Then, we focus on a paradigmatic model of active particles, i.e., run-and-tumble dynamics, which resembles the motion observed in several micro-organisms. Employing our framework, we compute the first three cumulants of the entropy production for a discrete version of this model. We also compare our analytical results with numerical simulations. We find that as the number of states increases, the distribution of entropy production deviates from a Gaussian. Finally, we extend our framework to a continuous state-space run-and-tumble model, using an appropriate scaling of the transition rates. The approach presented here might help uncover the features of nonequilibrium fluctuations of any current in biological systems operating out-of-equilibrium.

Figure 1

Schematic representation of a run-and-tumble particle. Top layer: $+$ regime. Bottom layer: $-$ regime. The particle hops forward and backward, respectively, with a rate $a$ ($b$) and $b$ ($a$) in the $+$ ($-$) regime, where $a>b$. Moreover, the particle switches states in between the two layers with a rate $r$.

Figure 2

Graphical representation for the computation of second-order correlation for the number of jumps. (a) $k^{\prime }>k$, (b) $k>k^{\prime }$, and (c) $k^{\prime }=k$. A circle indicates the set of states corresponding to the summation label, either $k$ or $k^{\prime }$. The arrow and equality, respectively, correspond to the transition probability from the left set of states to the right ones, and the Kronecker $\delta$'s equating the set of states.

Figure 3

Possible orderings in the graphical method for the third-order correlation of the number of jumps.

Figure 4

Scaled cumulants of entropy production. Dots: numerical simulation. Lines: theoretical predictions. Number of states in each regime, $N=8$, with transition rates $a=1.0$ and $b=0.1$. The inset shows the variation of ${\widehat{\kappa }}_{1,2.3}$ with different switching rates $r$. Here the averaging is performed over ${10}^5$ trajectories (generated using the Gillespie algorithm). In each panel, the color intensity increases with $r$.

Figure 5

(a) Scaling of cumulants (${\widehat{\kappa }}_{1,2,3}$) of entropy production for different numbers of states of the discrete run-and-tumble model. Points are obtained using analytical expressions (lines serve as a visual aid to connect the dots). The inset shows the evolution of scaled cumulants against time for the system with $N=2^5$. (b) Collapse of the complementary cumulative density function (c-cdf) of entropy production, $P^{>}\left({\mathrm{\Sigma }}_{\mathrm{env}}\right|N,T)$, for different numbers of nodes in the discrete run-and-tumble model. Entropy production at time $T=200$ is obtained from numerical simulation using ${10}^6$ trajectories initialized from a stationary state. The inset shows the uncollapsed c-cdf, without appropriate rescaling with the number of nodes. Parameters for both panels are $a=1.0$, $b=0.1$, and $r=0.05$.

Figure 6

Comparison between scaled cumulants of entropy production in the discrete and the continuous run-and-tumble model. The dashed line is the mean entropy production rate (EPR) given analytically in Ref. [97]. The dotted line is the variance of environmental entropy production calculated from the Langevin simulations of the continuous run-and-tumble model on a 1D box within $[-5,5]$ with velocity of the particle $v=1.0$, diffusion coefficient $D=0.5$, and switching rate $r=1.0$. The points are analytically calculated scaled cumulants of environmental entropy production in the discrete run-and-tumble model with scaling of transition rates given by (38).