Active Brownian particles: Entropy production and fluctuation response

Within the Rayleigh-Helmholtz model of active Brownian particles, activity is due to a nonlinear velocity-dependent force. In the presence of external trapping potential or constant force, the steady state of the system breaks detailed balance producing a net entropy. Using molecular dynamics simulations, we obtain the probability distributions of entropy production in these steady states. The distribution functions obey fluctuation theorems for entropy production. Using the simulation, we further show that the steady-state response function obeys a modified fluctuation-dissipation relation.

Figure 1

Steady-state probability distribution $P_s\left(v\right)$ for ABPs under a constant external force $f=0.2$. Points are from MD simulations, and the line is a plot of Eq. (9).

Figure 2

Probability distribution of total entropy production $\rho \left(\Delta s_t\right)$ calculated in the presence of an external force $f=0.2$. The calculations are performed after collecting data over ${\tau }_0=2.56,5.12,10.24,20.48,40.96\tau$.

Figure 3

Ratio of probability distributions of positive and negative entropy production $\rho (\Delta s_t=\Delta \sigma )/\rho (\Delta s_t=-\Delta \sigma )$ calculated from the data described in the legend of Fig. 2. The solid line is a plot of the function $exp(\Delta \sigma /k_B)$.

Figure 4

Probability distributions of total entropy productions $\rho \left(\Delta s_t\right)$ calculated in the presence of an external harmonic potential trap $U\left(x\right)=(1/2){\omega }_0^2{x}^2$ with ${\omega }_0^2=5$. The calculations are performed after collecting data over ${\tau }_0=2.56,5.12,10.24,20.48\tau$.

Figure 5

Ratio of probability distributions of positive and negative entropy productions $\rho (\Delta s_t=\Delta \sigma )/\rho (\Delta s_t=-\Delta \sigma )$ calculated from the data described in the legend of Fig. 4. The solid line shows a plot of $exp(\Delta \sigma /k_B)$.

Figure 6

Response functions and steady-state fluctuations. (a) Direct MD evaluation of response function for Rayleigh-Helmholtz ABPs, and passive Brownian particles within a harmonic trap of strength ${\omega }_0^2=5$. (b) Comparison of response function of ABPs $R_v\left(t\right)$ against steady-state fluctuations as given by the right-hand side of Eq. (25).