Stochastic resetting and first arrival subjected to Gaussian noise and Poisson white noise

We study the dynamics of an overdamped Brownian particle subjected to Poissonian stochastic resetting in a nonthermal bath, characterized by a Poisson white noise and a Gaussian noise. Applying the renewal theory we find an exact analytical expression for the spatial distribution at the steady state. Unlike the single exponential distribution as observed in the case of a purely thermal bath, the distribution is double exponential. Relaxation of the transient spatial distributions to the stationary one, for the limiting cases of Poissonian rate, is investigated carefully. In addition, we study the first-arrival properties of the system in the presence of a delta-function sink with strength $\kappa$, where $\kappa =0$ and $\kappa =\infty$ correspond to fully nonreactive and fully reactive sinks, respectively. We explore the effect of two competitive mechanisms: the diffusive spread in the presence of two noises and the increase in probability density around the initial position due to stochastic resetting. We show that there exists an optimal resetting rate, which minimizes the mean first-arrival time (MFAT) to the sink for a given value of the sink strength. We also explore the effect of the strength of the Poissonian noise on MFAT, in addition to sink strength. Our formalism generalizes the diffusion-limited reaction under resetting in a nonequilibrium bath and provides an efficient search strategy for a reactant to find a target site, relevant in a range of biophysical processes.

Figure 1

Plot of the probability distribution function of displacement at the steady state [given in Eq. (12)] for two different regimes of Poisson rates: (a) $r<\mu$, where $\mu =10$, and (b) $r>\mu$, where $\mu =5.0$. Here we have taken $D_A=D_T=1.0$. The curves with point symbols are the approximate analytical results given in Eqs. (14) and (16), and these equations are used for comparison in panels (a) and (b), respectively.

Figure 2

Plot of the probability distribution functions [Eq. (12)] as a function of displacements for different values of $D_A/D_T$. In panel (a) $D_T>D_A$, where we have taken $D_A=1.0$. In panel (b) $D_A>D_T$, where $D_T=1.0$. Other parameters are $r=\mu =5.0$. Approximate results of the PDF given in Eqs. (17) and (18) represented by the lines with point symbols are in good agreement with the exact one (solid lines) as shown in panels (a) and (b), respectively.

Figure 3

Comparative plots of analytical results of the complete probability distribution function (PDF) with the numerical one in logarithmic scale as a function of displacements at different times. The dotted curves are plotted using the data obtained from the numerical inverse Fourier transform of Eq. (8). The position $x^{*}\left(t\right)$ which distinguishes two regimes is marked by blue dashed vertical lines for different times. In panels (a) and (b) the plots are for two limiting cases: (a) $\mu \ll 1, D_T\gg D_A$ and (b) $\mu \gg r, D_A/D_T$ is finite. The values of parameters used for computations are given by (a) $D_T=20,D_A=0.1,\mu =0.001,r=0.5$ and (b) $D_T=1.0,D_A=10,\mu =40,r=1.0$. The solid lines corresponds to analytical results given in Eqs. (22) and (27) for two respective cases. The black solid (upper) line is for $x\left(t\right)<x^{*}\left(t\right)$ whereas other solid curves (magenta, red) represent the PDFs for large values of $t$ (here we have taken $t=2.5,5.0,10$) in the region $x\left(t\right)>x^{*}\left(t\right)$.

Figure 4

Logarithmic plot of first-passage times versus resetting rate $r$: (a) for different values of sink strength $\kappa$ taking $D_A=5.0,D_T=1.0$ and (b) for different $D_A$ values keeping the thermal diffusivity constant at $D_T=1.0$ in the case of complete absorption. The solid curves are drawn by solving Eq. (31) numerically taking $\mu =5000$ and assuming that the distance between the sink and the source is fixed at $|x_s-x_0|=2.50$. The dotted curves correspond to Eq. (32) which is a quite approximate result for the limit $\frac{D_A}{D_T}\frac{r}{\mu }\ll 1$. In panel (a) the solid curves agree well with the analytical results for the given range of Poisson rate $r$, whereas the dotted lines are good fits only for small values of $r$ if $D_A$ is large, as shown in panel (b).

Figure 5

(a) Plot of optimal rate $r^{*}$ as a function of the sink strength $\kappa$. The other parameters are $D_A=5.0,D_T=1.0$. (b) The optimal rate $r^{*}$ for complete absorption is plotted against $D_A$ considering $D_T$ constant at $D_T=1.0$. The curves are obtained by solving the equation $\frac{d}{dr}\langle {\mathbb{T}}_r\rangle {|}_{r=r^{*}}=0$. Solid lines correspond to the exact results computed with the aid of Eqs. (31) and (29), and approximate result given in Eq. (32) is used to draw the dashed lines. For both panels (a) and (b), the other parameters taken for numerical calculation are given here: $|x_S-x_0|=2.50,a_0=0.02236$.