Path-integral formalism for stochastic resetting: Exactly solved examples and shortcuts to confinement

We study the dynamics of overdamped Brownian particles diffusing in conservative force fields and undergoing stochastic resetting to a given location at a generic space-dependent rate of resetting. We present a systematic approach involving path integrals and elements of renewal theory that allows us to derive analytical expressions for a variety of statistics of the dynamics such as (i) the propagator prior to first reset, (ii) the distribution of the first-reset time, and (iii) the spatial distribution of the particle at long times. We apply our approach to several representative and hitherto unexplored examples of resetting dynamics. A particularly interesting example for which we find analytical expressions for the statistics of resetting is that of a Brownian particle trapped in a harmonic potential with a rate of resetting that depends on the instantaneous energy of the particle. We find that using energy-dependent resetting processes is more effective in achieving spatial confinement of Brownian particles on a faster time scale than performing quenches of parameters of the harmonic potential.

Figure 1

Stochastic resetting of Brownian particles moving in force fields and subject to stochastic resetting at a rate that may depend on the space. Illustration of the motion of a single overdamped Brownian particle (gray sphere) that diffuses in two dimensions $(x,y)$ in the presence of a conservative potential $V(x,y)$. The particle traces a stochastic trajectory (black curve) in the two-dimensional space until its position is reset at a random instant in time to its initial value (orange arrow); subsequent to the reset, the particle resumes its stochastic motion until the next reset happens. The rate of resetting $r(x,y)$ (left color bar) may depend on the position of the particle. We study three scenarios: (a) resetting of free Brownian particles under a space-independent rate of resetting; (b) resetting of free Brownian particles under a space-dependent rate of resetting; and (c) resetting of Brownian particles moving in force fields under a space-dependent rate of resetting.

Figure 2

Theory versus simulation results for the stationary spatial distribution of a free Brownian particle undergoing parabolic resetting. Circles denote simulation results, while the line represents the exact analytical expression given in Eq. (56). Numerical results were obtained from ${10}^4$ independent simulations of the Langevin dynamics described in Sec. 2a. Error bars associated with the data points are smaller than the symbol size. Parameter values are $D=1.0,\alpha =0.5$.

Figure 3

Illustration of the energy-dependent resetting of a Brownian particle moving in a harmonic potential. A Brownian particle (gray circle) immersed in a thermal bath at temperature $T$ moves with diffusion coefficient $D$, with its motion being confined by a harmonic potential $V\left(x\right)=\kappa x^2/2$ (green area), where $\kappa$ is the stiffness constant. Here, $x$ is the position of the particle with respect to the center of the potential. The particle, initially located in the trap center $(t^{\prime }=0$; left panel), diffuses at subsequent times in the energy landscape $(t^{\prime }<t$; middle panel), until a resetting event occurs at time $t^{\prime }=t$ (right panel). The black curve represents the history of the particle from $t^{\prime }=0$ up to the time corresponding to each snapshot. The rate of resetting (right color bar) is proportional to the instantaneous energy of the particle, and therefore, a reset is more likely to take place as the particle climbs up the potential.

Figure 4

Theory versus simulation results for the stationary spatial distribution of a Brownian particle trapped in a harmonic potential and undergoing energy-dependent resetting. Circles denote simulation results, while the line represents the exact analytical expression given in Eq. (72). Numerical results were obtained from ${10}^4$ independent simulations of the Langevin dynamics described in Sec. 2a. Error bars associated with the data points are smaller than the symbol size. Parameter values are $D=1.0,{\tau }_c=0.5$.

Figure 5

Shortcut to particle confinement. Confinement of a Brownian particle using harmonic potentials and space-dependent rates of resetting. Symbols represent simulation results for relaxation processes of ${10}^5$ noninteracting Brownian particles that are initially in equilibrium in a harmonic potential $V\left(x\right)=(1/2)\kappa x^2$. Blue circles represent the mean-squared displacement of the particle, in units of $\langle x^2\left(0\right)\rangle$, after an instantaneous quench to a harmonic potential with stiffness ${\kappa }^{\prime }=1.7\kappa$ (blue arrow in the inset). Red circles, on the other hand, represent the mean-squared displacement of the particle, in units of $\langle x^2\left(0\right)\rangle$, after an instantaneous quench of the rate of resetting from $r\left(x\right)=0$ to $r\left(x\right)=(3/4)({\mu }^2{\kappa }^2/D)x^2$ (red arrow in the inset). For the case in which the stiffness of the potential is quenched from $\kappa$ to ${\kappa }^{\prime }$, it is easily seen that $\langle x^2\left(t\right)\rangle =\langle x^2\left(0\right)\rangle exp(-2\mu k^{\prime }t)+(D/\left(\mu k^{\prime }\right))[1-exp(-2\mu k^{\prime }t)]$, thereby implying a relaxation time scale ${\tau }_{\mathrm{quench}}=1/\left(2\mu k^{\prime }\right)$ and yielding the blue curve in the figure. Note that the time on the $x$ axis is measured in units of ${\tau }_c=1/\mu \kappa$. The red curve, depicting the process of relaxation in the presence of resetting, may be fitted to a good approximation to $A+B{e}^{-t/{\tau }_{\mathrm{reset}}}$. One observes that ${\tau }_{\mathrm{reset}}\approx {\tau }_{\mathrm{quench}}/3$. Parameter values are $D=1,\kappa =1$, and $\mu =10$.