Path-integral formalism for stochastic resetting:
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 with a generic space-dependent rate of resetting. We present a systematic approach involving path integrals and elements of renewal theory that allows 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 timescale than by performing quenches of parameters of the harmonic potential.
pacs:05.40.-a, 02.50.-r, 05.70.Ln
Changes are inevitable in nature, and those that are most dramatic with often drastic consequences are the ones that occur all of a sudden. A particular class of such changes comprises those in which the system during its temporal evolution makes a sudden jump (a “reset”) to a fixed state or configuration. Many nonequilibrium processes are encountered across disciplines, e.g., in physics, biology, and information processing, which involve sudden transitions between different states or configurations. The erasure of a bit of information Landauer:1961 (); Bennett:1973 () by mesoscopic machines may be thought of as a physical process in which a memory device that is strongly affected by thermal fluctuations resets its state (0 or 1) to a prescribed erasure state Berut:2012 (); Mandal:2012 (); Roldan:2014 (); Koski:2014 (); Fuchs:2016 (). In biology, resetting plays an important role inter alia in sensing of extracellular ligands by single cells Mora:2015 (), and in transcription of genetic information by macromolecular enzymes called RNA polymerases Roldan:2016 (). During RNA transcription, the recovery of RNA polymerases from inactive transcriptional pauses is a result of a kinetic competition between diffusion and resetting of the polymerase to an active state via RNA cleavage Roldan:2016 (), as has been recently tested in high-resolution single-molecule experiments Lisica:2016 (). Also, there are ample examples of biochemical processes that initiate (i.e., reset) at random so-called stopping times Gillespie:2014 (); Hanggi:1990 (); Neri:2017 (), with the initiation at each instance occurring in different regions of space Julicher:1997 (). In addition, interactions play a key role in determining when and where a chemical reaction occurs Gillespie:2014 (), a fact that affects the statistics of the resetting process. For instance, in the above mentioned example of recovery of RNA polymerase by the process of resetting, the interaction of the hybrid DNA-RNA may alter the time that a polymerase takes to recover from its inactive state Zamft:2012 (). It is therefore quite pertinent and timely to study resetting of mesoscopic systems that evolve under the influence of external or conservative force fields.
Simple diffusion subject to resetting to a given location at random times has emerged in recent years as a convenient theoretical framework to discuss the phenomenon of stochastic resetting Evans:2011-1 (); Evans:2011-2 (); Evans:2014 (); Christou:2015 (); Eule:2016 (); Nagar:2016 (). The framework has later been generalized to consider different choices of the resetting position Boyer:2014 (); Majumdar:2015-2 (), resetting of continuous-time random walks Montero:2013 (); Mendez:2016 (), Lévy Kusmierz:2014 () and exponential constant-speed flights Campos:2015 (), time-dependent resetting of a Brownian particle Pal:2016 (), and in discussing memory effects Boyer:2017 () and phase transitions in reset processes Harris:2017 (). Stochastic resetting has also been invoked in the context of many-body dynamics, e.g., in reaction-diffusion models Durang:2014 (), fluctuating interfaces Gupta:2014 (); Gupta:2016 (), interacting Brownian motion Falcao:2017 (), and in discussing optimal search times in a crowded environment Kusmierz:2015 (); Reuveni:2016 (); Bhat:2016 (); Pal:2017 (). However, little is known about the statistics of stochastic resetting of Brownian particles that diffuse under the influence of force fields Pal:2015 (), and that too in presence of a rate of resetting that varies in space.
In this paper, we study the dynamics of overdamped Brownian particles immersed in a thermal environment, which diffuse under the influence of a force field, and whose position may be stochastically reset to a given spatial location with a rate of resetting that has an essential dependence on space. We use an approach that allows to obtain exact expressions for the transition probability prior to the first reset, the first reset-time distribution, and, most importantly, the stationary spatial distribution of the particle. The approach is based on a combination of the theory of renewals Cox:1962 () and the Feynman-Kac path-integral formalism of treating stochastic processes Feynman:2010 (); Schulman:1981 (); Kac:1949 (); Kac:1951 (), and consists in a mapping of the dynamics of the Brownian resetting problem to a suitable quantum mechanical evolution in imaginary time. We note that the Feynman-Kac formalism has been applied extensively in the past to discuss dynamical processes involving diffusion satya (), and has to the best of our knowledge not been applied to discuss stochastic resetting. To demonstrate the utility of the approach, we consider several different stochastic resetting problems, see Fig. 1: i) Free Brownian particles subject to a space-independent rate of resetting (Fig. 1a)); ii) Free Brownian particles subject to resetting with a rate that depends quadratically on the distance to the origin (Fig. 1b)); and iii) Brownian particles trapped in a harmonic potential and undergoing reset events with a rate that depends on the energy of the particle (Fig. 1c)). In this paper, we consider for purposes of illustration the corresponding scenarios in one dimension, although our general approach may be extended to higher dimensions. Remarkably, we obtain exact analytical expressions in all cases, and, notably, in cases ii) and iii), where a standard treatment of analytic solution by using the Fokker-Planck approach may appear daunting, and whose relevance in physics may be explored in the context of, e.g., optically-trapped colloidal particles and hopping processes in glasses and gels. We further explore the dynamical properties of case iii), and compare the relaxation properties of dynamics corresponding to potential energy quenches and due to sudden activation of space-dependent stochastic resetting.
Ii General formalism
ii.1 Model of study: resetting of Brownian particles diffusing in force fields
Consider an overdamped Brownian particle diffusing in one dimension in presence of a time-independent force field , with denoting a potential energy landscape. The dynamics of the particle is described by a Langevin equation of the form
where is the mobility of the particle, defined as the velocity per unit force. In Eq. (1), is a Gaussian white noise, with the properties
where denotes average over noise realizations, and is the diffusion coefficient of the particle, with the dimension of length-squared over time. We assume that the Einstein relation holds: , with being the temperature of the environment, and with being the Boltzmann constant. In addition to the dynamics (1), the particle is subject to a stochastic resetting dynamics with a space-dependent resetting rate , whereby, while at position at time , the particle in the ensuing infinitesimal time interval either follows the dynamics (1) with probability , or resets to a given reset destination with probability . Our analysis holds for any arbitrary reset function , with the only obvious constraint ; moreover, the formalism may be generalized to higher dimensions. In the following, we consider the reset location to be the same as the initial location of the particle, that is, .
A quantity of obvious interest and relevance is the spatial distribution of the particle: What is the probability that the particle is at position at time , given that its initial location is ? From the dynamics given in the preceding paragraph, it is straightforward to write the time evolution equation of :
where the first two terms on the right hand side account for the contribution from the diffusion of the particle in the force field , while the last two terms stand for the contribution owing to the resetting of the particle: the third term represents the loss in probability arising from the resetting of the particle to , while the fourth term denotes the gain in probability at the location owing to resetting from all locations . When exists, the stationary distribution satisfies
It is evident that solving for either the time-dependent distribution or the stationary distribution from Eqs. (3) and (4), respectively, is a formidable task even with , unless the function has simple forms. For example, in Ref. Evans:2011-2 (), the authors considered a solvable example with , where the function is zero in a window around and is constant outside the window.
In this work, we employ a different approach to solve for the stationary spatial distribution, by invoking the path integral formalism of quantum mechanics and by using elements of the theory of renewals. In this approach, we compute , the stationary distribution in presence of reset events, in terms of suitably-defined functions that take into account the occurrence of trajectories that evolve without undergoing any reset events in a given time, see Eq. (30) below. This approach provides a viable alternative to obtaining the stationary spatial distribution by solving the Fokker-Planck equation (4) that explicitly takes into account the occurrence of trajectories that evolve while undergoing reset events in a given time. As we will demonstrate below, the method allows to obtain exact expressions even in cases with nontrivial forms of and .
ii.2 Path-integral approach to stochastic resetting
Here, we invoke the well-established path-integral approach based on the Feynman-Kac formalism to discuss stochastic resetting. To proceed, let us first consider a representation of the dynamics in discrete times , with , and being a small time step. The dynamics in discrete times involves the particle at position at time to either reset and be at at the next time step with probability or follow the dynamics given by Eq. (1) with probability . The position of the particle at time is thus given by
where we have defined , and have used the Stratonovich rule in discretizing the dynamics (1), and where the time-discretized Gaussian, white noise satisfies
with a positive constant with the dimension of length-squared over time-squared. In particular, the joint probability distribution of occurrence of a given realization of the noise, with being a positive integer, is given by
In the absence of any resetting and forces, the displacement of the particle at time from the initial location is given by , so that the mean-squared displacement is . In the continuous-time limit, , keeping the product fixed and finite and equal to , the mean-squared displacement becomes , with .
ii.2.1 The propagator prior to first reset.
What is the probability of occurrence of particle trajectories that start at position and end at a given location at time without having undergone any reset event? From the discrete-time dynamics given by Eq. (5) and the joint distribution (7), the probability of occurrence of a given particle trajectory is given by
Here, the factor enforces the condition that the particle has not reset at any of the instants , while is the Jacobian matrix for the transformation , which is obtained from Eq. (5) as or equivalently
with primes denoting derivative with respect to . One thus has
where in obtaining the last step, we have used the smallness of . Thus, for small , we get
From Eq. (LABEL:eq:path-0), it follows by considering all possible trajectories that the probability density that the particle while starting at position ends at a given location at time without having undergone any reset event is given by
In the limit of continuous time, defining one gets the exact expression for the corresponding probability density as the following path integral:
where on the right hand side of Eq. (II.2.1), we have introduced the resetting action as
Invoking the Feynman-Kac formalism, we identify the path integral on the right hand side of Eq. (II.2.1) with the propagator of a quantum mechanical evolution in (negative) imaginary time due to a quantum Hamiltonian (see Appendix), to get
where the quantum Hamiltonian is
the mass in the equivalent quantum problem is
and the quantum potential is given by
Note that in the quantum propagator in Eq. (16), the Planck’s constant has been set to unity, , while the time of propagation is imaginary: Wick (). Since the Hamiltonian contains no explicit time dependence, the propagator is effectively a function of the time to propagate from the initial location to the final location , and not individually of the initial and final times. Let us note that on using , the prefactor equals , where is the heat absorbed by the particle from the environment along the trajectory Sekimoto:1998 (); Sekimoto:2000 ().
ii.2.2 Distribution of the first-reset time
Let us now ask for the probability of occurrence of trajectories that start at position and reset for the first time at time . In terms of , one gets this probability density as
since by the very definition of , a reset has to happen only at the final time when the particle has reached the location , where may in principle take any value in the interval . The probability density is normalized as .
ii.2.3 Spatial time-dependent probability distribution
Using renewal theory, we now show that knowing and is sufficient to obtain the spatial distribution of the particle at any time . The probability density that the particle is at at time while starting from is given by
where we have defined the probability density to reset at time as
One may easily understand Eq. (21) by invoking the theory of renewals Cox:1962 () and realizing that the dynamics is renewed each time the particle resets to . This may be seen as follows. The particle while starting from may reach at time by experiencing not a single reset; the corresponding contribution to the spatial distribution is given by the first term on the right hand side of Eq. (21). The particle may also reach at time by experiencing the last reset event (i.e., the last renewal) at time instant , with , and then propagating from the reset location to without experiencing any further reset, where the last reset may take place with rate from any location where the particle happened to be at time ; such contributions are represented by the second term on the right hand side of Eq. (21). The spatial distribution is normalized as for all possible values of and .
Multiplying both sides of Eq. (21) by , and then integrating over , we get
The square-bracketed quantity on the right hand side is nothing but , so that we get
Taking the Laplace transform on both sides of Eq. (24), we get
where and are respectively the Laplace transforms of and . Solving for from Eq. (25) yields
Next, taking the Laplace transform with respect to time on both sides of Eq. (21), we obtain
ii.2.4 Stationary spatial distribution
On applying the final value theorem, one may obtain the stationary spatial distribution as
provided the stationary distribution (i.e., ) exists. Now, since is normalized to unity, , we may expand its Laplace transform to leading orders in as , provided that the mean first-reset time , defined as
is finite. Similarly, we may expand to leading orders in as , provided that is finite. From Eq. (28), we thus find the stationary spatial distribution to be given by the integral over all times of the propagator prior to first reset divided by the mean first-reset time:
Iii Exactly Solved Examples
iii.1 Free particle with space-independent resetting
Let us first consider the simplest case of free diffusion with a space-independent rate of resetting , with a positive constant having the dimension of inverse time. Here, on using Eq. (15) with , we have
Since in the present situation, the effective quantum potential is space independent, we may rewrite Eq. (31) as:
where the quantum Hamiltonian is now that of a free particle:
Therefore, the statistics of resetting of a free particle under a space-independent rate of resetting may be found from the quantum propagator of a free particle, which is given by Schulman:1981 ()
and hence, the distribution of the first-reset time may be found on using Eq. (20):
which is normalized to unity: , as expected.
Next, the probability density that the particle is at at time , while starting from , is obtained on using Eq. (21) as
Taking the limit , we obtain the stationary spatial distribution as
which may also be obtained by using Eqs. (30) and (38), and also Eq. (39) that implies that . From Eq. (LABEL:eq:Pxt-constant-resetting), we obtain an exact expression for the time-dependent spatial distribution as
while Eq. (LABEL:eq;Pxstat-constant-resetting) yields the exact stationary distribution as
where is the complementary error function. The stationary distribution (43) may be put in the scaling form
where the scaling function is given by . For the particular case , Eq. (43) matches with the result derived in Ref. Evans:2011-1 (). Note that the steady state distribution (44) exhibits a cusp at the resetting location . Since the resetting location is taken to be the same as the initial location, the particle visits repeatedly in time the initial location, thereby keeping a memory of the latter that makes an explicit appearance even in the long-time stationary state.
iii.2 Free particle with “parabolic” resetting
We now study the dynamics of a free Brownian particle whose position is reset to the initial position with a rate of resetting that is proportional to the square of the current position of the particle. In this case, we have , with having the dimension of . From Eqs. (15) and (16), and given that in this case , we get
with the Hamiltonian obtained from Eq. (17) by setting :
We thus see that the statistics of resetting of a free particle subject to a “parabolic” rate of resetting may be found from the propagator of a quantum harmonic oscillator. Following Schulman Schulman:1981 (), a quantum harmonic oscillator with the Hamiltonian given by
with and being the mass and the frequency of the oscillator, has the quantum propagator
Using the parameters given in Eq. (46), and substituting and in Eq. (LABEL:eq:Gq-free-parabola), we have
We may now derive the statistics of resetting by using the propagator (LABEL:eq:Gq-free-parabola-1). Equation (LABEL:eq:parabolic-resetting-Pxt-0-x0) together with Eq. (LABEL:eq:Gq-free-parabola-1) imply
Integrating Eq. (LABEL:eq:parabolic-resetting-Pxt-x0) over , we get the distribution of the first-reset time as
For the case , Eqs. (LABEL:eq:parabolic-resetting-Pxt-x0) and (LABEL:eq:parabolic-resetting-Pt-x0) reduce to simpler expressions:
Equation (53) may be put in the scaling form
with . Equation (53) yields the mean first-reset time for to be given by
where is the th order modified Bessel function of the second kind. Equation (56) implies that the stationary distribution is symmetric around , which is expected since the resetting rate is symmetric around . The stationary distribution (56) may be put in the scaling form
where the scaling function is given by .