Diffusion–Advection Equations on a Comb: Resetting and Random Search

Sandev, Trifce; Domazetoski, Viktor; Iomin, Alexander; Kocarev, Ljupčo

doi:10.3390/math9030221

Cited by 21 publications

(25 citation statements)

References 70 publications

(111 reference statements)

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“…( 31) from Ref. [76] (µ → V , σ 2 → 2D, x 0 = 0), as it should be. Finally, the variance is given by var…”

Section: Moments Of Diffusion With Resettingmentioning

confidence: 87%

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Autocorrelation functions and ergodicity in diffusion with stochastic resetting

Stojkoski,

Sandev,

Kocarev

et al. 2021

Preprint

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Diffusion with stochastic resetting is a paradigm of resetting processes. Standard renewal or master equation approach are typically used to study steady state and other transport properties such as average, mean squared displacement etc. What remains less explored is the two time point correlation functions whose evaluation is often daunting since it requires the implementation of the exact time dependent probability density functions of the resetting processes which are unknown for most of the problems. We adopt a different approach that allows us to write a stochastic solution in the level of a single trajectory undergoing resetting. Moments and the autocorrelation functions between any two times along the trajectory can then be computed directly using the laws of total expectation. Estimation of autocorrelation functions turns out to be pivotal for investigating the ergodic properties of various observables for this canonical model. In particular, we investigate two observables (i) sample mean which is widely used in economics and (ii) time-averaged-mean-squareddisplacement (TAMSD) which is of acute interest in physics. We find that both diffusion and drift-diffusion processes are ergodic at the mean level unlike their reset-free counterparts. In contrast, resetting renders ergodicity breaking in the TAMSD while both the stochastic processes are ergodic when resetting is absent. We quantify these behaviors with detailed analytical study and corroborate with extensive numerical simulations. The current study provides an important baseline that unifies two different approaches, used ubiquitously in economics and physics, for studying the ergodic properties in diffusion with resetting. We believe that our results can be verified in single particle experimental set-ups and thus have strong implications in the field of resetting.

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“…( 31) from Ref. [76] (µ → V , σ 2 → 2D, x 0 = 0), as it should be. Finally, the variance is given by var…”

Section: Moments Of Diffusion With Resettingmentioning

confidence: 87%

“…is the PDF of the drift-diffusion equation without resetting (r = 0) and the initial position is x 0 = 0 [76]), i.e., P r (x, t = 0) = δ(x). The stationary PDF, P * r (x) = lim t→∞ P r (x, t) then takes the following form…”

Section: Preliminariesmentioning

confidence: 99%

Autocorrelation functions and ergodicity in diffusion with stochastic resetting

Stojkoski,

Sandev,

Kocarev

et al. 2021

Preprint

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“…We note that the case α = 1 in heterogeneous media does not necessarily imply that the process has a Gaussian probability density function (PDF), instead, for instance, exponential or stretched Gaussian forms may be observed [19,20]. We also note that the MSD may also grow exponentially, for a multiplicative noise such as geometric Brownian motion or heterogeneous diffusion processes [21,22,23,24], or logarithmically in strongly disordered environments [25,26].A by-now classical model for heterogeneous systems, popularised by Mandelbrot, are fractals, such as the Sierpiński gasket [27,28,29]. However, such ideal mathematical fractals are often insufficient to adequately describe real fractals such as networks of rivers, blood vessels, or nerve fibres, for which random fractals such as percolation clusters are more appropriate [27,28,29].…”

mentioning

confidence: 95%

“…SR can be phrased as a renewal [67,68] or a non-renewal [69] process. Moreover, SR dynamics was studied for motion in bounded domains [70,71], as well as in monotonic [23,72,73] and non-monotonic potentials [74,75], and under time-dependent resetting [76]. Efficient escape under resetting was investigated [77], and it was shown that interesting phase transitions occur in the parameter space for the optimal resetting rate [63,70,78,79,80].…”

Section: Introductionmentioning

confidence: 99%

Backbone diffusion and first-passage dynamics in a comb structure with confining branches under stochastic resetting

Singh

Sandev

Iomin

et al. 2021

Preprint

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We study the diffusive motion of a test particle in a two-dimensional comb structure consisting of a main backbone channel with continuously distributed side branches, in the presence of stochastic Markovian resetting to the initial position of the particle. We assume that the motion along the infinitely long branches is biased by a confining potential. The crossover to the steady state is quantified in terms of a large deviation function, which is derived for the first time for comb structures in present paper. We show that the relaxation region is demarcated by a nonlinear "light-cone" beyond which the system is evolving in time. We also investigate the first-passage times along the backbone and calculate the mean first-passage time and optimal resetting rate.Backbone diffusion and first-passage dynamics in a comb structure with confining branches. . . 2 [12,13,14,15]. Sometimes for α > 2 the term hyperdiffusion is used [16,17,18]. We note that the case α = 1 in heterogeneous media does not necessarily imply that the process has a Gaussian probability density function (PDF), instead, for instance, exponential or stretched Gaussian forms may be observed [19,20]. We also note that the MSD may also grow exponentially, for a multiplicative noise such as geometric Brownian motion or heterogeneous diffusion processes [21,22,23,24], or logarithmically in strongly disordered environments [25,26].A by-now classical model for heterogeneous systems, popularised by Mandelbrot, are fractals, such as the Sierpiński gasket [27,28,29]. However, such ideal mathematical fractals are often insufficient to adequately describe real fractals such as networks of rivers, blood vessels, or nerve fibres, for which random fractals such as percolation clusters are more appropriate [27,28,29]. In many cases such structures have a characteristic backbone from which various branches emerge [28,29]. A highly effective model addressing transport on such random loopless structures is a comb, in which infinite branches branch off the central backbone. The comb model was introduced to understand anomalous transport in percolation clusters [30,31,32,33,34]. Now, comb-like models are widely employed to describe various experimental applications. Comb-like structures are particularly important from a biophysical point of view as they provide a way to address transport along spiny dendrites [35,36,37], in which the transport properties crucially depend on the underlying geometry [38]. Similar approaches are being used in the modelling of river basins with their often very ramified geometry [39,40]. In fact, long time retention data of tracers in water catchments reveal scaling exponents consistent with comb dynamics [41,42].Depending on the specific setting the geometry of comb structures effects both subdiffusion [43,44,45], including ultraslow diffusion [46], and superdiffusion [17, 47, 48]. The nontrivial nature of transport along a comb is discernible from the fact that motion along the branches results in a long-range memory for motion along the...

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“…Motivated by this myriad of possible applications, several works have systematically investigated the combination of diffusion with stochastic resetting [21][22][23][24][25][26][27][28][29][30][31][32]. Despite this increasing interest in studying diffusion with stochastic resetting, much less attention has been paid towards considering the effect of heterogeneous media [33].…”

Section: Introductionmentioning

confidence: 99%

Transient anomalous diffusion in heterogeneous media with stochastic resetting

Lenzi,

Guilherme

et al. 2022

Preprint

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We investigate a diffusion process in heterogeneous media where particles stochastically reset to their initial positions at a constant rate. The heterogeneous media is modeled using a spatialdependent diffusion coefficient with a power-law dependence on particles' positions. We use the Green function approach to obtain exact solutions for the probability distribution of particles' positions and the mean square displacement. These results are further compared and agree with numerical simulations of a Langevin equation. We also study the first-passage time problem associated with this diffusion process and obtain an exact expression for the mean first-passage time. Our findings show that this system exhibits non-Gaussian distributions, transient anomalous diffusion (sub-or superdiffusion) and stationary states that simultaneously depend on the media heterogeneity and the resetting rate. We further demonstrate that the media heterogeneity non-trivially affect the mean first-passage time, yielding an optimal resetting rate for which this quantity displays a minimum.

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Diffusion–Advection Equations on a Comb: Resetting and Random Search

Cited by 21 publications

References 70 publications

Autocorrelation functions and ergodicity in diffusion with stochastic resetting

Autocorrelation functions and ergodicity in diffusion with stochastic resetting

Backbone diffusion and first-passage dynamics in a comb structure with confining branches under stochastic resetting

Transient anomalous diffusion in heterogeneous media with stochastic resetting

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