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Lawley SD, Johnson J. Slowest first passage times, redundancy, and menopause timing. J Math Biol 2023; 86:90. [PMID: 37148411 DOI: 10.1007/s00285-023-01921-9] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2022] [Revised: 04/01/2023] [Accepted: 04/17/2023] [Indexed: 05/08/2023]
Abstract
Biological events are often initiated when a random "searcher" finds a "target," which is called a first passage time (FPT). In some biological systems involving multiple searchers, an important timescale is the time it takes the slowest searcher(s) to find a target. For example, of the hundreds of thousands of primordial follicles in a woman's ovarian reserve, it is the slowest to leave that trigger the onset of menopause. Such slowest FPTs may also contribute to the reliability of cell signaling pathways and influence the ability of a cell to locate an external stimulus. In this paper, we use extreme value theory and asymptotic analysis to obtain rigorous approximations to the full probability distribution and moments of slowest FPTs. Though the results are proven in the limit of many searchers, numerical simulations reveal that the approximations are accurate for any number of searchers in typical scenarios of interest. We apply these general mathematical results to models of ovarian aging and menopause timing, which reveals the role of slowest FPTs for understanding redundancy in biological systems. We also apply the theory to several popular models of stochastic search, including search by diffusive, subdiffusive, and mortal searchers.
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Affiliation(s)
- Sean D Lawley
- Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA.
| | - Joshua Johnson
- Division of Reproductive Sciences, Reproductive Endocrinology and Infertility, Department of Obstetrics and Gynecology, University of Colorado-Anschutz Medical Campus, Aurora, CO, USA
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2
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Radice M. Effects of mortality on stochastic search processes with resetting. Phys Rev E 2023; 107:024136. [PMID: 36932537 DOI: 10.1103/physreve.107.024136] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2022] [Accepted: 02/10/2023] [Indexed: 06/18/2023]
Abstract
We study the first-passage time to the origin of a mortal Brownian particle, with mortality rate μ, diffusing in one dimension. The particle starts its motion from x>0 and it is subject to stochastic resetting with constant rate r. We first unveil the relation between the probability of reaching the target and the mean first-passage time of the corresponding problem in absence of mortality, which allows us to deduce under which conditions the former can be increased by adjusting the restart rate. We then consider the first-passage time conditioned on the event that the particle reaches the target before dying, and provide exact expressions for the mean and the variance as functions of r, corroborated by numerical simulations. By studying the impact of resetting for different mortality regimes, we also show that, if the average lifetime τ_{μ}=1/μ is long enough with respect to the diffusive time scale τ_{D}=x^{2}/(4D), there exist both a resetting rate r_{μ}^{*} that maximizes the probability and a rate r_{m} that minimizes the mean first-passage time. However, the two never coincide for positive μ, making the optimization problem highly nontrivial.
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Affiliation(s)
- Mattia Radice
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
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3
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Lawley SD. Extreme first-passage times for random walks on networks. Phys Rev E 2020; 102:062118. [PMID: 33465958 DOI: 10.1103/physreve.102.062118] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2020] [Accepted: 10/09/2020] [Indexed: 06/12/2023]
Abstract
Many biological, social, and communication systems can be modeled by "searchers" moving through a complex network. For example, intracellular cargo is transported on tubular networks, news and rumors spread through online social networks, and the rapid global spread of infectious diseases occurs through passengers traveling on the airport network. To understand the timescale of search (or "transport" or "spread"), one commonly studies the first-passage time (FPT) of a single searcher (or "transporter" or "spreader") to a target. However, in many scenarios the relevant timescale is not the FPT of a single searcher to a target, but rather the FPT of the fastest searcher to a target out of many searchers. For example, many processes in cell biology are triggered by the first molecule to find a target out of many, and the time it takes an infectious disease to reach a particular city depends on the first infected traveler to arrive out of potentially many infected travelers. Such fastest FPTs are called extreme FPTs. In this paper, we study extreme FPTs for a general class of continuous-time random walks on networks (which includes continuous-time Markov chains). In the limit of many searchers, we find explicit formulas for the probability distribution and all the moments of the kth fastest FPT for any fixed k≥1. These rigorous formulas depend only on network parameters along a certain geodesic path(s) from the starting location to the target since the fastest searchers take a direct route to the target. Hence, the extreme FPTs are independent of the details of the network outside this geodesic(s) and can be drastically faster and less variable than conventional FPTs of single searchers. Furthermore, our results allow one to estimate if a particular system is in a regime characterized by fast extreme FPTs. We also prove similar results for mortal searchers on a network that are conditioned to find the target before a fast inactivation time. We illustrate our results with numerical simulations and uncover potential pitfalls of modeling diffusive or subdiffusive processes involving extreme statistics. In particular, we find that the many searcher limit does not commute with the diffusion limit for random walks, and thus care must be taken when choosing spatially continuous versus spatially discrete diffusion models.
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Affiliation(s)
- Sean D Lawley
- Department of Mathematics, University of Utah, Salt Lake City, Utah 84112 USA
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4
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Balakrishnan V, Abad E, Abil T, Kozak JJ. First-passage properties of mortal random walks: Ballistic behavior, effective reduction of dimensionality, and scaling functions for hierarchical graphs. Phys Rev E 2019; 99:062110. [PMID: 31330722 DOI: 10.1103/physreve.99.062110] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/29/2019] [Indexed: 11/07/2022]
Abstract
We consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites. The configuration comprising the graph, the starting point of the walk, and the locations of the trap sites is taken to be exactly self-similar as one goes from one generation of the family to the next. Under these circumstances, the total probability that the walker hits a trap is determined exactly as a function of the single-step survival probability q of the mortal walker. On the nth generation graph of the family, this probability is shown to be given by the nth iterate of a certain scaling function or map q→f(q). The properties of the map then determine, in each case, the behavior of the trapping probability, the mean time to trapping, the temporal scaling factor governing the random walk dimension on the graph, and other related properties. The formalism is illustrated for the cases of a linear hierarchical lattice and the Sierpinski graphs in two and three Euclidean dimensions. We find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint. The relevance of this finding for experiments involving travel times of particles in diffusion-decay systems is discussed.
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Affiliation(s)
- V Balakrishnan
- Department of Physics, Indian Institute of Technology Madras Chennai 600 036, India
| | - E Abad
- Departamento de Física Aplicada and Instituto de Computación Científica Avanzada (ICCAEx) Centro Universitario de Mérida, Universidad de Extremadura, E-06800 Mérida, Spain
| | - Tim Abil
- College of Computing and Digital Media DePaul University, 243 South Wabash, Chicago, Illinois 60604-2301, USA
| | - John J Kozak
- Department of Chemistry DePaul University, Chicago, Illinois 6604-6116, USA
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5
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Grebenkov DS, Tupikina L. Heterogeneous continuous-time random walks. Phys Rev E 2018; 97:012148. [PMID: 29448342 DOI: 10.1103/physreve.97.012148] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2017] [Indexed: 11/07/2022]
Abstract
We introduce a heterogeneous continuous-time random walk (HCTRW) model as a versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as porous or disordered media, multiscale or crowded environments, weighted graphs or networks. We derive the exact form of the propagator and investigate the effects of spatiotemporal heterogeneities onto the diffusive dynamics via the spectral properties of the generalized transition matrix. In particular, we show how the distribution of first-passage times changes due to local and global heterogeneities of the medium. The HCTRW formalism offers a unified mathematical language to address various diffusion-reaction problems, with numerous applications in material sciences, physics, chemistry, biology, and social sciences.
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Affiliation(s)
- Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, 91128 Palaiseau, France.,Interdisciplinary Scientific Center Poncelet (ISCP), (UMI 2615 CNRS/IUM/IITP RAS/Steklov MI RAS/Skoltech/HSE), Bolshoy Vlasyevskiy Pereulok 11, 119002 Moscow, Russia
| | - Liubov Tupikina
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, 91128 Palaiseau, France
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6
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Abstract
We introduce and investigate the escape problem for random walkers that may eventually die, decay, bleach, or lose activity during their diffusion towards an escape or reactive region on the boundary of a confining domain. In the case of a first-order kinetics (i.e., exponentially distributed lifetimes), we study the effect of the associated death rate onto the survival probability, the exit probability, and the mean first passage time. We derive the upper and lower bounds and some approximations for these quantities. We reveal three asymptotic regimes of small, intermediate, and large death rates. General estimates and asymptotics are compared to several explicit solutions for simple domains and to numerical simulations. These results allow one to account for stochastic photobleaching of fluorescent tracers in bio-imaging, degradation of mRNA molecules in genetic translation mechanisms, or high mortality rates of spermatozoa in the fertilization process. Our findings provide a mathematical ground for optimizing storage containers and materials to reduce the risk of leakage of dangerous chemicals or nuclear wastes.
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Affiliation(s)
- D. S. Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS – Ecole Polytechnique, University Paris-Saclay, 91128 Palaiseau, France
| | - J.-F. Rupprecht
- Mechanobiology Institute, National University of Singapore, 5A Engineering Drive 1, Singapore 117411, Singapore
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Le Vot F, Abad E, Yuste SB. Continuous-time random-walk model for anomalous diffusion in expanding media. Phys Rev E 2017; 96:032117. [PMID: 29347028 DOI: 10.1103/physreve.96.032117] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/21/2017] [Indexed: 06/07/2023]
Abstract
Expanding media are typical in many different fields, e.g., in biology and cosmology. In general, a medium expansion (contraction) brings about dramatic changes in the behavior of diffusive transport properties such as the set of positional moments and the Green's function. Here, we focus on the characterization of such effects when the diffusion process is described by the continuous-time random-walk (CTRW) model. As is well known, when the medium is static this model yields anomalous diffusion for a proper choice of the probability density function (pdf) for the jump length and the waiting time, but the behavior may change drastically if a medium expansion is superimposed on the intrinsic random motion of the diffusing particle. For the case where the jump length and the waiting time pdfs are long-tailed, we derive a general bifractional diffusion equation which reduces to a normal diffusion equation in the appropriate limit. We then study some particular cases of interest, including Lévy flights and subdiffusive CTRWs. In the former case, we find an analytical exact solution for the Green's function (propagator). When the expansion is sufficiently fast, the contribution of the diffusive transport becomes irrelevant at long times and the propagator tends to a stationary profile in the comoving reference frame. In contrast, for a contracting medium a competition between the spreading effect of diffusion and the concentrating effect of contraction arises. In the specific case of a subdiffusive CTRW in an exponentially contracting medium, the latter effect prevails for sufficiently long times, and all the particles are eventually localized at a single point in physical space. This "big crunch" effect, totally absent in the case of normal diffusion, stems from inefficient particle spreading due to subdiffusion. We also derive a hierarchy of differential equations for the moments of the transport process described by the subdiffusive CTRW model in an expanding medium. From this hierarchy, the full time evolution of the second-order moment is obtained for some specific types of expansion. In the case of an exponential expansion, exact recurrence relations for the Laplace-transformed moments are obtained, whence the long-time behavior of moments of arbitrary order is subsequently inferred. Our analytical and numerical results for both Lévy flights and subdiffusive CTRWs confirm the intuitive expectation that the medium expansion hinders the mixing of diffusive particles occupying separate regions. In the case of Lévy flights, we quantify this effect by means of the so-called "Lévy horizon."
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Affiliation(s)
- F Le Vot
- Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEX), Universidad de Extremadura, E-06071 Badajoz, Spain
| | - E Abad
- Departamento de Física Aplicada and Instituto de Computación Científica Avanzada (ICCAEX), Centro Universitario de Mérida and Universidad de Extremadura, E-06800 Mérida, Spain
| | - S B Yuste
- Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEX), Universidad de Extremadura, E-06071 Badajoz, Spain
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8
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Lapeyre GJ, Dentz M. Reaction–diffusion with stochastic decay rates. Phys Chem Chem Phys 2017; 19:18863-18879. [DOI: 10.1039/c7cp02971c] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023]
Abstract
Microscopic physical and chemical fluctuations in a reaction–diffusion system lead to anomalous chemical kinetics and transport on the mesoscopic scale. Emergent non-Markovian effects lead to power-law reaction times and localization of reacting species.
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Affiliation(s)
- G. John Lapeyre
- Spanish National Research Council (IDAEA-CSIC)
- E-08034 Barcelona
- Spain
- ICFO–Institut de Ciències Fotòniques
- Mediterranean Technology Park
| | - Marco Dentz
- Spanish National Research Council (IDAEA-CSIC)
- E-08034 Barcelona
- Spain
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9
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Chupeau M, Bénichou O, Redner S. Universality classes of foraging with resource renewal. Phys Rev E 2016; 93:032403. [PMID: 27078386 DOI: 10.1103/physreve.93.032403] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/05/2015] [Indexed: 06/05/2023]
Abstract
We determine the impact of resource renewal on the lifetime of a forager that depletes its environment and starves if it wanders too long without eating. In the framework of a minimal starving random-walk model with resource renewal, there are three universal classes of behavior as a function of the renewal time. For sufficiently rapid renewal, foragers are immortal, while foragers have a finite lifetime otherwise. In the specific case of one dimension, there is a third regime, for sufficiently slow renewal, in which the lifetime of the forager is independent of the renewal time. We outline an enumeration method to determine the mean lifetime of the forager in the mortal regime.
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Affiliation(s)
- M Chupeau
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France
| | - O Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France
| | - S Redner
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
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10
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Campos D, Méndez V. Phase transitions in optimal search times: How random walkers should combine resetting and flight scales. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062115. [PMID: 26764640 DOI: 10.1103/physreve.92.062115] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/19/2015] [Indexed: 06/05/2023]
Abstract
Recent works have explored the properties of Lévy flights with resetting in one-dimensional domains and have reported the existence of phase transitions in the phase space of parameters which minimizes the mean first passage time (MFPT) through the origin [L. Kusmierz et al., Phys. Rev. Lett. 113, 220602 (2014)]. Here, we show how actually an interesting dynamics, including also phase transitions for the minimization of the MFPT, can also be obtained without invoking the use of Lévy statistics but for the simpler case of random walks with exponentially distributed flights of constant speed. We explore this dynamics both in the case of finite and infinite domains, and for different implementations of the resetting mechanism to show that different ways to introduce resetting consistently lead to a quite similar dynamics. The use of exponential flights has the strong advantage that exact solutions can be obtained easily for the MFPT through the origin, so a complete analytical characterization of the system dynamics can be provided. Furthermore, we discuss in detail how the phase transitions observed in random walks with resetting are closely related to several ideas recurrently used in the field of random search theory, in particular, to other mechanisms proposed to understand random search in space as mortal random walks or multiscale random walks. As a whole, we corroborate that one of the essential ingredients behind MFPT minimization lies in the combination of multiple movement scales (regardless of their specific origin).
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Affiliation(s)
- Daniel Campos
- Grup de Física Estadística, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
| | - Vicenç Méndez
- Grup de Física Estadística, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
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11
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Hamaneh MB, Haber J, Yu YK. Analytical solution and scaling of fluctuations in complex networks traversed by damped, interacting random walkers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:052803. [PMID: 26651740 PMCID: PMC5873644 DOI: 10.1103/physreve.92.052803] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/13/2015] [Indexed: 06/05/2023]
Abstract
A general model for random walks (RWs) on networks is proposed. It incorporates damping and time-dependent links, and it includes standard (undamped, noninteracting) RWs (SRWs), coalescing RWs, and coalescing-branching RWs as special cases. The exact, time-dependent solutions for the average numbers of visits (w) to nodes and their fluctuations (σ2) are given, and the long-term σ-w relation is studied. Although σ ∝ w(1/2) for SRWs, this power law can be fragile when coalescing-branching interaction is present. Damping, however, often strengthens it but with an exponent generally different from 1/2.
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12
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Meerson B, Redner S. Mortality, redundancy, and diversity in stochastic search. PHYSICAL REVIEW LETTERS 2015; 114:198101. [PMID: 26024200 DOI: 10.1103/physrevlett.114.198101] [Citation(s) in RCA: 38] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/23/2015] [Indexed: 06/04/2023]
Abstract
We investigate a stochastic search process in one dimension under the competing roles of mortality, redundancy, and diversity of the searchers. This picture represents a toy model for the fertilization of an oocyte by sperm. A population of N independent and mortal diffusing searchers all start at x=L and attempt to reach the target at x=0. When mortality is irrelevant, the search time scales as τ_{D}/lnN for lnN≫1, where τ_{D}~L^{2}/D is the diffusive time scale. Conversely, when the mortality rate μ of the searchers is sufficiently large, the search time scales as sqrt[τ_{D}/μ], independent of N. When searchers have distinct and high mortalities, a subpopulation with a nontrivial optimal diffusivity is most likely to reach the target. We also discuss the effect of chemotaxis on the search time and its fluctuations.
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Affiliation(s)
- Baruch Meerson
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - S Redner
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
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Campos D, Abad E, Méndez V, Yuste SB, Lindenberg K. Optimal search strategies of space-time coupled random walkers with finite lifetimes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:052115. [PMID: 26066127 DOI: 10.1103/physreve.91.052115] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2015] [Indexed: 06/04/2023]
Abstract
We present a simple paradigm for detection of an immobile target by a space-time coupled random walker with a finite lifetime. The motion of the walker is characterized by linear displacements at a fixed speed and exponentially distributed duration, interrupted by random changes in the direction of motion and resumption of motion in the new direction with the same speed. We call these walkers "mortal creepers." A mortal creeper may die at any time during its motion according to an exponential decay law characterized by a finite mean death rate ω(m). While still alive, the creeper has a finite mean frequency ω of change of the direction of motion. In particular, we consider the efficiency of the target search process, characterized by the probability that the creeper will eventually detect the target. Analytic results confirmed by numerical results show that there is an ω(m)-dependent optimal frequency ω=ω(opt) that maximizes the probability of eventual target detection. We work primarily in one-dimensional (d=1) domains and examine the role of initial conditions and of finite domain sizes. Numerical results in d=2 domains confirm the existence of an optimal frequency of change of direction, thereby suggesting that the observed effects are robust to changes in dimensionality. In the d=1 case, explicit expressions for the probability of target detection in the long time limit are given. In the case of an infinite domain, we compute the detection probability for arbitrary times and study its early- and late-time behavior. We further consider the survival probability of the target in the presence of many independent creepers beginning their motion at the same location and at the same time. We also consider a version of the standard "target problem" in which many creepers start at random locations at the same time.
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Affiliation(s)
- D Campos
- Grup de Física Estadística, Departament de Física, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
| | - E Abad
- Departamento de Física Aplicada and Instituto de Computación Científica Avanzada (ICCAEX), Centro Universitario de Mérida, Universidad de Extremadura, E-06800 Mérida, Spain
| | - V Méndez
- Grup de Física Estadística, Departament de Física, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
| | - S B Yuste
- Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEX), Universidad de Extremadura, E-06071 Badajoz, Spain
| | - K Lindenberg
- Department of Chemistry and Biochemistry, and BioCircuits Institute, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093-0340, USA
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Metzler R, Jeon JH, Cherstvy AG, Barkai E. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys Chem Chem Phys 2014; 16:24128-64. [DOI: 10.1039/c4cp03465a] [Citation(s) in RCA: 1046] [Impact Index Per Article: 104.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022]
Abstract
This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.
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Affiliation(s)
- Ralf Metzler
- Institute of Physics and Astronomy
- University of Potsdam
- Potsdam-Golm, Germany
- Physics Department
- Tampere University of Technology
| | - Jae-Hyung Jeon
- Physics Department
- Tampere University of Technology
- Tampere, Finland
- Korean Institute for Advanced Study (KIAS)
- Seoul, Republic of Korea
| | - Andrey G. Cherstvy
- Institute of Physics and Astronomy
- University of Potsdam
- Potsdam-Golm, Germany
| | - Eli Barkai
- Physics Department and Institute of Nanotechnology and Advanced Materials
- Bar-Ilan University
- Ramat Gan, Israel
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