1
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Shkilev VP. First-passage behavior of the random-barrier model. Phys Rev E 2023; 108:044122. [PMID: 37978623 DOI: 10.1103/physreve.108.044122] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/05/2023] [Accepted: 09/22/2023] [Indexed: 11/19/2023]
Abstract
The previously proposed transport equation for the random-barrier model, which is the diffusion equation with resetting to positions visited in the past, is used here to calculate the first-passage times. The results obtained are compared with those obtained using the normal diffusion equation with an effective diffusion coefficient. It is shown that, under certain conditions, the equation with the effective diffusion coefficient can greatly overestimate the time of the first passage. In particular, the rate constant of a bimolecular diffusion-controlled reaction calculated from this equation can be significantly lower than the actual rate. This result can serve as one of the possible explanations for the high rates of diffusion-controlled reactions observed in an experiment.
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Affiliation(s)
- V P Shkilev
- Chuiko Institute of Surface Chemistry, National Academy of Sciences of Ukraine, General Naumov Street 17, 03164 Kyiv, Ukraine
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2
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Kay T, Giuggioli L. Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman-Kac Approach. JOURNAL OF STATISTICAL PHYSICS 2023; 190:92. [PMID: 37128546 PMCID: PMC10140114 DOI: 10.1007/s10955-023-03105-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/24/2022] [Accepted: 04/03/2023] [Indexed: 05/03/2023]
Abstract
We derive, through subordination techniques, a generalized Feynman-Kac equation in the form of a time fractional Schrödinger equation. We relate such equation to a functional which we name the subordinated local time. We demonstrate through a stochastic treatment how this generalized Feynman-Kac equation describes subdiffusive processes with reactions. In this interpretation, the subordinated local time represents the number of times a specific spatial point is reached, with the amount of time spent there being immaterial. This distinction provides a practical advance due to the potential long waiting time nature of subdiffusive processes. The subordinated local time is used to formulate a probabilistic understanding of subdiffusion with reactions, leading to the well known radiation boundary condition. We demonstrate the equivalence between the generalized Feynman-Kac equation with a reflecting boundary and the fractional diffusion equation with a radiation boundary. We solve the former and find the first-reaction probability density in analytic form in the time domain, in terms of the Wright function. We are also able to find the survival probability and subordinated local time density analytically. These results are validated by stochastic simulations that use the subordinated local time description of subdiffusion in the presence of reactions.
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Affiliation(s)
- Toby Kay
- Department of Engineering Mathematics, University of Bristol, Bristol, BS8 1UB UK
| | - Luca Giuggioli
- Department of Engineering Mathematics, University of Bristol, Bristol, BS8 1UB UK
- Bristol Centre for Complexity Sciences, University of Bristol, Bristol, BS8 1UB UK
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3
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Wang X, Chen Y. Ergodic property of random diffusivity system with trapping events. Phys Rev E 2022; 105:014106. [PMID: 35193240 DOI: 10.1103/physreve.105.014106] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2021] [Accepted: 12/10/2021] [Indexed: 12/25/2022]
Abstract
A Brownian yet non-Gaussian phenomenon has recently been observed in many biological and active matter systems. The main idea of explaining this phenomenon is to introduce a random diffusivity for particles moving in inhomogeneous environment. This paper considers a Langevin system containing a random diffusivity and an α-stable subordinator with α<1. This model describes the particle's motion in complex media where both the long trapping events and random diffusivity exist. We derive the general expressions of ensemble- and time-averaged mean-squared displacements which only contain the values of the inverse subordinator and diffusivity. Further taking specific time-dependent diffusivity, we obtain the analytic expressions of ergodicity breaking parameter and probability density function of the time-averaged mean-squared displacement. The results imply the nonergodicity of the random diffusivity model with any kind of diffusivity, including the critical case where the model presents normal diffusion.
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Affiliation(s)
- Xudong Wang
- School of Science, Nanjing University of Science and Technology, Nanjing, 210094, P.R. China
| | - Yao Chen
- College of Sciences, Nanjing Agricultural University, Nanjing, 210094, P.R. China
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4
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Le Vot F, Yuste SB, Abad E, Grebenkov DS. First-encounter time of two diffusing particles in confinement. Phys Rev E 2020; 102:032118. [PMID: 33076026 DOI: 10.1103/physreve.102.032118] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/24/2020] [Accepted: 08/20/2020] [Indexed: 11/07/2022]
Abstract
We investigate how confinement may drastically change both the probability density of the first-encounter time and the associated survival probability in the case of two diffusing particles. To obtain analytical insights into this problem, we focus on two one-dimensional settings: a half-line and an interval. We first consider the case with equal particle diffusivities, for which exact results can be obtained for the survival probability and the associated first-encounter time density valid over the full time domain. We also evaluate the moments of the first-encounter time when they exist. We then turn to the case with unequal diffusivities and focus on the long-time behavior of the survival probability. Our results highlight the great impact of boundary effects in diffusion-controlled kinetics even for simple one-dimensional settings, as well as the difficulty of obtaining analytic results as soon as the translational invariance of such systems is broken.
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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
| | - S B Yuste
- 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 Universidad de Extremadura, E-06800 Mérida, Spain
| | - D S Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, IP Paris, 91128 Palaiseau, France and Institute of Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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5
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Lanoiselée Y, Moutal N, Grebenkov DS. Diffusion-limited reactions in dynamic heterogeneous media. Nat Commun 2018; 9:4398. [PMID: 30353010 PMCID: PMC6199324 DOI: 10.1038/s41467-018-06610-6] [Citation(s) in RCA: 80] [Impact Index Per Article: 13.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2018] [Accepted: 09/11/2018] [Indexed: 12/02/2022] Open
Abstract
Most biochemical reactions in living cells rely on diffusive search for target molecules or regions in a heterogeneous overcrowded cytoplasmic medium. Rapid rearrangements of the medium constantly change the effective diffusivity felt locally by a diffusing particle and thus impact the distribution of the first-passage time to a reaction event. Here, we investigate the effect of these dynamic spatiotemporal heterogeneities onto diffusion-limited reactions. We describe a general mathematical framework to translate many results for ordinary homogeneous Brownian motion to heterogeneous diffusion. In particular, we derive the probability density of the first-passage time to a reaction event and show how the dynamic disorder broadens the distribution and increases the likelihood of both short and long trajectories to reactive targets. While the disorder slows down reaction kinetics on average, its dynamic character is beneficial for a faster search and realization of an individual reaction event triggered by a single molecule. “Diffusing diffusivity” concept has been recently put forward to account for rapid structural rearrangements in soft matter and biological systems. Here the authors propose a general mathematical framework to compute the distribution of first-passage times in a dynamically heterogeneous medium.
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Affiliation(s)
- Yann Lanoiselée
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, University Paris-Saclay, 91128, Palaiseau, France
| | - Nicolas Moutal
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, University Paris-Saclay, 91128, Palaiseau, France
| | - Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, University Paris-Saclay, 91128, Palaiseau, France.
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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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7
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Grebenkov DS. First passage times for multiple particles with reversible target-binding kinetics. J Chem Phys 2017; 147:134112. [DOI: 10.1063/1.4996395] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022] Open
Affiliation(s)
- Denis S. Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS–Ecole Polytechnique, University Paris-Saclay, 91128 Palaiseau,
France and Interdisciplinary Scientific Center Poncelet (ISCP) (UMI 2615 CNRS/IUM/IITP RAS/Steklov MI RAS/Skoltech/HSE), Bolshoy
Vlasyevskiy Pereulok 11, 119002 Moscow, Russia
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8
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Grebenkov DS. Analytical representations of the spread harmonic measure density. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:052108. [PMID: 26066120 DOI: 10.1103/physreve.91.052108] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2015] [Indexed: 06/04/2023]
Abstract
We study the spread harmonic measure that characterizes the spatial distribution of reaction events on a partially reactive surface. For Euclidean domains in which Brownian motion can be split into independent lateral and transverse displacements, we derive analytical formulas for the spread harmonic measure density and analyze its asymptotic behavior. This analysis is applicable to slab domains, general cylindrical domains, and a half-space. We investigate the spreading effect due to multiple reflections on the surface, and the underlying role of finite reactivity. We discuss further extensions and applications of analytical results to describe Laplacian transfer phenomena such as permeation through semipermeable membranes, secondary current distribution on partially blocking electrodes, and surface relaxation in nuclear magnetic resonance.
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Affiliation(s)
- Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, 91128 Palaiseau, France and St. Petersburg National Research University of Information Technologies, Mechanics and Optics, 197101 St. Petersburg, Russia
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9
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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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10
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Massignan P, Manzo C, Torreno-Pina JA, García-Parajo MF, Lewenstein M, Lapeyre GJ. Nonergodic subdiffusion from Brownian motion in an inhomogeneous medium. PHYSICAL REVIEW LETTERS 2014; 112:150603. [PMID: 24785018 DOI: 10.1103/physrevlett.112.150603] [Citation(s) in RCA: 90] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2014] [Indexed: 06/03/2023]
Abstract
Nonergodicity observed in single-particle tracking experiments is usually modeled by transient trapping rather than spatial disorder. We introduce models of a particle diffusing in a medium consisting of regions with random sizes and random diffusivities. The particle is never trapped but rather performs continuous Brownian motion with the local diffusion constant. Under simple assumptions on the distribution of the sizes and diffusivities, we find that the mean squared displacement displays subdiffusion due to nonergodicity for both annealed and quenched disorder. The model is formulated as a walk continuous in both time and space, similar to the Lévy walk.
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Affiliation(s)
- P Massignan
- ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels, Spain
| | - C Manzo
- ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels, Spain
| | - J A Torreno-Pina
- ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels, Spain
| | - M F García-Parajo
- ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels, Spain and ICREA-Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain
| | - M Lewenstein
- ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels, Spain and ICREA-Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain
| | - G J Lapeyre
- ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels, Spain
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11
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Spendier K, Kenkre VM. Analytic solutions for some reaction-diffusion scenarios. J Phys Chem B 2013; 117:15639-50. [PMID: 23883422 DOI: 10.1021/jp406322t] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Motivated currently by the problem of coalescence of receptor clusters in mast cells in the general subject of immune reactions, and formerly by the investigation of exciton trapping and sensitized luminescence in molecular systems and aggregates, we present analytic expressions for survival probabilities of moving entities undergoing diffusion and reaction on encounter. Results we provide cover several novel situations in simple 1-d systems as well as higher-dimensional counterparts along with a useful compendium of such expressions in chemical physics and allied fields. We also emphasize the importance of the relationship of discrete sink term analysis to continuum boundary condition studies.
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Affiliation(s)
- Kathrin Spendier
- Consortium of the Americas for Interdisciplinary Science, University of New Mexico , Albuquerque, New Mexico 87131, United States
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12
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Abad E, Yuste SB, Lindenberg K. Evanescent continuous-time random walks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:062110. [PMID: 24483389 DOI: 10.1103/physreve.88.062110] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/29/2013] [Indexed: 06/03/2023]
Abstract
We study how an evanescence process affects the number of distinct sites visited by a continuous-time random walker in one dimension. We distinguish two very different cases, namely, when evanescence can only occur concurrently with a jump, and when evanescence can occur at any time. The first is characteristic of trapping processes on a lattice, whereas the second is associated with spontaneous death processes such as radioactive decay. In both of these situations we consider three different forms of the waiting time distribution between jumps, namely, exponential, long tailed, and ultraslow.
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Affiliation(s)
- E Abad
- Departamento de Física Aplicada, Centro Universitario de Mérida, Universidad de Extremadura, E-06800 Mérida, Spain
| | - S B Yuste
- Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain
| | - Katja Lindenberg
- Department of Chemistry and Biochemistry and BioCircuits Institute, University of California San Diego, La Jolla, California 92093-0340, USA
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13
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Eliazar II, Cohen MH. Topography of chance. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:052104. [PMID: 24329211 DOI: 10.1103/physreve.88.052104] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2013] [Indexed: 06/03/2023]
Abstract
We present a model of multiplicative Langevin dynamics that is based on two foundations: the Langevin equation and the notion of multiplicative evolution. The model is a nonlinear mechanism transforming a white-noise input to a dynamic-equilibrium output, using a single control: an underlying convex U-shaped potential function. The output is quantified by a stationary density which can attain a given number of shapes and a given number of randomness categories. The model generates each admissible combination of the output's shape and randomness in a universal and robust fashion. Moreover, practically all the probability distributions that are supported on the positive half-line, and that are commonly encountered and applied across the sciences, can be reverse engineered by this model. Hence, this model is a universal equilibrium mechanism, in the context of multiplicative dynamics, for the robust generation of "chance": the model's output. In turn, the properties of the produced "chance," the output's shape and randomness, are determined with mathematical precision by the control's landscape, its topography. Thus, a topographic map of chance is established. As a particular application, probability distributions with power-law tails are shown to be universally and robustly generated by controls on the "edge of convexity": convex U-shaped potential functions with asymptotically linear wings.
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Affiliation(s)
- Iddo I Eliazar
- School of Chemistry, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
| | - Morrel H Cohen
- Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA and Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
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Eliazar I. Poissonian renormalizations, exponentials, and power laws. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:052125. [PMID: 23767505 DOI: 10.1103/physreve.87.052125] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/25/2013] [Indexed: 06/02/2023]
Abstract
This paper presents a comprehensive "renormalization study" of Poisson processes governed by exponential and power-law intensities. These Poisson processes are of fundamental importance, as they constitute the very bedrock of the universal extreme-value laws of Gumbel, Fréchet, and Weibull. Applying the method of Poissonian renormalization we analyze the emergence of these Poisson processes, unveil their intrinsic dynamical structures, determine their domains of attraction, and characterize their structural phase transitions. These structural phase transitions are shown to be governed by uniform and harmonic intensities, to have universal domains of attraction, to uniquely display intrinsic invariance, and to be intimately connected to "white noise" and to "1/f noise." Thus, we establish a Poissonian explanation to the omnipresence of white and 1/f noises.
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Affiliation(s)
- Iddo Eliazar
- Holon Institute of Technology, P.O. Box 305, Holon 58102, Israel.
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15
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Abad E, Yuste SB, Lindenberg K. Survival probability of an immobile target in a sea of evanescent diffusive or subdiffusive traps: a fractional equation approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:061120. [PMID: 23367906 DOI: 10.1103/physreve.86.061120] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/03/2012] [Indexed: 06/01/2023]
Abstract
We calculate the survival probability of an immobile target surrounded by a sea of uncorrelated diffusive or subdiffusive evanescent traps (i.e., traps that disappear in the course of their motion). Our calculation is based on a fractional reaction-subdiffusion equation derived from a continuous time random walk model of the system. Contrary to an earlier method valid only in one dimension (d=1), the equation is applicable in any Euclidean dimension d and elucidates the interplay between anomalous subdiffusive transport, the irreversible evanescence reaction, and the dimension in which both the traps and the target are embedded. Explicit results for the survival probability of the target are obtained for a density ρ(t) of traps which decays (i) exponentially and (ii) as a power law. In the former case, the target has a finite asymptotic survival probability in all integer dimensions, whereas in the latter case there are several regimes where the values of the decay exponent for ρ(t) and the anomalous diffusion exponent of the traps determine whether or not the target has a chance of eternal survival in one, two, and three dimensions.
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Affiliation(s)
- E Abad
- Departamento de Física Aplicada, Centro Universitario de Mérida, Universidad de Extremadura, E-06800 Mérida, Spain
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16
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Eliazar I. Geometric theory for Weibull's distribution. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:031103. [PMID: 23030862 DOI: 10.1103/physreve.86.031103] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/16/2012] [Revised: 06/21/2012] [Indexed: 06/01/2023]
Abstract
Weibull's distribution is the principal phenomenological law of relaxation in the physical sciences and spans three different relaxation regimes: subexponential ("stretched exponential"), exponential, and superexponential. The probabilistic theory of extreme-value statistics asserts that the linear scaling limits of minima of ensembles of positive-valued random variables, which are independent and identically distributed, are universally governed by Weibull's distribution. However, this probabilistic theory does not take into account spatial geometry, which often plays a key role in the physical sciences. In this paper we present a general and versatile model of random reactions in random environments and establish a geometry-based theory for the universal emergence of Weibull's distribution.
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Affiliation(s)
- Iddo Eliazar
- Holon Institute of Technology, P. O. Box 305, Holon 58102, Israel.
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17
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Wang Y, Zhao N, Yan Y. Langevin dynamics of correlated subdiffusion and normal diffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:041142. [PMID: 22680452 DOI: 10.1103/physreve.85.041142] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/15/2011] [Revised: 11/29/2011] [Indexed: 06/01/2023]
Abstract
We analyze the dissipative dynamics of a particle governed by a two-dimensional generalized Langevin equation with coupled fractional Gaussian noise and white noise in its respective coordinates, assuming the lowest-order coupling form. Two situations are studied: In the first the particle is free from external force and in the second the particle is subject to a two-dimensional harmonic potential. We derive the general expressions for the mean values, variances, and velocity autocorrelation function and evaluate their temporal evolutions via the numerical Laplace inversion technique. Through the analytical results of the short-time and long-time behaviors, we also explicitly elucidate the effects of fluctuation correlation coupling and interoscillator coupling on the dynamic behaviors of the particle. It is shown that in both situations the couplings do not affect the short-time behavior of self-diffusions in each coordinate, and the subdiffusive and normal diffusive features of these processes resemble those in a one-dimensional system with fractional Gaussian noise and white noise, respectively. However, over a long time period, the fluctuation correlation extends the characteristic time scales for the self-diffusions of a free particle; while only the interoscillator coupling induces a retardation of the relaxation processes of a bounded particle toward equilibrium. Moreover, both couplings generate a cross diffusion, whose long-time approximation has two possible forms, the selection of which depends on the relevant time scales of self-diffusions in each coordinate.
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Affiliation(s)
- Yingxi Wang
- College of Chemistry, Sichuan University, Chengdu 610064, China
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18
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Eliazar I. Langevin dynamics, entropic crowding, and stochastic cloaking. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:061132. [PMID: 22304065 DOI: 10.1103/physreve.84.061132] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/22/2011] [Indexed: 05/31/2023]
Abstract
We consider a pack of independent probes--within a spatially inhomogeneous thermal bath consisting of a vast number of randomly moving particles--which are subjected to an external force. The stochastic dynamics of the probes are governed by Langevin's equation. The probes attain a steady state distribution which, in general, is different than the concentration of the particles in the spatially inhomogeneous thermal bath. In this paper we explore the state of "entropic crowding" in which the probes' distribution and the particles' concentration coincide--thus yielding maximal relative entropies of one with respect to the other. Entropic crowding can be attained by two scenarios which are analyzed in detail: (i) "entropically crowding thermal baths"--in which the particles crowd uniformly around the probes; (ii) "entropically crowding Langevin forces"--in which the probes crowd uniformly amongst the particles. Entropic crowding is equivalent to the optimal stochastic cloaking of the probes within the spatially inhomogeneous thermal bath.
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Affiliation(s)
- Iddo Eliazar
- Department of Technology Management, Holon Institute of Technology, PO Box 305, Holon 58102, Israel.
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Yuste SB, Abad E, Lindenberg K. Reaction-subdiffusion model of morphogen gradient formation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:061123. [PMID: 21230660 DOI: 10.1103/physreve.82.061123] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/01/2010] [Revised: 11/18/2010] [Indexed: 05/30/2023]
Abstract
We study gradient formation of subdiffusive morphogens. The morphogens are produced at a source point at a constant rate. From there they move subdiffusively and are also subject to degradation at a rate that may depend on location and on time. Our analysis is based on a reaction-subdiffusion equation obtained from a continuous time random-walk model with a long-tailed waiting time distribution that also incorporates an evanescence process. Spatially uniform degradation at a constant rate leads to an exponentially decreasing stationary concentration profile hardly distinguishable from that obtained with normal diffusion. On the other hand, with location-dependent degradation we find a rich gamut of profiles, some qualitatively quite different from those occurring with normal diffusion. We conclude that long-time morphogen concentration profiles are very sensitive to the spatial dependence of the reactivity and may also serve as a sensitive measure of the occurrence of anomalous diffusion.
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Affiliation(s)
- S B Yuste
- Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain
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Abad E, Yuste SB, Lindenberg K. Reaction-subdiffusion and reaction-superdiffusion equations for evanescent particles performing continuous-time random walks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:031115. [PMID: 20365705 DOI: 10.1103/physreve.81.031115] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/16/2009] [Indexed: 05/29/2023]
Abstract
Starting from a continuous-time random-walk (CTRW) model of particles that may evanesce as they walk, our goal is to arrive at macroscopic integrodifferential equations for the probability density for a particle to be found at point r at time t given that it started its walk from r_{0} at time t=0 . The passage from the CTRW to an integrodifferential equation is well understood when the particles are not evanescent. Depending on the distribution of stepping times and distances, one arrives at standard macroscopic equations that may be "normal" (diffusion) or "anomalous" (subdiffusion and/or superdiffusion). The macroscopic description becomes considerably more complicated and not particularly intuitive if the particles can die during their walk. While such equations have been derived for specific cases, e.g., for location-independent exponential evanescence, we present a more general derivation valid under less stringent constraints than those found in the current literature.
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Affiliation(s)
- E Abad
- Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain
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Yuste SB, Borrego R, Abad E. Divergent series and memory of the initial condition in the long-time solution of some anomalous diffusion problems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:021105. [PMID: 20365528 DOI: 10.1103/physreve.81.021105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/04/2009] [Indexed: 05/29/2023]
Abstract
We consider various anomalous d -dimensional diffusion problems in the presence of an absorbing boundary with radial symmetry. The motion of particles is described by a fractional diffusion equation. Their mean-square displacement is given by r(2) proportional, variant t(gamma)(0<gamma< or =1) , resulting in normal diffusive motion if gamma=1 and subdiffusive motion otherwise. For the subdiffusive case in sufficiently high dimensions, divergent series appear when the concentration or survival probabilities are evaluated via the method of separation of variables. While the solution for normal diffusion problems is, at most, divergent as t-->0 , the emergence of such series in the long-time domain is a specific feature of subdiffusion problems. We present a method to regularize such series, and, in some cases, validate the procedure by using alternative techniques (Laplace transform method and numerical simulations). In the normal diffusion case, we find that the signature of the initial condition on the approach to the steady state rapidly fades away and the solution approaches a single (the main) decay mode in the long-time regime. In remarkable contrast, long-time memory of the initial condition is present in the subdiffusive case as the spatial part Psi1(r) describing the long-time decay of the solution to the steady state is determined by a weighted superposition of all spatial modes characteristic of the normal diffusion problem, the weight being dependent on the initial condition. Interestingly, Psi1(r) turns out to be independent of the anomalous diffusion exponent gamma .
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Affiliation(s)
- S Bravo Yuste
- Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain
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Grebenkov DS. Subdiffusion in a bounded domain with a partially absorbing-reflecting boundary. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:021128. [PMID: 20365551 DOI: 10.1103/physreve.81.021128] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2009] [Revised: 01/15/2010] [Indexed: 05/29/2023]
Abstract
The exit time of a subdiffusive process from a bounded domain with a partially absorbing/reflecting boundary is considered. The short-time and long-time behaviors of the exit time probability density are investigated by using a spectral decomposition on the basis of the Laplace operator eigenfunctions. Rotation-invariant domains are analyzed in depth in order to illustrate the use of theoretical formulas and to compare them to numerical simulations. The asymptotic results obtained are relevant for describing subdiffusion inside a living cell with a semipermeable membrane, in a chemical reactor filled with catalytic grains of finite reactivity, or in mineral or biological samples which are probed by nuclear magnetic resonance measurements subject to surface relaxation.
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Affiliation(s)
- Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée, C.N.R.S.-Ecole Polytechnique, 91128 Palaiseau, France.
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Grebenkov DS. Searching for partially reactive sites: Analytical results for spherical targets. J Chem Phys 2010; 132:034104. [DOI: 10.1063/1.3294882] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Borrego R, Abad E, Yuste SB. Survival probability of a subdiffusive particle in a d-dimensional sea of mobile traps. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:061121. [PMID: 20365132 DOI: 10.1103/physreve.80.061121] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/22/2009] [Indexed: 05/29/2023]
Abstract
We investigate the long-time behavior of the survival probability P(t) of a mobile particle in d-dimensional continuous Euclidean media doped with noninteracting mobile traps. The particle is strictly subdiffusive, implying that its mean-square displacement grows as tgamma' with 0<gamma'<1. Initially, the traps are scattered randomly and their subsequent mean-square displacement grows as tgamma with 0<gamma<or=1. Instantaneous annihilation of the particle takes place upon contact with any of the traps. The solution to this problem is obtained by deriving lower and upper asymptotic bounds of the survival probability and showing that they converge to one another for long times, thereby unambiguously determining the long-time decay of P(t). For d>or=2 we find that at late times the survival probability is that of the pure target problem (the problem where the particle remains immobile) in agreement with previous studies for the d=1 case. These decay laws remain invariant over the whole gamma range as opposed to the dynamical crossover observed for the case of a purely diffusive particle (gamma'=1) where, for gamma<2/(2+d) , the survival probability becomes that of the so-called trapping problem (the problem where the particle moves in a sea of static traps). This behavior implies that for sufficiently low values of gamma(gamma<2/(2+d)) the survival probability becomes singular in the limit gamma'-->1: trappinglike for gamma'=1 and targetlike for any gamma'<1.
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Affiliation(s)
- R Borrego
- Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain
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Zaid IM, Lomholt MA, Metzler R. How subdiffusion changes the kinetics of binding to a surface. Biophys J 2009; 97:710-21. [PMID: 19651029 DOI: 10.1016/j.bpj.2009.05.022] [Citation(s) in RCA: 40] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/23/2009] [Revised: 02/23/2009] [Accepted: 05/05/2009] [Indexed: 11/29/2022] Open
Abstract
Under molecular crowding conditions, biopolymers have been reported to subdiffuse, (r(2)(t)) approximately = t(alpha), with 0 <alpha < 1. Here we study the exchange dynamics of such a subdiffusing particle with a reactive boundary using a continuous time random walk approach. We derive the generalized boundary condition and consider the unbinding from the boundary. An ensuing weak ergodicity breaking has profound consequences for material exchange between the boundary and bulk. We discuss the effects in biological contexts such as gene regulation or membrane-bulk exchange processes. We also suggest various methods to experimentally probe the subdiffusive behavior.
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Affiliation(s)
- Irwin M Zaid
- Physics Department, Technical University of Munich, Garching, Germany
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He Y, Burov S, Metzler R, Barkai E. Random time-scale invariant diffusion and transport coefficients. PHYSICAL REVIEW LETTERS 2008; 101:058101. [PMID: 18764430 DOI: 10.1103/physrevlett.101.058101] [Citation(s) in RCA: 287] [Impact Index Per Article: 17.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/29/2008] [Indexed: 05/26/2023]
Abstract
Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement delta2[over ] of individual particles remains a random variable while indicating that the particle motion is subdiffusive. We investigate this type of ergodicity breaking within the continuous time random walk model and show that delta2[over ] differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable delta2[over ]. Similarly we quantify the response to a constant external field, revealing a generalization of the Einstein relation. Consequences for the interpretation of single molecule tracking data are discussed.
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Affiliation(s)
- Y He
- Department of Physics, Bar Ilan University, Ramat-Gan, Israel
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Yuste SB, Oshanin G, Lindenberg K, Bénichou O, Klafter J. Survival probability of a particle in a sea of mobile traps: a tale of tails. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:021105. [PMID: 18850784 DOI: 10.1103/physreve.78.021105] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/19/2008] [Indexed: 05/26/2023]
Abstract
We study the long-time tails of the survival probability P(t) of an A particle diffusing in d-dimensional media in the presence of a concentration rho of traps B that move subdiffusively, such that the mean square displacement of each trap grows as tgamma with 0 < or = gamma < or =1. Starting from a continuous time random walk description of the motion of the particle and of the traps, we derive lower and upper bounds for P(t) and show that for gamma < or =2/(d+2) these bounds coincide asymptotically, thus determining asymptotically exact results. The asymptotic decay law in this regime is exactly that obtained for immobile traps. This means that for sufficiently subdiffusive traps, the moving A particle sees the traps as essentially immobile, and Lifshitz or trapping tails remain unchanged. For gamma >2/(d+2) and d< or =2 the upper and lower bounds again coincide, leading to a decay law equal to that of a stationary particle. Thus, in this regime the moving traps see the particle as essentially immobile. For d>2 , however, the upper and lower bounds in this gamma regime no longer coincide, and the decay law for the survival probability of the A particle remains ambiguous.
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Affiliation(s)
- S B Yuste
- Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain
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