1
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Baule A, Sollich P. Exponential increase of transition rates in metastable systems driven by non-Gaussian noise. Sci Rep 2023; 13:3853. [PMID: 36890184 PMCID: PMC9995508 DOI: 10.1038/s41598-023-30577-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/09/2022] [Accepted: 02/27/2023] [Indexed: 03/10/2023] Open
Abstract
Noise-induced escape from metastable states governs a plethora of transition phenomena in physics, chemistry, and biology. While the escape problem in the presence of thermal Gaussian noise has been well understood since the seminal works of Arrhenius and Kramers, many systems, in particular living ones, are effectively driven by non-Gaussian noise for which the conventional theory does not apply. Here we present a theoretical framework based on path integrals that allows the calculation of both escape rates and optimal escape paths for a generic class of non-Gaussian noises. We find that non-Gaussian noise always leads to more efficient escape and can enhance escape rates by many orders of magnitude compared with thermal noise, highlighting that away from equilibrium escape rates cannot be reliably modelled based on the traditional Arrhenius-Kramers result. Our analysis also identifies a new universality class of non-Gaussian noises, for which escape paths are dominated by large jumps.
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Affiliation(s)
- Adrian Baule
- School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, UK.
| | - Peter Sollich
- Institute for Theoretical Physics, Georg-August-University Göttingen, 37077, Göttingen, Germany.,Department of Mathematics, King's College London, London, WC2R 2LS, UK
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2
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van Kan A, Alexakis A, Brachet ME. Lévy on-off intermittency. Phys Rev E 2021; 103:052115. [PMID: 34134220 DOI: 10.1103/physreve.103.052115] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/12/2021] [Accepted: 04/27/2021] [Indexed: 11/07/2022]
Abstract
We present an alternative form of intermittency, Lévy on-off intermittency, which arises from multiplicative α-stable white noise close to an instability threshold. We study this problem in the linear and nonlinear regimes, both theoretically and numerically, for the case of a pitchfork bifurcation with fluctuating growth rate. We compute the stationary distribution analytically and numerically from the associated fractional Fokker-Planck equation in the Stratonovich interpretation. We characterize the system in the parameter space (α,β) of the noise, with stability parameter α∈(0,2) and skewness parameter β∈[-1,1]. Five regimes are identified in this parameter space, in addition to the well-studied Gaussian case α=2. Three regimes are located at 1<α<2, where the noise has finite mean but infinite variance. They are differentiated by β and all display a critical transition at the deterministic instability threshold, with on-off intermittency close to onset. Critical exponents are computed from the stationary distribution. Each regime is characterized by a specific form of the density and specific critical exponents, which differ starkly from the Gaussian case. A finite or infinite number of integer-order moments may converge, depending on parameters. Two more regimes are found at 0<α≤1. There, the mean of the noise diverges, and no critical transition occurs. In one case, the origin is always unstable, independently of the distance μ from the deterministic threshold. In the other case, the origin is conversely always stable, independently of μ. We thus demonstrate that an instability subject to nonequilibrium, power-law-distributed fluctuations can display substantially different properties than for Gaussian thermal fluctuations, in terms of statistics and critical behavior.
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Affiliation(s)
- Adrian van Kan
- Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
| | - Alexandros Alexakis
- Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
| | - Marc-Etienne Brachet
- Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
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3
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Dubkov AA, Dybiec B, Spagnolo B, Kharcheva A, Guarcello C, Valenti D. Statistics of residence time for Lévy flights in unstable parabolic potentials. Phys Rev E 2020; 102:042142. [PMID: 33212570 DOI: 10.1103/physreve.102.042142] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/21/2020] [Accepted: 09/22/2020] [Indexed: 11/07/2022]
Abstract
We analyze the residence time problem for an arbitrary Markovian process describing nonlinear systems without a steady state. We obtain exact analytical results for the statistical characteristics of the residence time. For diffusion in a fully unstable potential profile in the presence of Lévy noise we get the conditional probability density of the particle position and the average residence time. The noise-enhanced stability phenomenon is observed in the system investigated. Results from numerical simulations are in very good agreement with analytical ones.
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Affiliation(s)
- Alexander A Dubkov
- Radiophysics Department, Lobachevsky State University of Nizhni Novgorod, Gagarin Avenue 23, 603950 Nizhni Novgorod, Russia
| | - Bartłomiej Dybiec
- Institute of Theoretical Physics and Mark Kac Center for Complex Systems Research, Jagiellonian University, ul. St. Łojasiewicza 11, 30-348 Kraków, Poland
| | - Bernardo Spagnolo
- Radiophysics Department, Lobachevsky State University of Nizhni Novgorod, Gagarin Avenue 23, 603950 Nizhni Novgorod, Russia.,Dipartimento di Fisica e Chimica "Emilio Segrè," Group of Interdisciplinary Theoretical Physics, Università di Palermo and CNISM, Unità di Palermo, Viale delle Scienze, Edificio 18, I-90128 Palermo, Italy.,Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Via S. Sofia 64, I-90123 Catania, Italy
| | - Anna Kharcheva
- Radiophysics Department, Lobachevsky State University of Nizhni Novgorod, Gagarin Avenue 23, 603950 Nizhni Novgorod, Russia.,Dipartimento di Fisica e Chimica "Emilio Segrè," Group of Interdisciplinary Theoretical Physics, Università di Palermo and CNISM, Unità di Palermo, Viale delle Scienze, Edificio 18, I-90128 Palermo, Italy
| | - Claudio Guarcello
- Dipartimento di Fisica "E. R. Caianiello," Università di Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy.,INFN, Sezione di Napoli Gruppo Collegato di Salerno, Complesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy
| | - Davide Valenti
- Dipartimento di Fisica e Chimica "Emilio Segrè," Group of Interdisciplinary Theoretical Physics, Università di Palermo and CNISM, Unità di Palermo, Viale delle Scienze, Edificio 18, I-90128 Palermo, Italy.,CNR-IRIB, Consiglio Nazionale delle Ricerche-Istituto per la Ricerca e l'Innovazione Biomedica, Via Ugo La Malfa 153, 90146 Palermo, Italy
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4
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Capała K, Dybiec B, Gudowska-Nowak E. Peculiarities of escape kinetics in the presence of athermal noises. CHAOS (WOODBURY, N.Y.) 2020; 30:013127. [PMID: 32013513 DOI: 10.1063/1.5126263] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/31/2019] [Accepted: 12/19/2019] [Indexed: 06/10/2023]
Abstract
Stochastic evolution of various dynamic systems and reaction networks is commonly described in terms of noise assisted escape of an overdamped particle from a potential well, as devised by the paradigmatic Langevin equation in which additive Gaussian stochastic force reproduces effects of thermal fluctuations from the reservoir. When implemented for systems close to equilibrium, the approach correctly explains the emergence of the Boltzmann distribution for the ensemble of trajectories generated by the Langevin equation and relates the intensity of the noise strength to the mobility. This scenario can be further generalized to include effects of non-Gaussian, burstlike forcing modeled by Lévy noise. In this case, however, the pulsatile additive noise cannot be treated as the internal (thermal) since the relation between the strength of the friction and variance of the noise is violated. Heavy tails of Lévy noise distributions not only facilitate escape kinetics, but also, more importantly, change the escape protocol by altering the final stationary state to a non-Boltzmann, nonequilibrium form. As a result, contrary to the kinetics induced by a Gaussian white noise, escape rates in environments with Lévy noise are determined not by the barrier height, but instead by the barrier width. We further discuss consequences of simultaneous action of thermal and Lévy noises on statistics of passage times and population of reactants in double-well potentials.
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Affiliation(s)
- Karol Capała
- Marian Smoluchowski Institute of Physics and Mark Kac Center for Complex Systems Research, Jagiellonian University, ul. St. Łojasiewicza 11, 30-348 Kraków, Poland
| | - Bartłomiej Dybiec
- Marian Smoluchowski Institute of Physics and Mark Kac Center for Complex Systems Research, Jagiellonian University, ul. St. Łojasiewicza 11, 30-348 Kraków, Poland
| | - Ewa Gudowska-Nowak
- Marian Smoluchowski Institute of Physics and Mark Kac Center for Complex Systems Research, Jagiellonian University, ul. St. Łojasiewicza 11, 30-348 Kraków, Poland
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5
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Cieśla M, Capała K, Dybiec B. Multimodal stationary states under Cauchy noise. Phys Rev E 2019; 99:052118. [PMID: 31212488 DOI: 10.1103/physreve.99.052118] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/20/2019] [Indexed: 06/09/2023]
Abstract
A Lévy noise is an efficient description of out-of-equilibrium systems. The presence of Lévy flights results in a plenitude of noise-induced phenomena. Among others, Lévy flights can produce stationary states with more than one modal value in single-well potentials. Here we explore stationary states in special double-well potentials demonstrating that a sufficiently high potential barrier separating potential wells can produce bimodal stationary states in each potential well. Furthermore, we explore how the decrease in the barrier height affects the multimodality of stationary states. Finally, we explore the role of multimodality of stationary states on noise-induced escape over the static potential barrier.
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Affiliation(s)
- Michał Cieśla
- Marian Smoluchowski Institute of Physics, and Mark Kac Center for Complex Systems Research, Jagiellonian University, ul. St. Łojasiewicza 11, 30-348 Kraków, Poland
| | - Karol Capała
- Marian Smoluchowski Institute of Physics, and Mark Kac Center for Complex Systems Research, Jagiellonian University, ul. St. Łojasiewicza 11, 30-348 Kraków, Poland
| | - Bartłomiej Dybiec
- Marian Smoluchowski Institute of Physics, and Mark Kac Center for Complex Systems Research, Jagiellonian University, ul. St. Łojasiewicza 11, 30-348 Kraków, Poland
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6
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Guarcello C, Valenti D, Spagnolo B, Pierro V, Filatrella G. Anomalous transport effects on switching currents of graphene-based Josephson junctions. NANOTECHNOLOGY 2017; 28:134001. [PMID: 28164862 DOI: 10.1088/1361-6528/aa5e75] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
We explore the effect of noise on the ballistic graphene-based small Josephson junctions in the framework of the resistively and capacitively shunted model. We use the non-sinusoidal current-phase relation specific for graphene layers partially covered by superconducting electrodes. The noise induced escapes from the metastable states, when the external bias current is ramped, given the switching current distribution, i.e. the probability distribution of the passages to finite voltage from the superconducting state as a function of the bias current, that is the information more promptly available in the experiments. We consider a noise source that is a mixture of two different types of processes: a Gaussian contribution to simulate an uncorrelated ordinary thermal bath, and non-Gaussian, α-stable (or Lévy) term, generally associated to non-equilibrium transport phenomena. We find that the analysis of the switching current distribution makes it possible to efficiently detect a non-Gaussian noise component in a Gaussian background.
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Affiliation(s)
- Claudio Guarcello
- SPIN-CNR, Via Dodecaneso 33, I-16146 Genova, Italy. NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza S. Silvestro 12, I-56127 Pisa, Italy. Radiophysics Department, Lobachevsky State University, Nizhniy Novgorod, Russia
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7
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The Switch in a Genetic Toggle System with Lévy Noise. Sci Rep 2016; 6:31505. [PMID: 27539010 PMCID: PMC4990962 DOI: 10.1038/srep31505] [Citation(s) in RCA: 67] [Impact Index Per Article: 8.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2016] [Accepted: 07/21/2016] [Indexed: 11/08/2022] Open
Abstract
A bistable toggle switch is a paradigmatic model in the field of biology. The dynamics of the system induced by Gaussian noise has been intensively investigated, but Gaussian noise cannot incorporate large bursts typically occurring in real experiments. This paper aims to examine effects of variations from one protein imposed by a non-Gaussian Lévy noise, which is able to describe even large jumps, on the coherent switch and the on/off switch via the steady-state probability density, the joint steady-state probability density, and the mean first passage time. We find that a large burst of one protein due to the Lévy noises can induce coherent switches even with small noise intensities in contrast to the Gaussian case which requires large intensities for this. The influences of the stability index, skewness parameter and noise intensity on the on/off switch are analyzed, leading to an adjustment of the concentrations of both proteins and a decision which stable point to stay most. The mean first passage times show complex effects under Lévy noise, especially the stability index and skewness parameter. Our results also imply that the presence of non-Gaussian Lévy noises has fundamentally changed the escape mechanism in such a system compared with Gaussian noise.
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8
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Srokowski T. Escape process in systems characterized by stable noises and position-dependent resting times. Phys Rev E 2016; 93:062142. [PMID: 27415243 DOI: 10.1103/physreve.93.062142] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2016] [Indexed: 11/07/2022]
Abstract
Stochastic systems characterized by a random driving in a form of the general stable noise are considered. The particle experiences long rests due to the traps the density of which is position dependent and obeys a power-law form attributed to the underlying self-similar structure. Both the one- and two-dimensional cases are analyzed. The random walk description involves a position-dependent waiting time distribution. On the other hand, the stochastic dynamics is formulated in terms of the subordination technique where the random time generator is position dependent. The first passage time problem is addressed by evaluating a first passage time density distribution and an escape rate. The influence of the medium nonhomogeneity on those quantities is demonstrated; moreover, the dependence of the escape rate on the stability index and the memory parameter is evaluated. Results indicate essential differences between the Gaussian case and the case involving Lévy flights.
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Affiliation(s)
- Tomasz Srokowski
- Institute of Nuclear Physics, Polish Academy of Sciences, PL - 31-342 Kraków, Poland
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9
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Martin RJ, Craster RV, Kearney MJ. Infinite product expansion of the Fokker–Planck equation with steady-state solution. Proc Math Phys Eng Sci 2015; 471:20150084. [PMID: 26346100 PMCID: PMC4528656 DOI: 10.1098/rspa.2015.0084] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/07/2015] [Accepted: 05/22/2015] [Indexed: 11/24/2022] Open
Abstract
We present an analytical technique for solving Fokker–Planck equations that have a steady-state solution by representing the solution as an infinite product rather than, as usual, an infinite sum. This method has many advantages: automatically ensuring positivity of the resulting approximation, and by design exactly matching both the short- and long-term behaviour. The efficacy of the technique is demonstrated via comparisons with computations of typical examples.
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Affiliation(s)
- R. J. Martin
- Apollo Global Management International LLP, 25 St George Street, London W1S 1FS, UK
- Department of Mathematics, Imperial College London, South Kensington, London SW7 2AZ, UK
| | - R. V. Craster
- Department of Mathematics, Imperial College London, South Kensington, London SW7 2AZ, UK
| | - M. J. Kearney
- Faculty of Engineering and Physical Sciences, University of Surrey, Guildford GU2 7XH, UK
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10
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Lisowski B, Valenti D, Spagnolo B, Bier M, Gudowska-Nowak E. Stepping molecular motor amid Lévy white noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042713. [PMID: 25974533 DOI: 10.1103/physreve.91.042713] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2014] [Indexed: 06/04/2023]
Abstract
We consider a model of a stepping molecular motor consisting of two connected heads. Directional motion of the stepper takes place along a one-dimensional track. Each head is subject to a periodic potential without spatial reflection symmetry. When the potential for one head is switched on, it is switched off for the other head. Additionally, the system is subject to the influence of symmetric, white Lévy noise that mimics the action of external random forcing. The stepper exhibits motion with a preferred direction which is examined by analyzing the median of the displacement of a midpoint between the positions of the two heads. We study the modified dynamics of the stepper by numerical simulations. We find flux reversals as noise parameters are changed. Speed and direction appear to very sensitively depend on characteristics of the noise.
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Affiliation(s)
- Bartosz Lisowski
- M. Smoluchowski Institute of Physics Jagiellonian University, Kraków, Poland and Unit of Pharmacoepidemiology and Pharmacoeconomics, Faculty of Pharmacy, Jagiellonian University Medical College, Kraków, Poland
| | - Davide Valenti
- Dipartimento di Fisica e Chimica, Group of Interdisciplinary Theoretical Physics, Viale delle Scienze, Ed. 18, Università di Palermo and CNISM, Unità di Palermo, Palermo I-90128, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Via S. Sofia 64, I-90123 Catania, Italy
| | - Bernardo Spagnolo
- Dipartimento di Fisica e Chimica, Group of Interdisciplinary Theoretical Physics, Viale delle Scienze, Ed. 18, Università di Palermo and CNISM, Unità di Palermo, Palermo I-90128, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Via S. Sofia 64, I-90123 Catania, Italy
| | - Martin Bier
- Department of Physics, East Carolina University, Greenville, North Carolina 27858, USA
| | - Ewa Gudowska-Nowak
- Mark Kac Center for Complex Systems Research and Malopolska Center of Biotechnology, Jagiellonian University, Kraków, Poland
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11
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12
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Szczepaniec K, Dybiec B. Quantifying a resonant-activation-like phenomenon in non-Markovian systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:042138. [PMID: 24827223 DOI: 10.1103/physreve.89.042138] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2014] [Indexed: 06/03/2023]
Abstract
Resonant activation is an effect of a noise-induced escape over a modulated potential barrier. The modulation of an energy landscape facilitates the escape kinetics and makes it optimal as measured by the mean first-passage time. A canonical example of resonant activation is a Brownian particle moving in a time-dependent potential under action of Gaussian white noise. Resonant activation is observed not only in typical Markovian-Gaussian systems but also in far-from-equilibrium and far-from-Markovianity regimes. We demonstrate that using an alternative to the mean first-passage time, robust measures of resonant activation, the signature of this effect can be observed in general continuous-time random walks in modulated potentials, even in situations when the mean first-passage time diverges.
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Affiliation(s)
- Krzysztof Szczepaniec
- Marian Smoluchowski Institute of Physics, and Mark Kac Center for Complex Systems Research, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, Poland
| | - Bartłomiej Dybiec
- Marian Smoluchowski Institute of Physics, and Mark Kac Center for Complex Systems Research, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, Poland
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13
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Janakiraman D, Sebastian KL. Path-integral formulation for Lévy flights: evaluation of the propagator for free, linear, and harmonic potentials in the over- and underdamped limits. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:061105. [PMID: 23367891 DOI: 10.1103/physreve.86.061105] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2012] [Indexed: 06/01/2023]
Abstract
Lévy flights can be described using a Fokker-Planck equation, which involves a fractional derivative operator in the position coordinate. Such an operator has its natural expression in the Fourier domain. Starting with this, we show that the solution of the equation can be written as a Hamiltonian path integral. Though this has been realized in the literature, the method has not found applications as the path integral appears difficult to evaluate. We show that a method in which one integrates over the position coordinates first, after which integration is performed over the momentum coordinates, can be used to evaluate several path integrals that are of interest. Using this, we evaluate the propagators for (a) free particle, (b) particle subjected to a linear potential, and (c) harmonic potential. In all the three cases, we have obtained results for both overdamped and underdamped cases.
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Affiliation(s)
- Deepika Janakiraman
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India
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14
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Srokowski T. Anomalous diffusion in systems driven by the stable Lévy noise with a finite noise relaxation time and inertia. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:021118. [PMID: 22463164 DOI: 10.1103/physreve.85.021118] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2011] [Revised: 11/23/2011] [Indexed: 05/31/2023]
Abstract
Dynamical systems driven by a general Lévy stable noise are considered. The inertia is included and the noise, represented by a generalized Ornstein-Uhlenbeck process, has a finite relaxation time. A general linear problem (the additive noise) is solved: the resulting distribution converges with time to the distribution for the white-noise, massless case. Moreover, a multiplicative noise is discussed. It can make the distribution steeper and the variance, which is finite, depends sublinearly on time (subdiffusion). For a small mass, a white-noise limit corresponds to the Stratonovich interpretation. On the other hand, the distribution tails agree with the Itô interpretation if the inertia is very large. An escape time from the potential well is calculated.
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Affiliation(s)
- Tomasz Srokowski
- Institute of Nuclear Physics, Polish Academy of Sciences, PL-31-342 Kraków, Poland
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15
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PAVLYUKEVICH ILYA. FIRST EXIT TIMES OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY MULTIPLICATIVE LÉVY NOISE WITH HEAVY TAILS. STOCH DYNAM 2011. [DOI: 10.1142/s0219493711003413] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, we study first exit times from a bounded domain of a gradient dynamical system Ẏt = -∇U(Yt) perturbed by a small multiplicative Lévy noise with heavy tails. A special attention is paid to the way the multiplicative noise is introduced. In particular, we determine the asymptotics of the first exit time of solutions of Itô, Stratonovich and Marcus canonical SDEs.
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Affiliation(s)
- ILYA PAVLYUKEVICH
- Institut für Stochastik, Fakultät für Mathematik und Informatik, Friedrich–Schiller–Universität Jena, Ernst–Abbe–Platz 2, Jena, 07743, Germany
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16
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Garber A, Moloney NR, Kantz H. Hopping over a heat barrier. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:031134. [PMID: 21517481 DOI: 10.1103/physreve.83.031134] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/16/2010] [Revised: 12/08/2010] [Indexed: 05/30/2023]
Abstract
We analyze diffusion in a finite domain with a position-dependent diffusion coefficient in terms of a stochastic hopping process. Via a coordinate transformation, we map the original system onto a problem with constant diffusion but nontrivial potential. In this way we show that a regime with enhanced diffusion acts as a potential barrier. We compute first-passage time distributions, hopping rates, and eigenvalues of the Fokker-Planck operator, and thereby verify that diffusion with a heat barrier is equivalent to a hopping process between metastable states.
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Affiliation(s)
- Anja Garber
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, D01187 Dresden, Germany
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17
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Baura A, Sen MK, Goswami G, Bag BC. Colored non-Gaussian noise driven open systems: Generalization of Kramers’ theory with a unified approach. J Chem Phys 2011; 134:044126. [DOI: 10.1063/1.3521394] [Citation(s) in RCA: 30] [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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18
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Dybiec B. Escape from the potential well: competition between long jumps and long waiting times. J Chem Phys 2010; 133:244114. [PMID: 21197983 DOI: 10.1063/1.3511722] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Within a concept of the fractional diffusion equation and subordination, the paper examines the influence of a competition between long waiting times and long jumps on the escape from the potential well. Applying analytical arguments and numerical methods, we demonstrate that the presence of long waiting times distributed according to a power-law distribution with a diverging mean leads to very general asymptotic properties of the survival probability. The observed survival probability asymptotically decays like a power law whose form is not affected by the value of the exponent characterizing the power law jump length distribution. It is demonstrated that this behavior is typical of and generic for systems exhibiting long waiting times. We also show that the survival probability has a universal character not only asymptotically, but also at small times. Finally, it is indicated which properties of the first passage time density are sensitive to the exact value of the exponent characterizing the jump length distribution.
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Affiliation(s)
- Bartłomiej Dybiec
- Marian Smoluchowski Institute of Physics, and Mark Kac Center for Complex Systems Research, Jagellonian University, ul. Reymonta 4, 30-059 Kraków, Poland.
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19
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Sheu SY, Yang DY. Determination of Protein Surface Hydration Shell Free Energy of Water Motion: Theoretical Study and Molecular Dynamics Simulation. J Phys Chem B 2010; 114:16558-66. [DOI: 10.1021/jp105164t] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Sheh-Yi Sheu
- Department of Life Sciences and Institute of Genome Sciences and Institute of Biomedical Informatics, National Yang-Ming University, Taipei 112, Taiwan, Institute of Atomic and Molecular Science, Academia Sinica, Taipei 106, Taiwan, and Division of Biomolecular Sensing, Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan
| | - Dah-Yen Yang
- Department of Life Sciences and Institute of Genome Sciences and Institute of Biomedical Informatics, National Yang-Ming University, Taipei 112, Taiwan, Institute of Atomic and Molecular Science, Academia Sinica, Taipei 106, Taiwan, and Division of Biomolecular Sensing, Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan
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Getfert S, Reimann P. Thermally activated escape far from equilibrium: A unified path-integral approach. Chem Phys 2010. [DOI: 10.1016/j.chemphys.2010.04.028] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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21
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Srokowski T. Nonlinear stochastic equations with multiplicative Lévy noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:051110. [PMID: 20866188 DOI: 10.1103/physreve.81.051110] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/28/2009] [Indexed: 05/29/2023]
Abstract
The Langevin equation with a multiplicative Lévy white noise is solved. The noise amplitude and the drift coefficient have a power-law form. A validity of ordinary rules of the calculus for the Stratonovich interpretation is discussed. The solution has the algebraic asymptotic form and the variance may assume a finite value for the case of the Stratonovich interpretation. The problem of escaping from a potential well is analyzed numerically; predictions of different interpretations of the stochastic integral are compared.
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Affiliation(s)
- Tomasz Srokowski
- Institute of Nuclear Physics, Polish Academy of Sciences, PL-31-342 Kraków, Poland
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22
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Srokowski T. Multiplicative Lévy processes: Itô versus Stratonovich interpretation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:051113. [PMID: 20364953 DOI: 10.1103/physreve.80.051113] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2009] [Revised: 10/12/2009] [Indexed: 05/29/2023]
Abstract
Langevin equation with a multiplicative stochastic force is considered. That force is uncorrelated, it has the Lévy distribution and the power-law intensity. The Fokker-Planck equations, which correspond both to the Itô and Stratonovich interpretation, are presented. They are solved for the case without drift and for the harmonic oscillator potential. The variance is evaluated; it is always infinite for the Itô case whereas for the Stratonovich one it can be finite and rise with time slower that linearly, which indicates subdiffusion. Analytical results are compared with numerical simulations.
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Affiliation(s)
- Tomasz Srokowski
- Institute of Nuclear Physics, Polish Academy of Sciences, PL-31-342 Kraków, Poland
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23
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Dybiec B. Lévy noises: double stochastic resonance in a single-well potential. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:041111. [PMID: 19905277 DOI: 10.1103/physreve.80.041111] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2009] [Revised: 08/23/2009] [Indexed: 05/28/2023]
Abstract
We study properties of a single-well fourth-order potential perturbed by a periodically modulated stable noise. Periodic modulation of the stable noise asymmetry results in an occurrence of the dynamical hysteresis which is the manifestation of the stochastic resonance in the system at hand. We show that the single-well potential with time modulated stable driving is a minimalistic setup, allowing the occurrence of the stochastic resonance (as measured by the hysteresis loop area). Finally, we demonstrate that the observed stochastic resonance is of the double type, i.e., the system efficiency measured by the hysteresis loop area depends in a nonmonotonous way both on the scale parameter (noise intensity) and on the stability exponent characterizing tails asymptotic of noise pulses.
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Affiliation(s)
- Bartłomiej Dybiec
- Marian Smoluchowski Institute of Physics, and Mark Kac Center for Complex Systems Research, Jagellonian University, ul. Reymonta 4, 30-059 Kraków, Poland.
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24
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Lubashevsky I, Friedrich R, Heuer A. Continuous-time multidimensional Markovian description of Lévy walks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:031148. [PMID: 19905103 DOI: 10.1103/physreve.80.031148] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2009] [Indexed: 05/28/2023]
Abstract
The paper presents a multidimensional model for nonlinear Markovian random walks that generalizes the one we developed previously [I. Lubashevsky, R. Friedrich, and A. Heuer, Phys. Rev. E 79, 011110 (2009)] in order to describe the Lévy-type stochastic processes in terms of continuous trajectories of walker motion. This approach may open a way to treat Lévy flights or Lévy random walks in inhomogeneous media or systems with boundaries in the future. The proposed model assumes the velocity of a wandering particle to be affected by a linear friction and a nonlinear Langevin force whose intensity is proportional to the magnitude of the velocity for its large values. Based on the singular perturbation technique, the corresponding Fokker-Planck equation is analyzed and the relationship between the system parameters and the Lévy exponent is found. Following actually the previous paper we demonstrate also that anomalously long displacements of the wandering particle are caused by extremely large fluctuations in the particle velocity whose duration is determined by the system parameters rather than the duration of the observation interval. In this way we overcome the problem of ascribing to Lévy random-walk non-Markov properties.
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Affiliation(s)
- Ihor Lubashevsky
- A.M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilov Str 38, 119991 Moscow, Russia
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25
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Grigoriu M. Numerical solution of stochastic differential equations with Poisson and Lévy white noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:026704. [PMID: 19792278 DOI: 10.1103/physreve.80.026704] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2009] [Indexed: 05/28/2023]
Abstract
A fixed time step method is developed for integrating stochastic differential equations (SDE's) with Poisson white noise (PWN) and Lévy white noise (LWN). The method for integrating SDE's with PWN has the same structure as that proposed by Kim [Phys. Rev. E 76, 011109 (2007)], but is established by using different arguments. The integration of SDE's with LWN is based on a representation of Lévy processes by sums of scaled Brownian motions and compound Poisson processes. It is shown that the numerical solutions of SDE's with PWN and LWN converge weakly to the exact solutions of these equations, so that they can be used to estimate not only marginal properties but also distributions of functionals of the exact solutions. Numerical examples are used to demonstrate the applications and the accuracy of the proposed integration algorithms.
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Affiliation(s)
- M Grigoriu
- Cornell University, Ithaca, New York 14853-3501, USA
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26
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Imkeller P, Pavlyukevich I, Wetzel T. First exit times for Lévy-driven diffusions with exponentially light jumps. ANN PROBAB 2009. [DOI: 10.1214/08-aop412] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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27
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Lubashevsky I, Friedrich R, Heuer A. Realization of Lévy walks as Markovian stochastic processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:011110. [PMID: 19257004 DOI: 10.1103/physreve.79.011110] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2007] [Indexed: 05/27/2023]
Abstract
Based on multivariate Langevin processes we present a realization of Lévy flights as a continuous process. For the simple case of a particle moving under the influence of friction and a velocity-dependent stochastic force we explicitly derive the generalized Langevin equation and the corresponding generalized Fokker-Planck equation describing Lévy flights. Our procedure is similar to the treatment of the Kramers-Fokker-Planck equation in the Smoluchowski limit. The proposed approach may open a way to treat Lévy flights in inhomogeneous media or systems with boundaries in the future.
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Affiliation(s)
- Ihor Lubashevsky
- A.M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilov Strasse 38, 119991 Moscow, Russia
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28
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Dybiec B, Gudowska-Nowak E, Sokolov IM. Transport in a Lévy ratchet: group velocity and distribution spread. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:011117. [PMID: 18763929 DOI: 10.1103/physreve.78.011117] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2008] [Indexed: 05/26/2023]
Abstract
We consider the motion of an overdamped particle in a periodic potential lacking spatial symmetry under the influence of symmetric, white, Lévy noise, being a minimal setup for a "Lévy ratchet." Due to the nonthermal character of the Lévy noise, the particle exhibits a motion with a preferred direction even in the absence of whatever additional time-dependent forces. The examination of the Lévy ratchet has to be based on the characteristics of directionality which are different from typically used measures such as mean current and the dispersion of particle positions, since these become inappropriate when the moments of the noise diverge. To overcome this problem, we discuss robust measures of directionality of transport such as the position of the median of the particle displacement distribution characterizing the group velocity and the interquantile distance giving the measure of the distribution width. Moreover, we analyze the behavior of splitting probabilities for leaving an interval of a given length, unveiling qualitative differences between the noises with Lévy indices below and above unity.
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Affiliation(s)
- B Dybiec
- M Smoluchowski Institute of Physics and Mark Kac Center for Complex Systems Research, Jagellonian University, ul. Reymonta 4, Kraków, Poland.
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29
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Denisov SI, Horsthemke W, Hänggi P. Steady-state Lévy flights in a confined domain. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:061112. [PMID: 18643222 DOI: 10.1103/physreve.77.061112] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2008] [Revised: 05/15/2008] [Indexed: 05/26/2023]
Abstract
We derive the generalized Fokker-Planck equation associated with a Langevin equation driven by arbitrary additive white noise. We apply our result to study the distribution of symmetric and asymmetric Lévy flights in an infinitely deep potential well. The fractional Fokker-Planck equation for Lévy flights is derived and solved analytically in the steady state. It is shown that Lévy flights are distributed according to the beta distribution, whose probability density becomes singular at the boundaries of the well. The origin of the preferred concentration of flying objects near the boundaries in nonequilibrium systems is clarified.
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Affiliation(s)
- S I Denisov
- Institut für Physik, Universität Augsburg, Universitätsstrasse 1, Augsburg, Germany
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30
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Koren T, Lomholt MA, Chechkin AV, Klafter J, Metzler R. Leapover lengths and first passage time statistics for Lévy flights. PHYSICAL REVIEW LETTERS 2007; 99:160602. [PMID: 17995231 DOI: 10.1103/physrevlett.99.160602] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2007] [Indexed: 05/25/2023]
Abstract
Exact results for the first passage time and leapover statistics of symmetric and one-sided Lévy flights (LFs) are derived. LFs with a stable index alpha are shown to have leapover lengths that are asymptotically power law distributed with an index alpha for one-sided LFs and, surprisingly, with an index alpha/2 for symmetric LFs. The first passage time distribution scales like a power law with an index 1/2 as required by the Sparre-Andersen theorem for symmetric LFs, whereas one-sided LFs have a narrow distribution of first passage times. The exact analytic results are confirmed by extensive simulations.
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Affiliation(s)
- Tal Koren
- School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel
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31
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Dybiec B, Gudowska-Nowak E, Sokolov IM. Stationary states in Langevin dynamics under asymmetric Lévy noises. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:041122. [PMID: 17994951 DOI: 10.1103/physreve.76.041122] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2007] [Indexed: 05/25/2023]
Abstract
Properties of systems driven by white non-Gaussian noises can be very different from these of systems driven by the white Gaussian noise. We investigate stationary probability densities for systems driven by alpha-stable Lévy-type noises, which provide natural extension to the Gaussian noise having, however, a new property, namely a possibility of being asymmetric. Stationary probability densities are examined for a particle moving in parabolic, quartic, and in generic double well potential models subjected to the action of alpha-stable noises. Relevant solutions are constructed by methods of stochastic dynamics. In situations where analytical results are known they are compared with numerical results. Furthermore, the problem of estimation of the parameters of stationary densities is investigated.
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Affiliation(s)
- B Dybiec
- M. Smoluchowski Institute of Physics and Mark Kac Center for Complex Systems Research, Jagellonian University, ul. Reymonta 4, 30-059 Kraków, Poland.
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32
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Chaudhury S, Cherayil BJ. Dynamic disorder in single-molecule Michaelis-Menten kinetics: The reaction-diffusion formalism in the Wilemski-Fixman approximation. J Chem Phys 2007; 127:105103. [PMID: 17867782 DOI: 10.1063/1.2768059] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Single-molecule equations for the Michaelis-Menten [Biochem. Z. 49, 333 (1913)] mechanism of enzyme action are analyzed within the Wilemski-Fixman [J. Chem. Phys. 58, 4009 (1973); 60, 866 (1974)] approximation after the effects of dynamic disorder--modeled by the anomalous diffusion of a particle in a harmonic well--are incorporated into the catalytic step of the reaction. The solution of the Michaelis-Menten equations is used to calculate the distribution of waiting times between successive catalytic turnovers in the enzyme beta-galactosidase. The calculated distribution is found to agree qualitatively with experimental results on this enzyme obtained at four different substrate concentrations. The calculations are also consistent with measurements of correlations in the fluctuations of the fluorescent light emitted during the course of catalysis, and with measurements of the concentration dependence of the randomness parameter.
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Affiliation(s)
- Srabanti Chaudhury
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India
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33
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Kim C, Lee EK, Hänggi P, Talkner P. Numerical method for solving stochastic differential equations with Poissonian white shot noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:011109. [PMID: 17677412 DOI: 10.1103/physreve.76.011109] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/10/2007] [Indexed: 05/16/2023]
Abstract
We propose a numerical integration scheme to solve stochastic differential equations driven by Poissonian white shot noise. Our formula, which is based on an integral equation, which is equivalent to the stochastic differential equation, utilizes a discrete time approximation with fixed integration time step. We show that our integration formula approaches the Euler formula if the Poissonian noise approaches the Gaussian white noise. The accuracy and efficiency of the proposed algorithm are examined by studying the dynamics of an overdamped particle driven by Poissonian white shot noise in a spatially periodic potential. We find that the accuracy of the proposed algorithm only weakly depends on the parameters characterizing the Poissonian white shot noise; this holds true even if the limit of Gaussian white noise is approached.
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Affiliation(s)
- Changho Kim
- Department of Chemistry and School of Molecular Science (BK21), Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea
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34
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Chechkin AV, Sliusarenko OY, Metzler R, Klafter J. Barrier crossing driven by Lévy noise: universality and the role of noise intensity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:041101. [PMID: 17500859 DOI: 10.1103/physreve.75.041101] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/24/2006] [Indexed: 05/15/2023]
Abstract
We study the barrier crossing of a particle driven by white symmetric Lévy noise of index alpha and intensity D for three different generic types of potentials: (a) a bistable potential, (b) a metastable potential, and (c) a truncated harmonic potential. For the low noise intensity regime we recover the previously proposed algebraic dependence on D of the characteristic escape time, T_{esc} approximately C(alpha)D;{mu(alpha)} , where C(alpha) is a coefficient. It is shown that the exponent mu(alpha) remains approximately constant, mu approximately 1 for 0<alpha<2 ; at alpha=2 the power-law form of T_{esc} changes into the known exponential dependence on 1D ; it exhibits a divergencelike behavior as alpha approaches 2. In this regime we observe a monotonous increase of the escape time T_{esc} with increasing alpha (keeping the noise intensity D constant). The probability density of the escape time decays exponentially. In addition, for low noise intensities the escape times correspond to barrier crossing by multiple Lévy steps. For high noise intensities, the escape time curves collapse for all values of alpha . At intermediate noise intensities, the escape time exhibits nonmonotonic dependence on the index alpha , while still retaining the exponential form of the escape time density.
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Affiliation(s)
- Aleksei V Chechkin
- Institute for Theoretical Physics NSC KIPT, Akademicheskaya st. 1, 61108 Kharkov, Ukraine
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