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Bebon R, Godec A. Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime. PHYSICAL REVIEW LETTERS 2023; 131:237101. [PMID: 38134782 DOI: 10.1103/physrevlett.131.237101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/20/2023] [Revised: 10/18/2023] [Accepted: 10/31/2023] [Indexed: 12/24/2023]
Abstract
We derive general bounds on the probability that the empirical first-passage time τ[over ¯]_{n}≡∑_{i=1}^{n}τ_{i}/n of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct nonasymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.
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Affiliation(s)
- Rick Bebon
- Mathematical bioPhysics Group, Max Planck Institute for Multidisciplinary Sciences, 37077 Göttingen, Germany
| | - Aljaž Godec
- Mathematical bioPhysics Group, Max Planck Institute for Multidisciplinary Sciences, 37077 Göttingen, Germany
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2
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Everlasting impact of initial perturbations on first-passage times of non-Markovian random walks. Nat Commun 2022; 13:5319. [PMID: 36085151 PMCID: PMC9463153 DOI: 10.1038/s41467-022-32280-6] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2022] [Accepted: 07/25/2022] [Indexed: 11/09/2022] Open
Abstract
Persistence, defined as the probability that a signal has not reached a threshold up to a given observation time, plays a crucial role in the theory of random processes. Often, persistence decays algebraically with time with non trivial exponents. However, general analytical methods to calculate persistence exponents cannot be applied to the ubiquitous case of non-Markovian systems relaxing transiently after an imposed initial perturbation. Here, we introduce a theoretical framework that enables the non-perturbative determination of persistence exponents of Gaussian non-Markovian processes with non stationary dynamics relaxing to a steady state after an initial perturbation. Two situations are analyzed: either the system is subjected to a temperature quench at initial time, or its past trajectory is assumed to have been observed and thus known. Our theory covers the case of spatial dimension higher than one, opening the way to characterize non-trivial reaction kinetics for complex systems with non-equilibrium initial conditions.
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3
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Ghosh A, Chakraborty D. Persistence in Brownian motion of an ellipsoidal particle in two dimensions. J Chem Phys 2020; 152:174901. [PMID: 32384838 DOI: 10.1063/5.0004134] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We investigate the persistence probability p(t) of the position of a Brownian particle with shape asymmetry in two dimensions. The persistence probability is defined as the probability that a stochastic variable has not changed its sign in the given time interval. We explicitly consider two cases-diffusion of a free particle and that of a harmonically trapped particle. The latter is particularly relevant in experiments that use trapping and tracking techniques to measure the displacements. We provide analytical expressions of p(t) for both the scenarios and show that in the absence of the shape asymmetry, the results reduce to the case of an isotropic particle. The analytical expressions of p(t) are further validated against numerical simulation of the underlying overdamped dynamics. We also illustrate that p(t) can be a measure to determine the shape asymmetry of a colloid and the translational and rotational diffusivities can be estimated from the measured persistence probability. The advantage of this method is that it does not require the tracking of the orientation of the particle.
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Affiliation(s)
- Anirban Ghosh
- Indian Institute of Science Education and Research Mohali, Sec. 81, S.A.S. Nagar, Knowledge City, Manauli, Punjab 140306, India
| | - Dipanjan Chakraborty
- Indian Institute of Science Education and Research Mohali, Sec. 81, S.A.S. Nagar, Knowledge City, Manauli, Punjab 140306, India
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4
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Gomes WP, Penna ALA, Oliveira FA. From cellular automata to growth dynamics: The Kardar-Parisi-Zhang universality class. Phys Rev E 2019; 100:020101. [PMID: 31574642 DOI: 10.1103/physreve.100.020101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/28/2019] [Indexed: 06/10/2023]
Abstract
We demonstrate that in the continuous limit the etching mechanism yields the Kardar-Parisi-Zhang (KPZ) equation in a (d+1)-dimensional space. We show that the parameters ν, associated with the surface tension, and λ, associated with the nonlinear term of the KPZ equation, are not phenomenological, but rather they stem from a new probability distribution function. The Galilean invariance is recovered independently of d, and we illustrate this via very precise numerical simulations. We obtain firsthand the coupling parameter as a function of the probabilities. In addition, we strengthen the argument that there is no upper critical limit for the KPZ equation.
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Affiliation(s)
| | - André L A Penna
- Instituto de Física, Universidade de Brasília, Brazil
- International Center for Condensed Matter Physics, CP 04455, 70919-970 Brasília DF, Brazil
| | - Fernando A Oliveira
- Instituto de Física, Universidade de Brasília, Brazil
- International Center for Condensed Matter Physics, CP 04455, 70919-970 Brasília DF, Brazil
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5
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Dentz M, Neuweiler I, Méheust Y, Tartakovsky DM. Noise-driven interfaces and their macroscopic representation. Phys Rev E 2016; 94:052802. [PMID: 27967189 DOI: 10.1103/physreve.94.052802] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2015] [Indexed: 11/07/2022]
Abstract
We study the macroscopic representation of noise-driven interfaces in stochastic interface growth models in (1+1) dimensions. The interface is characterized macroscopically by saturation, which represents the fluctuating sharp interface by a smoothly varying phase field with values between 0 and 1. We determine the one-point interface height statistics for the Edwards-Wilkinson (EW) and Kadar-Paris-Zhang (KPZ) models in order to determine explicit deterministic equations for the phase saturation for each of them. While we obtain exact results for the EW model, we develop a Gaussian closure approximation for the KPZ model. We identify an interface compression term, which is related to mass transfer perpendicular to the growth direction, and a diffusion term that tends to increase the interface width. The interface compression rate depends on the mesoscopic mass transfer process along the interface and in this sense provides a relation between meso- and macroscopic interface dynamics. These results shed light on the relation between mesoscale and macroscale interface models, and provide a systematic framework for the upscaling of stochastic interface dynamics.
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Affiliation(s)
- Marco Dentz
- Spanish National Research Council (IDAEA-CSIC), 08034 Barcelona, Spain
| | - Insa Neuweiler
- Institute of Fluid Mechanics in Civil Engineering, Leibniz Universität Hannover, Hanover, Germany
| | - Yves Méheust
- Géosciences Rennes, UMR CNRS 6118, Université de Rennes 1, Rennes, France
| | - Daniel M Tartakovsky
- Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA
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Alves WS, Rodrigues EA, Fernandes HA, Mello BA, Oliveira FA, Costa IVL. Analysis of etching at a solid-solid interface. Phys Rev E 2016; 94:042119. [PMID: 27841509 DOI: 10.1103/physreve.94.042119] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/02/2016] [Indexed: 06/06/2023]
Abstract
We present a method to derive an analytical expression for the roughness of an eroded surface whose dynamics are ruled by cellular automaton. Starting from the automaton, we obtain the time evolution of the height average and height variance (roughness). We apply this method to the etching model in 1+1 dimensions, and then we obtain the roughness exponent. Using this in conjunction with the Galilean invariance we obtain the other exponents, which perfectly match the numerical results obtained from simulations. These exponents are exact, and they are the same as those exhibited by the Kardar-Parisi-Zhang (KPZ) model for this dimension. Therefore, our results provide proof for the conjecture that the etching and KPZ models belong to the same universality class. Moreover, the method is general, and it can be applied to other cellular automata models.
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Affiliation(s)
- Washington S Alves
- Graduate Program in Material Science, Faculdade UnB Planaltina, Universidade de Brasília, CEP 73300-000, Planaltina, DF, Brazil
- Instituto de Física, Universidade de Brasília, CP 04513, CEP 70919-970, Brasília, DF, Brazil
| | - Evandro A Rodrigues
- Instituto de Física, Universidade de Brasília, CP 04513, CEP 70919-970, Brasília, DF, Brazil
| | - Henrique A Fernandes
- Universidade Federal de Goiás, Campus Jataí, Br 364, Km 192, 3800, Parque Industrial, CEP 75801-615, Jataí, Goiás, Brazil
| | - Bernardo A Mello
- IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598, USA
| | - Fernando A Oliveira
- Instituto de Física, Universidade de Brasília, CP 04513, CEP 70919-970, Brasília, DF, Brazil
- Korea Institute for Advanced Study, Seoul 130722, South Korea
| | - Ismael V L Costa
- Graduate Program in Material Science, Faculdade UnB Planaltina, Universidade de Brasília, CEP 73300-000, Planaltina, DF, Brazil
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7
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Adler Y, Givli S. Closing the loop: lamellipodia dynamics from the perspective of front propagation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:042708. [PMID: 24229214 DOI: 10.1103/physreve.88.042708] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/19/2013] [Revised: 07/25/2013] [Indexed: 06/02/2023]
Abstract
We develop a simple physical model that captures the large-scale lamellipodia dynamics in crawling cells and explains the observed spectrum of fish keratocytes behavior. The main ingredients in this description are the geometrical evolution of the lamellipodium leading edge, the dynamic remodeling of the actin network, and the interconnection between them. We deviate from existing theoretical works and consider the lamellipodium leading edge as a propagating front. The agreement of our model with experimental works suggests that the large-scale morphological and migration features exhibited by keratocyte cells are a direct consequence of the closed feedback loop between the shape of the leading edge and the density of the actin network.
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Affiliation(s)
- Yair Adler
- Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
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8
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Efraim Y, Taitelbaum H. Persistence in reactive-wetting interfaces. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:050602. [PMID: 22181361 DOI: 10.1103/physreve.84.050602] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2011] [Indexed: 05/31/2023]
Abstract
In this paper, we report on persistence results of reactive-wetting advancing interfaces performed with mercury on silver at room temperature. Earlier kinetic roughening studies of reactive-wetting systems at room temperature as well as at high temperatures revealed some limited information on the spatiotemporal behavior of these systems. However, by calculating the persistence exponent, we were able to identify two distinct kinetic time regimes in this process. In the first one, while the interface is moving but its width is not yet growing, the persistence exponent is θ=0.55±0.05, which is typical for a random, noisy behavior. In the second regime, there is an effective growth of the interface width with a growth exponent β=0.67±0.06 followed by saturation, according to the Family-Vicsek description of interface growth. The persistence exponent in this regime is θ=0.37±0.05, which indicates that the relation θ=1-β seems to hold even for this nonlinear experimental system.
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Affiliation(s)
- Yael Efraim
- Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel.
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9
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Fort J. Vertical cultural transmission effects on demic front propagation: theory and application to the Neolithic transition in Europe. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:056124. [PMID: 21728622 DOI: 10.1103/physreve.83.056124] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/01/2011] [Revised: 03/15/2011] [Indexed: 05/31/2023]
Abstract
It is shown that Lotka-Volterra interaction terms are not appropriate to describe vertical cultural transmission. Appropriate interaction terms are derived and used to compute the effect of vertical cultural transmission on demic front propagation. They are also applied to a specific example, the Neolithic transition in Europe. In this example, it is found that the effect of vertical cultural transmission can be important (about 30%). On the other hand, simple models based on differential equations can lead to large errors (above 50%). Further physical, biophysical, and cross-disciplinary applications are outlined.
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Affiliation(s)
- Joaquim Fort
- Complex Systems Laboratory and Departament de Física, Universitat de Girona, ES-17071 Girona, Catalonia, Spain
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10
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Soriano J, Braslavsky I, Xu D, Krichevsky O, Stavans J. Universality of persistence exponents in two-dimensional Ostwald ripening. PHYSICAL REVIEW LETTERS 2009; 103:226101. [PMID: 20366113 DOI: 10.1103/physrevlett.103.226101] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/01/2009] [Indexed: 05/29/2023]
Abstract
We measured persistence exponents theta(phi) of Ostwald ripening in two dimensions, as a function of the area fraction phi occupied by coarsening domains. The values of theta(phi) in two systems, succinonitrile and brine, quenched to their liquid-solid coexistence region, compare well with one another, providing compelling evidence for the universality of the one-parameter family of exponents. For small phi, theta(phi) approximately = 0.39phi, as predicted by a model that assumes no correlations between evolving domains. These constitute the first measurements of persistence exponents in the case of phase transitions with a conserved order parameter.
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Affiliation(s)
- Jordi Soriano
- Departament d'ECM, Facultat de Física, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain
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11
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Isern N, Fort J. Time-delayed reaction-diffusion fronts. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:057103. [PMID: 20365098 DOI: 10.1103/physreve.80.057103] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/22/2009] [Indexed: 05/29/2023]
Abstract
A time-delayed second-order approximation for the front speed in reaction-dispersion systems was obtained by Fort and Méndez [Phys. Rev. Lett. 82, 867 (1999)]. Here we show that taking proper care of the effect of the time delay on the reactive process yields a different evolution equation and, therefore, an alternate equation for the front speed. We apply the new equation to the Neolithic transition. For this application the new equation yields speeds about 10% slower than the previous one.
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Affiliation(s)
- Neus Isern
- Departament de Física, Universitat de Girona, 17071 Girona, Catalonia, Spain
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12
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Fort J, Pérez-Losada J, Isern N. Fronts from integrodifference equations and persistence effects on the Neolithic transition. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:031913. [PMID: 17930277 DOI: 10.1103/physreve.76.031913] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/05/2007] [Revised: 07/31/2007] [Indexed: 05/25/2023]
Abstract
We extend a previous model of the Neolithic transition in Europe [J. Fort and V. Méndez, Phys. Rev. Lett. 82, 867 (1999)] by taking two effects into account: (i) we do not use the diffusion approximation (which corresponds to second-order Taylor expansions), and (ii) we take proper care of the fact that parents do not migrate away from their children (we refer to this as a time-order effect, in the sense that it implies that children grow up with their parents, before they become adults and can survive and migrate). We also derive a time-ordered, second-order equation, which we call the sequential reaction-diffusion equation, and use it to show that effect (ii) is the most important one, and that both of them should in general be taken into account to derive accurate results. As an example, we consider the Neolithic transition: the model predictions agree with the observed front speed, and the corrections relative to previous models are important (up to 70%).
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Affiliation(s)
- Joaquim Fort
- Departament de Física, Universitat de Girona, 17071 Girona, Catalonia, Spain
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13
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Tao CG, Cullen WG, Williams ED, Dasgupta C. Generalized survival in step fluctuations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:021601. [PMID: 17930045 DOI: 10.1103/physreve.76.021601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2006] [Indexed: 05/25/2023]
Abstract
The properties of the generalized survival probability, that is, the probability of not crossing an arbitrary location R during relaxation, have been investigated experimentally (via scanning tunneling microscope observations) and numerically. The results confirm that the generalized survival probability decays exponentially with a time constant tau(s) (R). The distance dependence of the time constant is shown to be tau(s) (R) = tau(s0) exp[-R/w (T)], where w2 (T) is the material-dependent mean-squared width of the step fluctuations. The result reveals the dependence on the physical parameters of the system inherent in the prior prediction of the time constant scaling with R/L(alpha), with L the system size and alpha the roughness exponent. The survival behavior is also analyzed using a contrasting concept, the generalized inside survival S(in) (t,R), which involves fluctuations to an arbitrary location R further from the average. Numerical simulations of the inside survival probability also show an exponential time dependence, and the extracted time constant empirically shows (R/w)(lambda) behavior, with lambda varying over 0.6 to 0.8 as the sampling conditions are changed. The experimental data show similar behavior, and can be well fit with lambda = 1.0 for T = 300 K, and 0.5 < lambda < 1 for T = 460 K. Over this temperature range, the ratio of the fixed sampling time to the underlying physical time constant, and thus the true correlation time, increases by a factor of approximately 10(3). Preliminary analysis indicates that the scaling effect due to the true correlation time is relevant in the parameter space of the experimental observations.
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Affiliation(s)
- C G Tao
- Department of Physics and Materials Research Science and Engineering Center, University of Maryland, College Park, Maryland 20742-4111, USA
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14
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Huang YH, I L. Memory and persistence correlation of microstructural fluctuations in two-dimensional dusty plasma liquids. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:016403. [PMID: 17677575 DOI: 10.1103/physreve.76.016403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2007] [Revised: 05/09/2007] [Indexed: 05/16/2023]
Abstract
We investigate the temporal memory and persistence time correlations of microstructural order fluctuations in quasi-two-dimensional dusty plasma liquids through directly monitoring particle positions. The persistence length of the ordered and the disordered microstates both follow power-law distribution for the cold liquid. The correlation probabilities of the persistence lengths in different pairs of fluctuating cycles show that an order-disorder-order and a disorder-order-disorder sequence are more likely to have the long-short-long or short-long-short persistent lengths combination. The persistence time correlation lasts for several fluctuating cycles. The correlation memory is stored in the surrounding heterogeneous network through mutual coupling. Increasing thermal noise level deteriorates the memory and leads to the less correlated stretched exponential distribution of the persistent length.
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Affiliation(s)
- Yu-Hsuan Huang
- Department of Physics and Center for Complex Systems, National Central University, Jhongli, Taiwan 32001, Republic of China
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15
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Rost M, Laurson L, Dubé M, Alava M. Fluctuations in fluid invasion into disordered media. PHYSICAL REVIEW LETTERS 2007; 98:054502. [PMID: 17358864 DOI: 10.1103/physrevlett.98.054502] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/02/2006] [Indexed: 05/14/2023]
Abstract
Interfaces moving in a disordered medium exhibit stochastic velocity fluctuations obeying universal scaling relations related to the presence or absence of conservation laws. For fluid invasion of porous media, we show that the fluctuations of the velocity are governed by a geometry-dependent length scale arising from fluid conservation. This result is compared to the statistics resulting from a nonequilibrium (depinning) transition between a moving interface and a stationary, pinned one.
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Affiliation(s)
- Martin Rost
- Theoretische Biologie, IZMB, Universität Bonn, 53115 Bonn, Germany
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16
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Conrad BR, Cullen WG, Dougherty DB, Lyubinetsky I, Williams ED. Spatial first-passage statistics of Al/Si(111)-(square root 3 x square root 3) step fluctuations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:021603. [PMID: 17358349 DOI: 10.1103/physreve.75.021603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/12/2006] [Revised: 12/11/2006] [Indexed: 05/14/2023]
Abstract
Spatial step edge fluctuations on a multicomponent surface of Al/Si(111)-(square root 3 x square root 3) were measured via scanning tunneling microscopy over a temperature range of 720-1070 K, for step lengths of L=65-160 nm. Even though the time scale of fluctuations of steps on this surface varies by orders of magnitude over the indicated temperature range, measured first-passage spatial persistence and survival probabilities are temperature independent. The power law functional form for spatial persistence probabilities is confirmed and the symmetric spatial persistence exponent is measured to be theta=0.498+/-0.062 in agreement with the theoretical prediction theta=1/2. The survival probability is found to scale directly with y/L, where y is the distance along the step edge. The form of the survival probabilities agrees quantitatively with the theoretical prediction, which yields exponential decay in the limit of small y/L. The decay constant is found experimentally to be y(s)/L=0.076+/-0.033 for y/L<or=0.2.
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Affiliation(s)
- B R Conrad
- Department of Physics and MRSEC, University of Maryland, College Park, Maryland 20742-4111, USA
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17
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Cakir R, Grigolini P, Krokhin AA. Dynamical origin of memory and renewal. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:021108. [PMID: 17025394 DOI: 10.1103/physreve.74.021108] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/24/2005] [Revised: 05/05/2006] [Indexed: 05/12/2023]
Abstract
We show that the dynamic approach to fractional Brownian motion (FBM) establishes a link between a non-Poisson renewal process with abrupt jumps resetting to zero the system's memory and correlated dynamic processes, whose individual trajectories keep a nonvanishing memory of their past time evolution. It is well known that the recrossings of the origin by an ordinary one-dimensional diffusion trajectory generates a Lévy (and thus renewal) process of index theta = 1/2 . We prove with theoretical and numerical arguments that this is the special case of a more general condition, insofar as the recrossings produced by the dynamic FBM generates a Lévy process with 0 < theta < 1. This result is extended to produce a satisfactory model for the fluorescent signal of blinking quantum dots.
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Affiliation(s)
- R Cakir
- Center for Nonlinear Science, University of North Texas, PO Box 311427, Denton, Texas 76203, USA
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18
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Majumdar SN, Dasgupta C. Spatial survival probability for one-dimensional fluctuating interfaces in the steady state. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:011602. [PMID: 16486156 DOI: 10.1103/physreve.73.011602] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/31/2005] [Revised: 10/24/2005] [Indexed: 05/06/2023]
Abstract
We report numerical and analytic results for the spatial survival probability for fluctuating one-dimensional interfaces with Edwards-Wilkinson or Kardar-Parisi-Zhang dynamics in the steady state. Our numerical results are obtained from analysis of steady-state profiles generated by integrating a spatially discretized form of the Edwards-Wilkinson equation to long times. We show that the survival probability exhibits scaling behavior in its dependence on the system size and the "sampling interval" used in the measurement for both "steady-state" and "finite" initial conditions. Analytic results for the scaling functions are obtained from a path-integral treatment of a formulation of the problem in terms of one-dimensional Brownian motion. A "deterministic approximation" is used to obtain closed-form expressions for survival probabilities from the formally exact analytic treatment. The resulting approximate analytic results provide a fairly good description of the numerical data.
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Affiliation(s)
- Satya N Majumdar
- Laboratoire de Physique Theorique et Modeles Statistiques, Universite Paris-Sud, Bat. 100, 91405 ORSAY cedex, France
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19
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Balankin AS, Matamoros DM. Anomalous roughness with system-size-dependent local roughness exponent. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:056102. [PMID: 16089597 DOI: 10.1103/physreve.71.056102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2005] [Indexed: 05/03/2023]
Abstract
We note that in a system far from equilibrium the interface roughening may depend on the system size which plays the role of control parameter. To detect the size effect on the interface roughness, we study the scaling properties of rough interfaces formed in paper combustion experiments. Using paper sheets of different width lambda L0, we found that the turbulent flame fronts display anomalous multiscaling characterized by nonuniversal global roughness exponent alpha and the system-size-dependent spectrum of local roughness exponents zeta(q) (lambda) = zeta(1) (1) q(-omega) lambda(phi) <alpha, whereas the burning fronts possess conventional multiaffine scaling. The structure factor of turbulent flame fronts also exhibit unconventional scaling dependence on lambda. These results are expected to apply to a broad range of far from equilibrium systems, when the kinetic energy fluctuations exceed a certain critical value.
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Constantin M, Das Sarma S. Mapping spatial persistent large deviations of nonequilibrium surface growth processes onto the temporal persistent large deviations of stochastic random walk processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:041602. [PMID: 15600416 DOI: 10.1103/physreve.70.041602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/29/2004] [Revised: 06/18/2004] [Indexed: 05/24/2023]
Abstract
Spatial persistent large deviations probability of surface growth processes governed by the Edwards-Wilkinson dynamics, Px(x,s), with -1< or =s< or =1 is mapped isomorphically onto the temporal persistent large deviations probability Pt(t,s) associated with the stochastic Markovian random walk problem. We show using numerical simulations that the infinite family of spatial persistent large deviations exponents thetax(s) characterizing the power-law decay of Px(x,s ) agrees, as predicted on theoretical grounds by Phys. Rev. Lett. 86, 3700 (2001)], with the numerical measurements of thetat(s), the continuous family of exponents characterizing the long-time power law behavior of Pt(t,s). We also discuss the simulations of the spatial persistence probability corresponding to a discrete model in the Mullins-Herring universality class, where our discrete simulations do not agree well with the theoretical predictions perhaps because of the severe finite-size corrections which are known to strongly inhibit the manifestation of the asymptotic continuum behavior in discrete models involving large values of the dynamical exponent and the associated extremely slow convergence to the asymptotic regime.
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Affiliation(s)
- M Constantin
- Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
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Failla R, Grigolini P, Ignaccolo M, Schwettmann A. Random growth of interfaces as a subordinated process. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:010101. [PMID: 15324032 DOI: 10.1103/physreve.70.010101] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/19/2004] [Indexed: 05/24/2023]
Abstract
We study the random growth of surfaces from within the perspective of a single column, namely, the fluctuation of the column height around the mean value, y (t) identical with h (t)-<h (t) >, which is depicted as being subordinated to a standard fluctuation-dissipation process with friction gamma. We argue that the main properties of Kardar-Parisi-Zhang theory, in one dimension, are derived by identifying the distribution of return times to y (0) =0, which is a truncated inverse power law, with the distribution of subordination times. The agreement of the theoretical prediction with the numerical treatment of the (1+1) -dimensional model of ballistic deposition is remarkably good, in spite of the finite-size effects affecting this model.
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Affiliation(s)
- R Failla
- Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, Texas 76203-1427, USA
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Constantin M, Sarma SD. Generalized survival in equilibrium step fluctuations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:052601. [PMID: 15244864 DOI: 10.1103/physreve.69.052601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2003] [Indexed: 05/24/2023]
Abstract
We investigate the dynamics of a generalized survival probability S(t,R) defined with respect to an arbitrary reference level R (rather than the average) in equilibrium step fluctuations. The exponential decay at large time scales of the generalized survival probability is numerically analyzed. S(t,R) is shown to exhibit simple scaling behavior as a function of system size L, sampling time deltat, and the reference level R. The generalized survival time scale tau(s) (R) associated with S(t,R) is shown to decay exponentially as a function of R.
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Affiliation(s)
- M Constantin
- Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
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Constantin M, Sarma SD, Dasgupta C. Spatial persistence and survival probabilities for fluctuating interfaces. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:051603. [PMID: 15244825 DOI: 10.1103/physreve.69.051603] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/18/2003] [Indexed: 05/24/2023]
Abstract
We report the results of numerical investigations of the steady-state (SS) and finite-initial-conditions (FIC) spatial persistence and survival probabilities for (1+1) -dimensional interfaces with dynamics governed by the nonlinear Kardar-Parisi-Zhang equation and the linear Edwards-Wilkinson (EW) equation with both white (uncorrelated) and colored (spatially correlated) noise. We study the effects of a finite sampling distance on the measured spatial persistence probability and show that both SS and FIC persistence probabilities exhibit simple scaling behavior as a function of the system size and the sampling distance. Analytical expressions for the exponents associated with the power-law decay of SS and FIC spatial persistence probabilities of the EW equation with power-law correlated noise are established and numerically verified.
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Affiliation(s)
- M Constantin
- Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA.
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Dasgupta C, Constantin M, Das Sarma S, Majumdar SN. Survival in equilibrium step fluctuations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:022101. [PMID: 14995501 DOI: 10.1103/physreve.69.022101] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/03/2003] [Indexed: 05/24/2023]
Abstract
We report the results of analytic and numerical investigations of the time scale of survival or non-zero-crossing probability S(t) in equilibrium step fluctuations described by Langevin equations appropriate for attachment/detachment and edge-diffusion limited kinetics. An exact relation between long-time behaviors of the survival probability and the autocorrelation function is established and numerically verified. S(t) is shown to exhibit a simple scaling behavior as a function of system size and sampling time. Our theoretical results are in agreement with those obtained from an analysis of experimental dynamical scanning tunneling microscopy data on step fluctuations on Al/Si(111) and Ag(111) surfaces.
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Affiliation(s)
- C Dasgupta
- Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
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