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Stanislavsky AA, Weron A. Confined modes of single-particle trajectories induced by stochastic resetting. Phys Rev E 2023; 108:044130. [PMID: 37978668 DOI: 10.1103/physreve.108.044130] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/22/2023] [Accepted: 09/25/2023] [Indexed: 11/19/2023]
Abstract
Random trajectories of single particles in living cells contain information about the interaction between particles, as well as with the cellular environment. However, precise consideration of the underlying stochastic properties, beyond normal diffusion, remains a challenge as applied to each particle trajectory separately. In this paper, we show how positions of confined particles in living cells can obey not only the Laplace distribution, but the Linnik one. This feature is detected in experimental data for the motion of G proteins and coupled receptors in cells, and its origin is explained in terms of stochastic resetting. This resetting process generates power-law waiting times, giving rise to the Linnik statistics in confined motion, and also includes exponentially distributed times as a limit case leading to the Laplace one. The stochastic process, which is affected by the resetting, can be Brownian motion commonly found in cells. Other possible models producing similar effects are discussed.
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Affiliation(s)
| | - Aleksander Weron
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
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Kosztołowicz T. Subdiffusion equation with fractional Caputo time derivative with respect to another function in modeling transition from ordinary subdiffusion to superdiffusion. Phys Rev E 2023; 107:064103. [PMID: 37464604 DOI: 10.1103/physreve.107.064103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2022] [Accepted: 05/11/2023] [Indexed: 07/20/2023]
Abstract
We use a subdiffusion equation with fractional Caputo time derivative with respect to another function g (g-subdiffusion equation) to describe a smooth transition from ordinary subdiffusion to superdiffusion. Ordinary subdiffusion is described by the equation with the "ordinary" fractional Caputo time derivative, superdiffusion is described by the equation with a fractional Riesz-type spatial derivative. We find the function g for which the solution (Green's function, GF) to the g-subdiffusion equation takes the form of GF for ordinary subdiffusion in the limit of small time and GF for superdiffusion in the limit of long time. To solve the g-subdiffusion equation we use the g-Laplace transform method. It is shown that the scaling properties of the GF for g-subdiffusion and the GF for superdiffusion are the same in the long time limit. We conclude that for a sufficiently long time the g-subdiffusion equation describes superdiffusion well, despite a different stochastic interpretation of the processes. Then, paradoxically, a subdiffusion equation with a fractional time derivative describes superdiffusion. The superdiffusive effect is achieved here not by making anomalously long jumps by a diffusing particle, but by greatly increasing the particle jump frequency which is derived by means of the g-continuous-time random walk model. The g-subdiffusion equation is shown to be quite general, it can be used in modeling of processes in which a kind of diffusion change continuously over time. In addition, some methods used in modeling of ordinary subdiffusion processes, such as the derivation of local boundary conditions at a thin partially permeable membrane, can be used to model g-subdiffusion processes, even if this process is interpreted as superdiffusion.
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Affiliation(s)
- Tadeusz Kosztołowicz
- Institute of Physics, Jan Kochanowski University, Uniwersytecka 7, 25-406 Kielce, Poland
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Kosztołowicz T, Dutkiewicz A. Composite subdiffusion equation that describes transient subdiffusion. Phys Rev E 2022; 106:044119. [PMID: 36397481 DOI: 10.1103/physreve.106.044119] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2022] [Accepted: 09/27/2022] [Indexed: 06/16/2023]
Abstract
A composite subdiffusion equation with fractional Caputo time derivative with respect to another function g is used to describe a process of a continuous transition from subdiffusion with parameters α and D_{α} to subdiffusion with parameters β and D_{β}. The parameters are defined by the time evolution of the mean square displacement of diffusing particle σ^{2}(t)=2D_{i}t^{i}/Γ(1+i), i=α,β. The function g controls the process at intermediate times. The composite subdiffusion equation is more general than the ordinary fractional subdiffusion equation with constant parameters; it has potentially wide application in modeling diffusion processes with changing parameters.
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Affiliation(s)
- Tadeusz Kosztołowicz
- Institute of Physics, Jan Kochanowski University, Uniwersytecka 7, 25-406 Kielce, Poland
| | - Aldona Dutkiewicz
- Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
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Kosztołowicz T. First-passage time for the g-subdiffusion process of vanishing particles. Phys Rev E 2022; 106:L022104. [PMID: 36110021 DOI: 10.1103/physreve.106.l022104] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2022] [Accepted: 08/05/2022] [Indexed: 06/15/2023]
Abstract
Subdiffusion equation and molecule survival equation, both with Caputo fractional time derivatives with respect to other functions g_{1} and g_{2}, respectively, are used to describe diffusion of a molecule that can disappear at any time with a constant probability. The process can be interpreted as an "ordinary" subdiffusion and "ordinary" molecule survival process in which timescales are changed by the functions g_{1} and g_{2}. We derive the first-passage time distribution for the process. The mutual influence of subdiffusion and molecule-vanishing processes can be included in the model when the functions g_{1} and g_{2} are related to each other. As an example, we consider the processes in which subdiffusion and molecule survival are highly related, which corresponds to the case of g_{1}≡g_{2}.
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Affiliation(s)
- Tadeusz Kosztołowicz
- Institute of Physics, Jan Kochanowski University, Uniwersytecka 7, 25-406 Kielce, Poland
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Wang W, Metzler R, Cherstvy AG. Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and models. Phys Chem Chem Phys 2022; 24:18482-18504. [DOI: 10.1039/d2cp01741e] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
How does a systematic time-dependence of the diffusion coefficient $D (t)$ affect the ergodic and statistical characteristics of fractional Brownian motion (FBM)? Here, we examine how the behavior of the...
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Stanislavsky A, Weron A. Optimal non-Gaussian search with stochastic resetting. Phys Rev E 2021; 104:014125. [PMID: 34412216 DOI: 10.1103/physreve.104.014125] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2021] [Accepted: 06/29/2021] [Indexed: 11/07/2022]
Abstract
In this paper we reveal that each subordinated Brownian process, leading to subdiffusion, under Poissonian resetting has a stationary state with the Laplace distribution. Its location parameter is defined only by the position to which the particle resets, and its scaling parameter is dependent on the Laplace exponent of the random process directing Brownian motion as a parent process. From the analysis of the scaling parameter the probability density function of the stochastic process, subject to reset, can be restored. In this case the mean time for the particle to reach a target is finite and has a minimum, optimal for the resetting rate. If the Brownian process is replaced by the Lévy motion (superdiffusion), then its stationary state obeys the Linnik distribution which belongs to the class of generalized Laplace distributions.
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Affiliation(s)
- Aleksander Stanislavsky
- Institute of Radio Astronomy, 4 Mystetstv Street, 61002 Kharkiv, Ukraine.,Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
| | - Aleksander Weron
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
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Kosztołowicz T, Dutkiewicz A. Subdiffusion equation with Caputo fractional derivative with respect to another function. Phys Rev E 2021; 104:014118. [PMID: 34412326 DOI: 10.1103/physreve.104.014118] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/11/2021] [Accepted: 06/23/2021] [Indexed: 12/12/2022]
Abstract
We show an application of a subdiffusion equation with Caputo fractional time derivative with respect to another function g to describe subdiffusion in a medium having a structure evolving over time. In this case a continuous transition from subdiffusion to other type of diffusion may occur. The process can be interpreted as "ordinary" subdiffusion with fixed subdiffusion parameter (subdiffusion exponent) α in which timescale is changed by the function g. As an example, we consider the transition from "ordinary" subdiffusion to ultraslow diffusion. The g-subdiffusion process generates the additional aging process superimposed on the "standard" aging generated by "ordinary" subdiffusion. The aging process is analyzed using coefficient of relative aging of g-subdiffusion with respect to "ordinary" subdiffusion. The method of solving the g-subdiffusion equation is also presented.
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Affiliation(s)
- Tadeusz Kosztołowicz
- Institute of Physics, Jan Kochanowski University, Uniwersytecka 7, 25-406 Kielce, Poland
| | - Aldona Dutkiewicz
- Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
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Chechkin A, Sokolov IM. Relation between generalized diffusion equations and subordination schemes. Phys Rev E 2021; 103:032133. [PMID: 33862700 DOI: 10.1103/physreve.103.032133] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2021] [Accepted: 02/26/2021] [Indexed: 11/07/2022]
Abstract
Generalized (non-Markovian) diffusion equations with different memory kernels and subordination schemes based on random time change in the Brownian diffusion process are popular mathematical tools for description of a variety of non-Fickian diffusion processes in physics, biology, and earth sciences. Some of such processes (notably, the fluid limits of continuous time random walks) allow for either kind of description, but other ones do not. In the present work we discuss the conditions under which a generalized diffusion equation does correspond to a subordination scheme, and the conditions under which a subordination scheme does possess the corresponding generalized diffusion equation. Moreover, we discuss examples of random processes for which only one, or both kinds of description are applicable.
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Affiliation(s)
- A Chechkin
- Institute of Physics and Astronomy, Potsdam University, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany and Akhiezer Institute for Theoretical Physics, Akademicheskaya Strasse 1, 61108 Kharkow, Ukraine
| | - I M Sokolov
- Institut für Physik and IRIS Adlershof, Humboldt Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
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Stanislavsky A, Weron A. Look at Tempered Subdiffusion in a Conjugate Map: Desire for the Confinement. ENTROPY (BASEL, SWITZERLAND) 2020; 22:E1317. [PMID: 33287082 PMCID: PMC7712244 DOI: 10.3390/e22111317] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/27/2020] [Revised: 11/15/2020] [Accepted: 11/16/2020] [Indexed: 11/17/2022]
Abstract
The Laplace distribution of random processes was observed in numerous situations that include glasses, colloidal suspensions, live cells, and firm growth. Its origin is not so trivial as in the case of Gaussian distribution, supported by the central limit theorem. Sums of Laplace distributed random variables are not Laplace distributed. We discovered a new mechanism leading to the Laplace distribution of observable values. This mechanism changes the contribution ratio between a jump and a continuous parts of random processes. Our concept uses properties of Bernstein functions and subordinators connected with them.
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Affiliation(s)
- Aleksander Stanislavsky
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wroclaw, Poland;
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Górska K, Horzela A, Lenzi EK, Pagnini G, Sandev T. Generalized Cattaneo (telegrapher's) equations in modeling anomalous diffusion phenomena. Phys Rev E 2020; 102:022128. [PMID: 32942420 DOI: 10.1103/physreve.102.022128] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2020] [Accepted: 07/20/2020] [Indexed: 11/07/2022]
Abstract
We study generalized Cattaneo (telegrapher's) equations involving memory effects introduced by smearing the time derivatives. Consistency conditions where the smearing functions obey restrict freedom in their choice but the proposed scheme goes beyond the approach based on using fractional derivatives. We find conditions under which solutions of the equations considered so far can be recognized as probability distributions, i.e., are normalizable and nonnegative on their domains. Nonnegativity of solutions is demonstrated by methods of positive definite and completely monotonic functions with the Bernstein theorem being the cornerstone of the ongoing proofs. Analysis of exactly solvable examples and relevant mean-squared displacements enables us to classify diffusion processes described by such got solutions and to identify them with either ordinary or anomalous diffusion which character may change over time. To complete the present research we compare our results with those obtained using the continuous-time random-walk and the continuous-time persistent random-walk approaches.
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Affiliation(s)
- K Górska
- Institute of Nuclear Physics, Polish Academy of Sciences, PL-31342 Kraków, Poland
| | - A Horzela
- Institute of Nuclear Physics, Polish Academy of Sciences, PL-31342 Kraków, Poland
| | - E K Lenzi
- Departamento de Fisica, Universidade Estadual de Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil
| | - G Pagnini
- BCAM-Basque Centre for Applied Mathematics, 48009 Bilbao, Basque Country Spain and Ikerbasque-Basque Foundation for Science, 48013 Bilbao, Basque Country, Spain
| | - T Sandev
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, 1000 Skopje, Macedonia, Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany and Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss Cyril and Methodius University, 1000 Skopje, Macedonia
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