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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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Šiler M, Ornigotti L, Brzobohatý O, Jákl P, Ryabov A, Holubec V, Zemánek P, Filip R. Diffusing up the Hill: Dynamics and Equipartition in Highly Unstable Systems. PHYSICAL REVIEW LETTERS 2018; 121:230601. [PMID: 30576167 DOI: 10.1103/physrevlett.121.230601] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/19/2018] [Revised: 08/20/2018] [Indexed: 06/09/2023]
Abstract
Stochastic motion of particles in a highly unstable potential generates a number of diverging trajectories leading to undefined statistical moments of the particle position. This makes experiments challenging and breaks down a standard statistical analysis of unstable mechanical processes and their applications. A newly proposed approach takes advantage of the local characteristics of the most probable particle motion instead of the divergent averages. We experimentally verify its theoretical predictions for a Brownian particle moving near an inflection in a highly unstable cubic optical potential. The most likely position of the particle atypically shifts against the force, despite the trajectories diverging in the opposite direction. The local uncertainty around the most likely position saturates even for strong diffusion and enables well-resolved position detection. Remarkably, the measured particle distribution quickly converges to a quasistationary one with the same atypical shift for different initial particle positions. The demonstrated experimental confirmation of the theoretical predictions approves the utility of local characteristics for highly unstable systems which can be exploited in thermodynamic processes to uncover energetics of unstable systems.
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Affiliation(s)
- Martin Šiler
- Institute of Scientific Instruments of the Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic
| | - Luca Ornigotti
- Department of Optics, Palacký University, 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic
| | - Oto Brzobohatý
- Institute of Scientific Instruments of the Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic
| | - Petr Jákl
- Institute of Scientific Instruments of the Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic
| | - Artem Ryabov
- Charles University, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách 2, 180 00 Praha 8, Czech Republic
| | - Viktor Holubec
- Charles University, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách 2, 180 00 Praha 8, Czech Republic
- Universität Leipzig, Institut für Theoretische Physik, Postfach 100 920, D-04009 Leipzig, Germany
| | - Pavel Zemánek
- Institute of Scientific Instruments of the Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic
| | - Radim Filip
- Department of Optics, Palacký University, 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic
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Ornigotti L, Ryabov A, Holubec V, Filip R. Brownian motion surviving in the unstable cubic potential and the role of Maxwell's demon. Phys Rev E 2018; 97:032127. [PMID: 29776045 DOI: 10.1103/physreve.97.032127] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2017] [Indexed: 06/08/2023]
Abstract
The trajectories of an overdamped particle in a highly unstable potential diverge so rapidly, that the variance of position grows much faster than its mean. A description of the dynamics by moments is therefore not informative. Instead, we propose and analyze local directly measurable characteristics, which overcome this limitation. We discuss the most probable particle position (position of the maximum of the probability density) and the local uncertainty in an unstable cubic potential, V(x)∼x^{3}, both in the transient regime and in the long-time limit. The maximum shifts against the acting force as a function of time and temperature. Simultaneously, the local uncertainty does not increase faster than the observable shift. In the long-time limit, the probability density naturally attains a quasistationary form. We interpret this process as a stabilization via the measurement-feedback mechanism, the Maxwell demon, which works as an entropy pump. The rules for measurement and feedback naturally arise from the basic properties of the unstable dynamics. All reported effects are inherent in any unstable system. Their detailed understanding will stimulate the development of stochastic engines and amplifiers and, later, their quantum counterparts.
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Affiliation(s)
- Luca Ornigotti
- Department of Optics, Palacký University, 17. listopadu 1192/12, CZ-771 46 Olomouc, Czech Republic
| | - Artem Ryabov
- Charles University, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách 2, CZ-180 00 Praha 8, Czech Republic
| | - Viktor Holubec
- Charles University, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách 2, CZ-180 00 Praha 8, Czech Republic
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany
| | - Radim Filip
- Department of Optics, Palacký University, 17. listopadu 1192/12, CZ-771 46 Olomouc, Czech Republic
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Thermally induced micro-motion by inflection in optical potential. Sci Rep 2017; 7:1697. [PMID: 28490794 PMCID: PMC5432010 DOI: 10.1038/s41598-017-01848-4] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2017] [Accepted: 04/04/2017] [Indexed: 11/16/2022] Open
Abstract
Recent technological progress in a precise control of optically trapped objects allows much broader ventures to unexplored territory of thermal motion in non-linear potentials. In this work, we exploit an experimental set-up of holographic optical tweezers to experimentally investigate Brownian motion of a micro-particle near the inflection point of the cubic optical potential. We present two complementary views on the non-linear Brownian motion. On an ensemble of stochastic trajectories, we simultaneously determine (i) the detailed short-time position statistics and (ii) the long-distance first-passage time statistics. We evaluate specific statistical moment ratios demonstrating strongly non-linear stochastic dynamics. This is a crucial step towards a possible massive exploitation of the broad class of complex non-linear stochastic effects with objects of more complex structure and shape including living ones.
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