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Cichos F, Xia T, Yang H, Zijlstra P. The ever-expanding optics of single-molecules and nanoparticles. J Chem Phys 2024; 161:010401. [PMID: 38949895 DOI: 10.1063/5.0221680] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/02/2024] [Accepted: 06/10/2024] [Indexed: 07/03/2024] Open
Affiliation(s)
- F Cichos
- Peter Debye Institute for Soft Matter Physics, Leipzig University, Leipzig, Germany
| | - T Xia
- Institute for Immunology, School of Medicine, Tsinghua University, Beijing, China
| | - H Yang
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
| | - P Zijlstra
- Department of Applied Physics and Science Education, Eindhoven University of Technology (TU/e), Eindhoven, The Netherlands
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2
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Barkai E, Flaquer-Galmés R, Méndez V. Ergodic properties of Brownian motion under stochastic resetting. Phys Rev E 2023; 108:064102. [PMID: 38243500 DOI: 10.1103/physreve.108.064102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2023] [Accepted: 09/26/2023] [Indexed: 01/21/2024]
Abstract
We study the ergodic properties of one-dimensional Brownian motion with resetting. Using generic classes of statistics of times between resets, we find respectively for thin- or fat-tailed distributions the normalized or non-normalized invariant density of this process. The former case corresponds to known results in the resetting literature and the latter to infinite ergodic theory. Two types of ergodic transitions are found in this system. The first is when the mean waiting time between resets diverges, when standard ergodic theory switches to infinite ergodic theory. The second is when the mean of the square root of time between resets diverges and the properties of the invariant density are drastically modified. We then find a fractional integral equation describing the density of particles. This finite time tool is particularly useful close to the ergodic transition where convergence to asymptotic limits is logarithmically slow. Our study implies rich ergodic behaviors for this nonequilibrium process which should hold far beyond the case of Brownian motion analyzed here.
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Affiliation(s)
- E Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel
| | - R Flaquer-Galmés
- Grup de Física Estadística, Departament de Física, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Barcelona, Spain
| | - V Méndez
- Grup de Física Estadística, Departament de Física, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Barcelona, Spain
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3
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Samama A, Barkai E. Statistics of long-range force fields in random environments: Beyond Holtsmark. Phys Rev E 2023; 108:044116. [PMID: 37978642 DOI: 10.1103/physreve.108.044116] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/04/2022] [Accepted: 09/18/2023] [Indexed: 11/19/2023]
Abstract
Since the times of Holtsmark (1911), statistics of fields in random environments have been widely studied, for example in astrophysics, active matter, and line-shape broadening. The power-law decay of the two-body interaction of the form 1/|r|^{δ}, and assuming spatial uniformity of the medium particles exerting the forces, imply that the fields are fat-tailed distributed, and in general are described by stable Lévy distributions. With this widely used framework, the variance of the field diverges, which is nonphysical, due to finite size cutoffs. We find a complementary statistical law to the Lévy-Holtsmark distribution describing the large fields in the problem, which is related to the finite size of the tracer particle. We discover biscaling with a sharp statistical transition of the force moments taking place when the order of the moment is d/δ, where d is the dimension. The high-order moments, including the variance, are described by the framework presented in this paper, which is expected to hold for many systems. The new scaling solution found here is nonnormalized similar to infinite invariant densities found in dynamical systems.
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Affiliation(s)
- Avraham Samama
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
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4
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Wen B, Li MG, Liu J, Bao JD. Ergodic Measure and Potential Control of Anomalous Diffusion. ENTROPY (BASEL, SWITZERLAND) 2023; 25:1012. [PMID: 37509959 PMCID: PMC10377995 DOI: 10.3390/e25071012] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2023] [Revised: 06/15/2023] [Accepted: 06/27/2023] [Indexed: 07/30/2023]
Abstract
In statistical mechanics, the ergodic hypothesis (i.e., the long-time average is the same as the ensemble average) accompanying anomalous diffusion has become a continuous topic of research, being closely related to irreversibility and increasing entropy. While measurement time is finite for a given process, the time average of an observable quantity might be a random variable, whose distribution width narrows with time, and one wonders how long it takes for the convergence rate to become a constant. This is also the premise of ergodic establishment, because the ensemble average is always equal to the constant. We focus on the time-dependent fluctuation width for the time average of both the velocity and kinetic energy of a force-free particle described by the generalized Langevin equation, where the stationary velocity autocorrelation function is considered. Subsequently, the shortest time scale can be estimated for a system transferring from a stationary state to an effective ergodic state. Moreover, a logarithmic spatial potential is used to modulate the processes associated with free ballistic diffusion and the control of diffusion, as well as the minimal realization of the whole power-law regime. The results presented suggest that non-ergodicity mimics the sparseness of the medium and reveals the unique role of logarithmic potential in modulating diffusion behavior.
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Affiliation(s)
- Bao Wen
- Institutes of Science and Development, Chinese Academy of Sciences, Beijing 100190, China
- School of Public Policy and Management, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Ming-Gen Li
- Department of Physics, Beijing Normal University, Beijing 100875, China
| | - Jian Liu
- Department of Physics, Beijing Technology and Business University, Beijing 100048, China
| | - Jing-Dong Bao
- Department of Physics, Beijing Normal University, Beijing 100875, China
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5
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Sakai I, Akimoto T. Non-self-averaging of current in a totally asymmetric simple exclusion process with quenched disorder. Phys Rev E 2023; 107:L052103. [PMID: 37329050 DOI: 10.1103/physreve.107.l052103] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/26/2022] [Accepted: 05/01/2023] [Indexed: 06/18/2023]
Abstract
We investigate the current properties in the totally asymmetric simple exclusion process (TASEP) on a quenched random energy landscape. In low- and high-density regimes, the properties are characterized by single-particle dynamics. In the intermediate one, the current becomes constant and is maximized. Based on the renewal theory, we derive accurate results for the maximum current. The maximum current significantly depends on a disorder realization, i.e., non-self-averaging (SA). We demonstrate that the disorder average of the maximum current decreases with the system size, and the sample-to-sample fluctuations of the maximum current exceed those of current in the low- and high-density regimes. We find a significant difference between single-particle dynamics and the TASEP. In particular, the non-SA behavior of the maximum current is always observed, whereas the transition from non-SA to SA for current in single-particle dynamics exists.
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Affiliation(s)
- Issei Sakai
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
| | - Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
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6
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Cubero D. Brillouin propagation modes of cold atoms undergoing Sisyphus cooling. Phys Rev E 2023; 107:034102. [PMID: 37072942 DOI: 10.1103/physreve.107.034102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/18/2022] [Accepted: 02/13/2023] [Indexed: 04/20/2023]
Abstract
An exact expression for the average velocity of cold atoms in a driven, dissipative optical lattice in terms of the amplitudes of atomic density waves is derived from semiclassical equations for the phase space densities of the Zeeman ground-state sublevels. The calculations are for a J_{g}=1/2→J_{e}=3/2 transition, as it is customary in theoretical studies of Sisyphus cooling. While the driver, an additional beam of small amplitude, sets the atoms into directed motion, the new expression permits the quantification of the contribution to the atomic motion of a specific atomic wave, revealing unexpected counterpropagating contributions from many modes. Additionally, the method is shown to provide the generic threshold for the transition into the regime of infinite density, regardless of the details, or even the presence, of driving.
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Affiliation(s)
- David Cubero
- Departamento de Física Aplicada I, Escuela Politécnica Superior, Universidad de Sevilla, Calle Virgen de África 7, 41011 Sevilla, Spain
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7
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Defaveri L, Barkai E, Kessler DA. Brownian particles in periodic potentials: Coarse-graining versus fine structure. Phys Rev E 2023; 107:024122. [PMID: 36932490 DOI: 10.1103/physreve.107.024122] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2022] [Accepted: 01/24/2023] [Indexed: 06/18/2023]
Abstract
We study the motion of an overdamped particle connected to a thermal heat bath in the presence of an external periodic potential in one dimension. When we coarse-grain, i.e., bin the particle positions using bin sizes that are larger than the periodicity of the potential, the packet of spreading particles, all starting from a common origin, converges to a normal distribution centered at the origin with a mean-squared displacement that grows as 2D^{*}t, with an effective diffusion constant that is smaller than that of a freely diffusing particle. We examine the interplay between this coarse-grained description and the fine structure of the density, which is given by the Boltzmann-Gibbs (BG) factor e^{-V(x)/k_{B}T}, the latter being nonnormalizable. We explain this result and construct a theory of observables using the Fokker-Planck equation. These observables are classified as those that are related to the BG fine structure, like the energy or occupation times, while others, like the positional moments, for long times, converge to those of the large-scale description. Entropy falls into a special category as it has a coarse-grained and a fine structure description. The basic thermodynamic formula F=TS-E is extended to this far-from-equilibrium system. The ergodic properties are also studied using tools from infinite ergodic theory.
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Affiliation(s)
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat Gan 52900, Israel
| | - David A Kessler
- Department of Physics, Bar-Ilan University, Ramat Gan 52900, Israel
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8
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Kimura M, Akimoto T. Occupation time statistics of the fractional Brownian motion in a finite domain. Phys Rev E 2022; 106:064132. [PMID: 36671174 DOI: 10.1103/physreve.106.064132] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2022] [Accepted: 12/09/2022] [Indexed: 06/17/2023]
Abstract
We study statistics of occupation times for a fractional Brownian motion (fBm), which is a typical model of a non-Markov process. Due to the non-Markovian nature, recurrence times to the origin depend on the history. Numerical simulations indicate that dependence on the sum of successive recurrence times becomes weak. As a result, the distribution of the occupation time in a finite domain follows the Mittag-Leffler distribution when the Hurst exponent of the fBm is close to 1/2. We show this distributional behavior of a time-averaged observable by renewal theory. This result is an extension of the distributional limit theorem known as the Darling-Kac theorem in general Markov processes to non-Markov processes.
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Affiliation(s)
- Mutsumi Kimura
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
| | - Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
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9
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Akimoto T, Barkai E, Radons G. Infinite ergodic theory for three heterogeneous stochastic models with application to subrecoil laser cooling. Phys Rev E 2022; 105:064126. [PMID: 35854593 DOI: 10.1103/physreve.105.064126] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/02/2022] [Accepted: 05/25/2022] [Indexed: 06/15/2023]
Abstract
We compare ergodic properties of the kinetic energy for three stochastic models of subrecoil-laser-cooled gases. One model is based on a heterogeneous random walk (HRW), another is an HRW with long-range jumps (the exponential model), and the other is a mean-field-like approximation of the exponential model (the deterministic model). All the models show an accumulation of the momentum at zero in the long-time limit, and a formal steady state cannot be normalized, i.e., there exists an infinite invariant density. We obtain the exact form of the infinite invariant density and the scaling function for the exponential and deterministic models, and we devise a useful approximation for the momentum distribution in the HRW model. While the models are kinetically nonidentical, it is natural to wonder whether their ergodic properties share common traits, given that they are all described by an infinite invariant density. We show that the answer to this question depends on the type of observable under study. If the observable is integrable, the ergodic properties, such as the statistical behavior of the time averages, are universal as they are described by the Darling-Kac theorem. In contrast, for nonintegrable observables, the models in general exhibit nonidentical statistical laws. This implies that focusing on nonintegrable observables, we discover nonuniversal features of the cooling process, which hopefully can lead to a better understanding of the particular model most suitable for a statistical description of the process. This result is expected to hold true for many other systems, beyond laser cooling.
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Affiliation(s)
- Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
| | - Eli Barkai
- Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel
| | - Günter Radons
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
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10
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Bao JD, Li Y, Marchesoni F. Consistent Hamiltonian models for space-momentum diffusion. Phys Rev E 2022; 105:L052105. [PMID: 35706220 DOI: 10.1103/physreve.105.l052105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2021] [Accepted: 05/15/2022] [Indexed: 06/15/2023]
Abstract
We develop a unified Hamiltonian approach to the diffusion of a particle coupled to a dissipative environment, an archetypal model widely invoked to interpret condensed phase phenomena, such as polymerization and cold-atom diffusion in optical lattices. By appropriate choices of the coupling functions, we reformulate phenomenological diffusion models by adding otherwise ignored space-momentum terms. We thus numerically predict a variety of diffusion regimes, from diffusion saturation to superballistic diffusion. With reference to ultracold atoms in optical lattices, we also show that time correlated external noises prevent superdiffusion from exceeding Richardson's law. Some of these results are unexpected and call for experimental validation.
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Affiliation(s)
- Jing-Dong Bao
- Department of Physics, Beijing Normal University, Beijing 100875, China
| | - Yunyun Li
- Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
| | - Fabio Marchesoni
- Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
- Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italy
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11
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Barkai E, Radons G, Akimoto T. Gas of sub-recoiled laser cooled atoms described by infinite ergodic theory. J Chem Phys 2022; 156:044118. [DOI: 10.1063/5.0076552] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 52900, Israel
| | - Günter Radons
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany and Institute of Mechatronics, 09126 Chemnitz, Germany
| | - Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
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12
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Radice M. One-dimensional telegraphic process with noninstantaneous stochastic resetting. Phys Rev E 2021; 104:044126. [PMID: 34781456 DOI: 10.1103/physreve.104.044126] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2021] [Accepted: 10/08/2021] [Indexed: 06/13/2023]
Abstract
In this paper, we consider the one-dimensional dynamical evolution of a particle traveling at constant speed and performing, at a given rate, random reversals of the velocity direction. The particle is subject to stochastic resetting, meaning that at random times it is forced to return to the starting point. Here we consider a return mechanism governed by a deterministic law of motion, so that the time cost required to return is correlated to the position occupied at the time of the reset. We show that in such conditions the process reaches a stationary state which, for some kinds of deterministic return dynamics, is independent of the return phase. Furthermore, we investigate the first-passage properties of the system and provide explicit formulas for the mean first-hitting time. Our findings are supported by numerical simulations.
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Affiliation(s)
- Mattia Radice
- Dipartimento di Scienza e Alta Tecnologia and Center for Nonlinear and Complex Systems, Università degli studi dell'Insubria, Via Valleggio 11, 22100 Como, Italy and I.N.F.N. Sezione di Milano, Via Celoria 16, 20133 Milano, Italy
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