1
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Escaff D. Anti-aligning interaction between active particles induces a finite wavelength instability: The dancing hexagons. Phys Rev E 2024; 109:024602. [PMID: 38491588 DOI: 10.1103/physreve.109.024602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2023] [Accepted: 01/11/2024] [Indexed: 03/18/2024]
Abstract
By considering a simple model for self-propelled particle interaction, we show that anti-aligning forces induce a finite wavelength instability. Consequently, the system exhibits pattern formation. The formed pattern involves, let us say, a choreographic movement of the active entities. At the level of particle density, the system oscillates between a stripe pattern and a hexagonal one. The underlying dynamics of these density oscillations consists of two counterpropagating and purely hexagonal traveling waves. They are assembling and disassembling a global hexagonal structure and a striped lineup of particles. This self-assembling process becomes quite erratic for long-time simulations, seeming aperiodic.
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Affiliation(s)
- Daniel Escaff
- Universidad de los Andes, Chile, Avenida Monseñor Álvaro del Portillo N° 12.455, Las Condes, Santiago 7620060, Chile
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2
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Zheng C, Tönjes R. Noise-induced swarming of active particles. Phys Rev E 2022; 106:064601. [PMID: 36671170 DOI: 10.1103/physreve.106.064601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/03/2022] [Accepted: 11/09/2022] [Indexed: 06/17/2023]
Abstract
We report on the effect of spatially correlated noise on the velocities of self-propelled particles. Correlations in the random forces acting on self-propelled particles can induce directed collective motion, i.e., swarming. Even with repulsive coupling in the velocity directions, which favors a disordered state, strong correlations in the fluctuations can align the velocities locally leading to a macroscopic, turbulent velocity field. On the other hand, while spatially correlated noise is aligning the velocities locally, the swarming transition to globally directed motion is inhibited when the correlation length of the noise is nonzero, but smaller than the system size. We analyze the swarming transition in d-dimensional space in a mean field model of globally coupled velocity vectors.
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Affiliation(s)
- Chunming Zheng
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany
| | - Ralf Tönjes
- Institute of Physics and Astronomy, Potsdam University, 14476 Potsdam-Golm, Germany
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3
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Bhattacharya K, Chakraborty A. Aggregation of self-propelled particles with sensitivity to local order. Phys Rev E 2022; 105:044124. [PMID: 35590585 DOI: 10.1103/physreve.105.044124] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/02/2021] [Accepted: 03/16/2022] [Indexed: 06/15/2023]
Abstract
We study a system of self-propelled particles (SPPs) in which individual particles are allowed to switch between a fast aligning and a slow nonaligning state depending upon the degree of the alignment in the neighborhood. The switching is modeled using a threshold for the local order parameter. This additional attribute gives rise to a mixed phase, in contrast to the ordered phases found in clean SPP systems. As the threshold is increased from zero, we find the sudden appearance of clusters of nonaligners. Clusters of nonaligners coexist with moving clusters of aligners with continual coalescence and fragmentation. The behavior of the system with respect to the clustering of nonaligners appears to be very different for values of low and high global densities. In the low density regime, for an optimal value of the threshold, the largest cluster of nonaligners grows in size up to a maximum that varies logarithmically with the total number of particles. However, on further increasing the threshold the size decreases. In contrast, for the high density regime, an initial abrupt rise is followed by the appearance of a giant cluster of nonaligners. The latter growth can be characterized as a continuous percolation transition. In addition, we find that the speed differences between aligners and nonaligners is necessary for the segregation of aligners and nonaligners.
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Affiliation(s)
- Kunal Bhattacharya
- Department of Industrial Engineering and Management, Aalto University School of Science, 00076 Aalto, Finland
- Department of Computer Science, Aalto University School of Science, 00076 Aalto, Finland
| | - Abhijit Chakraborty
- Complexity Science Hub Vienna, Josefstaedter Strasse 39, 1080 Vienna, Austria
- Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University, 1 Nakaadachi-cho, Yoshida, Sakyo-ku, Kyoto 606-8306, Japan
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4
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de Almeida RM, Giardini GS, Vainstein M, Glazier JA, Thomas GL. Exact solution for the Anisotropic Ornstein-Uhlenbeck process. PHYSICA A 2022; 587:126526. [PMID: 36937094 PMCID: PMC10022481 DOI: 10.1016/j.physa.2021.126526] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
Active-Matter models commonly consider particles with overdamped dynamics subject to a force (speed) with constant modulus and random direction. Some models also include random noise in particle displacement (a Wiener process), resulting in diffusive motion at short time scales. On the other hand, Ornstein-Uhlenbeck processes apply Langevin dynamics to the particles' velocity and predict motion that is not diffusive at short time scales. Experiments show that migrating cells have gradually varying speeds at intermediate and long time scales, with short-time diffusive behavior. While Ornstein-Uhlenbeck processes can describe the moderate-and long-time speed variation, Active-Matter models for over-damped particles can explain the short-time diffusive behavior. Isotropic models cannot explain both regimes, because short-time diffusion renders instantaneous velocity ill-defined, and prevents the use of dynamical equations that require velocity time-derivatives. On the other hand, both models correctly describe some of the different temporal regimes seen in migrating biological cells and must, in the appropriate limit, yield the same observable predictions. Here we propose and solve analytically an Anisotropic Ornstein-Uhlenbeck process for polarized particles, with Langevin dynamics governing the particle's movement in the polarization direction and a Wiener process governing displacement in the orthogonal direction. Our characterization provides a theoretically robust way to compare movement in dimensionless simulations to movement in experiments in which measurements have meaningful space and time units. We also propose an approach to deal with inevitable finite-precision effects in experiments and simulations.
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Affiliation(s)
- Rita M.C. de Almeida
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil
- Instituto Nacional de Ciência e Tecnologia, Sistemas Complexos, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil
- Program de Pós Graduação em Bioinformática, Universidade Federal do Rio Grande do Norte, Natal, RN, Brazil
| | | | - Mendeli Vainstein
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil
| | - James A. Glazier
- Biocomplexity Institute and Department of Intelligent Systems Engineering, Indiana University, Bloomington, IN, United States of America
| | - Gilberto L. Thomas
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil
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5
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Martin D, Chaté H, Nardini C, Solon A, Tailleur J, Van Wijland F. Fluctuation-Induced Phase Separation in Metric and Topological Models of Collective Motion. PHYSICAL REVIEW LETTERS 2021; 126:148001. [PMID: 33891435 DOI: 10.1103/physrevlett.126.148001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/04/2020] [Accepted: 02/01/2021] [Indexed: 06/12/2023]
Abstract
We study the role of noise on the nature of the transition to collective motion in dry active matter. Starting from field theories that predict a continuous transition at the deterministic level, we show that fluctuations induce a density-dependent shift of the onset of order, which in turn changes the nature of the transition into a phase-separation scenario. Our results apply to a range of systems, including models in which particles interact with their "topological" neighbors that have been believed so far to exhibit a continuous onset of order. Our analytical predictions are confirmed by numerical simulations of fluctuating hydrodynamics and microscopic models.
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Affiliation(s)
- David Martin
- Université de Paris, Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS, F-75205 Paris, France
| | - Hugues Chaté
- Service de Physique de l'Etat Condensé, CEA, CNRS Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France
- Computational Science Research Center, Beijing 100193, China
| | - Cesare Nardini
- Service de Physique de l'Etat Condensé, CEA, CNRS Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France
| | - Alexandre Solon
- Sorbonne Université, CNRS, Laboratoire Physique Théorique de la Matière Condensée, 75005 Paris, France
| | - Julien Tailleur
- Université de Paris, Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS, F-75205 Paris, France
| | - Frédéric Van Wijland
- Université de Paris, Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS, F-75205 Paris, France
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6
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Martin D, O'Byrne J, Cates ME, Fodor É, Nardini C, Tailleur J, van Wijland F. Statistical mechanics of active Ornstein-Uhlenbeck particles. Phys Rev E 2021; 103:032607. [PMID: 33862678 DOI: 10.1103/physreve.103.032607] [Citation(s) in RCA: 68] [Impact Index Per Article: 22.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/29/2020] [Accepted: 12/23/2020] [Indexed: 06/12/2023]
Abstract
We study the statistical properties of active Ornstein-Uhlenbeck particles (AOUPs). In this simplest of models, the Gaussian white noise of overdamped Brownian colloids is replaced by a Gaussian colored noise. This suffices to grant this system the hallmark properties of active matter, while still allowing for analytical progress. We study in detail the steady-state distribution of AOUPs in the small persistence time limit and for spatially varying activity. At the collective level, we show AOUPs to experience motility-induced phase separation both in the presence of pairwise forces or due to quorum-sensing interactions. We characterize both the instability mechanism leading to phase separation and the resulting phase coexistence. We probe how, in the stationary state, AOUPs depart from their thermal equilibrium limit by investigating the emergence of ratchet currents and entropy production. In the small persistence time limit, we show how fluctuation-dissipation relations are recovered. Finally, we discuss how the emerging properties of AOUPs can be characterized from the dynamics of their collective modes.
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Affiliation(s)
- David Martin
- Université de Paris, Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS,F-75205 Paris, France
| | - Jérémy O'Byrne
- Université de Paris, Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS,F-75205 Paris, France
| | - Michael E Cates
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
| | - Étienne Fodor
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
- Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg
| | - Cesare Nardini
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
- Service de Physique de l'État Condensé, CNRS UMR 3680, CEA-Saclay, 91191 Gif-sur-Yvette, France
| | - Julien Tailleur
- Université de Paris, Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS,F-75205 Paris, France
| | - Frédéric van Wijland
- Université de Paris, Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS,F-75205 Paris, France
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7
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Escaff D, Delpiano R. Flocking transition within the framework of Kuramoto paradigm for synchronization: Clustering and the role of the range of interaction. CHAOS (WOODBURY, N.Y.) 2020; 30:083137. [PMID: 32872818 DOI: 10.1063/5.0006218] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2020] [Accepted: 07/30/2020] [Indexed: 06/11/2023]
Abstract
A Kuramoto-type approach to address flocking phenomena is presented. First, we analyze a simple generalization of the Kuramoto model for interacting active particles, which is able to show the flocking transition (the emergence of coordinated movements in a group of interacting self-propelled agents). In the case of all-to-all interaction, the proposed model reduces to the Kuramoto model for phase synchronization of identical motionless noisy oscillators. In general, the nature of this non-equilibrium phase transition depends on the range of interaction between the particles. Namely, for a small range of interaction, the transition is first order, while for a larger range of interaction, it is a second order transition. Moreover, for larger interaction ranges, the system exhibits the same features as in the case of all-to-all interaction, showing a spatially homogeneous flux when flocking phenomenon has emerged, while for lower interaction ranges, the flocking transition is characterized by cluster formation. We compute the phase diagram of the model, where we distinguish three phases as a function of the range of interaction and the effective coupling strength: a disordered phase, a spatially homogeneous flocking phase, and a cluster-flocking phase. Then, we present a general discussion about the applicability of this way of modeling to more realistic and general situations, ending with a brief presentation of a second example (a second model with a conservative interaction) where the flocking transition may be studied within the framework that we are proposing.
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Affiliation(s)
- Daniel Escaff
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Monse nor Alvaro del Portillo 12455, Las Condes, Santiago, Chile
| | - Rafael Delpiano
- Transportation Studies Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Monse nor Alvaro del Portillo 12455, Las Condes, Santiago, Chile
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8
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Lee JS, Park JM, Park H. Thermodynamic uncertainty relation for underdamped Langevin systems driven by a velocity-dependent force. Phys Rev E 2019; 100:062132. [PMID: 31962517 DOI: 10.1103/physreve.100.062132] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2019] [Indexed: 06/10/2023]
Abstract
Recently, it has been shown that there is a trade-off relation between thermodynamic cost and current fluctuations, referred to as the thermodynamic uncertainty relation (TUR). The TUR has been derived for various processes, such as discrete-time Markov jump processes and overdamped Langevin dynamics. For underdamped dynamics, it has recently been reported that some modification is necessary for application of the TUR. However, the previous TUR for underdamped dynamics is not applicable to a system driven by a velocity-dependent force. In this study, we present a TUR, applicable to a system driven by a velocity-dependent force in the context of underdamped Langevin dynamics, by extending the theory of Vu and Hasegawa [Phys. Rev. E 100, 032130 (2019)2470-004510.1103/PhysRevE.100.032130]. We show that our TUR accurately describes the trade-off properties of a molecular refrigerator (cold damping), Brownian dynamics in a magnetic field, and an active particle system.
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Affiliation(s)
- Jae Sung Lee
- School of Physics and Quantum Universe Center, Korea Institute for Advanced Study, Seoul 02455, Korea
| | - Jong-Min Park
- School of Physics and Quantum Universe Center, Korea Institute for Advanced Study, Seoul 02455, Korea
| | - Hyunggyu Park
- School of Physics and Quantum Universe Center, Korea Institute for Advanced Study, Seoul 02455, Korea
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9
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Zumaya M, Larralde H, Aldana M. Delay in the dispersal of flocks moving in unbounded space using long-range interactions. Sci Rep 2018; 8:15872. [PMID: 30367121 PMCID: PMC6203710 DOI: 10.1038/s41598-018-34208-x] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/06/2018] [Accepted: 10/03/2018] [Indexed: 01/18/2023] Open
Abstract
Since the pioneering work by Vicsek and his collaborators on the motion of self-propelled particles, most of the subsequent studies have focused on the onset of ordered states through a phase transition driven by particle density and noise. Usually, the particles in these systems are placed within periodic boundary conditions and interact via short-range velocity alignment forces. However, when the periodic boundaries are eliminated, letting the particles move in open space, the system is not able to organize into a coherently moving group since even small amounts of noise cause the flock to break apart. While the phase transition has been thoroughly studied, the conditions to keep the flock cohesive in open space are still poorly understood. Here we extend the Vicsek model of collective motion by introducing long-range alignment interactions between the particles. We show that just a small number of these interactions is enough for the system to build up long lasting ordered states of collective motion in open space and in the presence of noise. This finding was verified for other models in addition to the Vicsek one, suggesting its generality and revealing the importance that long-range interactions can have for the cohesion of the flock.
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Affiliation(s)
- Martín Zumaya
- Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Avenida Universidad s/n, Colonia Chamilpa, Código Postal, 62210, Cuernavaca, Morelos, Mexico
- Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, Ciudad de México, Mexico
| | - Hernán Larralde
- Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Avenida Universidad s/n, Colonia Chamilpa, Código Postal, 62210, Cuernavaca, Morelos, Mexico
| | - Maximino Aldana
- Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Avenida Universidad s/n, Colonia Chamilpa, Código Postal, 62210, Cuernavaca, Morelos, Mexico.
- Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, Ciudad de México, Mexico.
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10
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Abstract
A theoretical analysis of active motion on curved surfaces is presented in terms of a generalization of the telegrapher equation. Such a generalized equation is explicitly derived as the polar approximation of the hierarchy of equations obtained from the corresponding Fokker-Planck equation of active particles diffusing on curved surfaces. The general solution to the generalized telegrapher equation is given for a pulse with vanishing current as initial data. Expressions for the probability density and the mean squared geodesic displacement are given in the limit of weak curvature. As an explicit example of the formulated theory, the case of active motion on the sphere is presented, where oscillations observed in the mean squared geodesic displacement are explained.
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Affiliation(s)
- Pavel Castro-Villarreal
- Facultad de Ciencias en Física y Matemáticas, Universidad Autónoma de Chiapas, Carretera Emiliano Zapata, Kilómetro 8, Rancho San Francisco, 29050 Tuxtla Gutiérrez, Chiapas, México
| | - Francisco J Sevilla
- Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000, Ciudad de México, México
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11
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Chun HM, Durang X, Noh JD. Emergence of nonwhite noise in Langevin dynamics with magnetic Lorentz force. Phys Rev E 2018; 97:032117. [PMID: 29776022 DOI: 10.1103/physreve.97.032117] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2017] [Indexed: 06/08/2023]
Abstract
We investigate the low mass limit of Langevin dynamics for a charged Brownian particle driven by a magnetic Lorentz force. In the low mass limit, velocity variables relaxing quickly are coarse-grained out to yield effective dynamics for position variables. Without the Lorentz force, the low mass limit is equivalent to the high friction limit. Both cases share the same Langevin equation that is obtained by setting the mass to zero. The equivalence breaks down in the presence of the Lorentz force. The low mass limit cannot be achieved by setting the mass to zero. The limit is also distinct from the large friction limit. We derive the effective equations of motion in the low mass limit. The resulting stochastic differential equation involves a nonwhite noise whose correlation matrix has antisymmetric components. We demonstrate the importance of the nonwhite noise by investigating the heat dissipation by a driven Brownian particle, where the emergent nonwhite noise has a physically measurable effect.
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Affiliation(s)
- Hyun-Myung Chun
- Department of Physics, University of Seoul, Seoul 02504, Korea
| | - Xavier Durang
- Department of Physics, University of Seoul, Seoul 02504, Korea
| | - Jae Dong Noh
- Department of Physics, University of Seoul, Seoul 02504, Korea
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
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12
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Lee HK, Lahiri S, Park H. Nonequilibrium steady states in Langevin thermal systems. Phys Rev E 2017; 96:022134. [PMID: 28950478 DOI: 10.1103/physreve.96.022134] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/28/2016] [Indexed: 06/07/2023]
Abstract
Equilibrium is characterized by its fundamental properties, such as the detailed balance, the fluctuation-dissipation relation, and no heat dissipation. Based on the stochastic thermodynamics, we show that these three properties are equivalent to each other in conventional Langevin thermal systems with microscopic reversibility. Thus, a conventional steady state has either all three properties (equilibrium) or none of them (nonequilibrium). In contrast, with velocity-dependent forces breaking the microscopic reversibility, we prove that the detailed balance and the fluctuation-dissipation relation mutually exclude each other, and no equivalence relation is possible between any two of the three properties. This implies that a steady state of Langevin systems with velocity-dependent forces may maintain some equilibrium properties but not all of them. Our results are illustrated with a few example systems.
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Affiliation(s)
- Hyun Keun Lee
- Department of Physics, Sungkyunkwan University, Suwon 16419, Korea
| | - Sourabh Lahiri
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
| | - Hyunggyu Park
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
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13
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Abstract
The diffusion of chiral active Brownian particles in three-dimensional space is studied analytically, by consideration of the corresponding Fokker-Planck equation for the probability density of finding a particle at position x and moving along the direction v[over ̂] at time t, and numerically, by the use of Langevin dynamics simulations. The analysis is focused on the marginal probability density of finding a particle at a given location and at a given time (independently of its direction of motion), which is found from an infinite hierarchy of differential-recurrence relations for the coefficients that appear in the multipole expansion of the probability distribution, which contains the whole kinematic information. This approach allows the explicit calculation of the time dependence of the mean-squared displacement and the time dependence of the kurtosis of the marginal probability distribution, quantities from which the effective diffusion coefficient and the "shape" of the positions distribution are examined. Oscillations between two characteristic values were found in the time evolution of the kurtosis, namely, between the value that corresponds to a Gaussian and the one that corresponds to a distribution of spherical shell shape. In the case of an ensemble of particles, each one rotating around a uniformly distributed random axis, evidence is found of the so-called effect "anomalous, yet Brownian, diffusion," for which particles follow a non-Gaussian distribution for the positions yet the mean-squared displacement is a linear function of time.
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Affiliation(s)
- Francisco J Sevilla
- Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000, México D.F., Mexico
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14
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Shim PS, Chun HM, Noh JD. Macroscopic time-reversal symmetry breaking at a nonequilibrium phase transition. Phys Rev E 2016; 93:012113. [PMID: 26871030 DOI: 10.1103/physreve.93.012113] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2015] [Indexed: 06/05/2023]
Abstract
We study the entropy production in a globally coupled Brownian particles system that undergoes an order-disorder phase transition. Entropy production is a characteristic feature of nonequilibrium dynamics with broken detailed balance. We find that the entropy production rate is subextensive in the disordered phase and extensive in the ordered phase. It is found that the entropy production rate per particle vanishes in the disordered phase and becomes positive in the ordered phase following critical scaling laws. We derive the scaling relations for associated critical exponents. The disordered phase exemplifies a case where the entropy production is subextensive with the broken detailed balance.
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Affiliation(s)
- Pyoung-Seop Shim
- Department of Physics, University of Seoul, Seoul 130-743, Korea
| | - Hyun-Myung Chun
- Department of Physics, University of Seoul, Seoul 130-743, Korea
| | - Jae Dong Noh
- Department of Physics, University of Seoul, Seoul 130-743, Korea
- School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
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