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Biroli M, Larralde H, Majumdar SN, Schehr G. Exact extreme, order, and sum statistics in a class of strongly correlated systems. Phys Rev E 2024; 109:014101. [PMID: 38366495 DOI: 10.1103/physreve.109.014101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2023] [Accepted: 11/27/2023] [Indexed: 02/18/2024]
Abstract
Even though strongly correlated systems are abundant, only a few exceptional cases admit analytical solutions. In this paper we present a large class of solvable systems with strong correlations. We consider a set of N independent and identically distributed random variables {X_{1},X_{2},...,X_{N}} whose common distribution has a parameter Y (or a set of parameters) which itself is random with its own distribution. For a fixed value of this parameter Y, the X_{i} variables are independent and we call them conditionally independent and identically distributed. However, once integrated over the distribution of the parameter Y, the X_{i} variables get strongly correlated yet retain a solvable structure for various observables, such as for the sum and the extremes of X_{i}^{'}s. This provides a simple procedure to generate a class of solvable strongly correlated systems. We illustrate how this procedure works via three physical examples where N particles on a line perform independent (i) Brownian motions, (ii) ballistic motions with random initial velocities, and (iii) Lévy flights, but they get strongly correlated via simultaneous resetting to the origin. Our results are verified in numerical simulations. This procedure can be used to generate an endless variety of solvable strongly correlated systems.
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Affiliation(s)
- Marco Biroli
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Hernán Larralde
- Instituto de Ciencias Físicas, UNAM, CP 62210 Cuernavaca Morelos, México
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, 75252 Paris Cedex 05, France
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Singh RK, Sandev T, Singh S. Bernoulli trial under restarts: A comparative study of resetting transitions. Phys Rev E 2023; 108:L052106. [PMID: 38115400 DOI: 10.1103/physreve.108.l052106] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/02/2023] [Accepted: 10/23/2023] [Indexed: 12/21/2023]
Abstract
A Bernoulli trial describing the escape behavior of a lamb to a safe haven in pursuit by a lion is studied under restarts. The process ends in two ways: either the lamb makes it to the safe haven (success) or is captured by the lion (failure). We study the first passage properties of this Bernoulli trial and find that only mean first passage time exists. Considering Poisson and sharp resetting, we find that the success probability is a monotonically decreasing function of the restart rate. The mean time, however, exhibits a nonmonotonic dependence on the restart rate taking a minimal value at an optimal restart rate. Furthermore, for sharp restart, the mean time possesses a local and a global minima. As a result, the optimal restart rate exhibits a continuous transition for Poisson resetting while it exhibits a discontinuous transition for sharp resetting as a function of the relative separation of the lion and the lamb. We also find that the distribution of first passage times under sharp resetting exhibits a periodic behavior.
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Affiliation(s)
- R K Singh
- Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel
| | - T Sandev
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
- Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany
- Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
| | - Sadhana Singh
- The Avram and Stella Goldstein-Goren Department of Biotechnology Engineering, Ben-Gurion University of the Negev, Be'er Sheva 84105, Israel
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Di Bello C, Hartmann AK, Majumdar SN, Mori F, Rosso A, Schehr G. Current fluctuations in stochastically resetting particle systems. Phys Rev E 2023; 108:014112. [PMID: 37583217 DOI: 10.1103/physreve.108.014112] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/07/2023] [Accepted: 04/28/2023] [Indexed: 08/17/2023]
Abstract
We consider a system of noninteracting particles on a line with initial positions distributed uniformly with density ρ on the negative half-line. We consider two different models: (i) Each particle performs independent Brownian motion with stochastic resetting to its initial position with rate r and (ii) each particle performs run-and-tumble motion, and with rate r its position gets reset to its initial value and simultaneously its velocity gets randomized. We study the effects of resetting on the distribution P(Q,t) of the integrated particle current Q up to time t through the origin (from left to right). We study both the annealed and the quenched current distributions and in both cases, we find that resetting induces a stationary limiting distribution of the current at long times. However, we show that the approach to the stationary state of the current distribution in the annealed and the quenched cases are drastically different for both models. In the annealed case, the whole distribution P_{an}(Q,t) approaches its stationary limit uniformly for all Q. In contrast, the quenched distribution P_{qu}(Q,t) attains its stationary form for QQ_{crit}(t). We show that Q_{crit}(t) increases linearly with t for large t. On the scale where Q∼Q_{crit}(t), we show that P_{qu}(Q,t) has an unusual large deviation form with a rate function that has a third-order phase transition at the critical point. We have computed the associated rate functions analytically for both models. Using an importance sampling method that allows to probe probabilities as tiny as 10^{-14000}, we were able to compute numerically this nonanalytic rate function for the resetting Brownian dynamics and found excellent agreement with our analytical prediction.
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Affiliation(s)
- Costantino Di Bello
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | | | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Francesco Mori
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom
| | - Alberto Rosso
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, 75252 Paris Cedex 05, France
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Biroli M, Majumdar SN, Schehr G. Critical number of walkers for diffusive search processes with resetting. Phys Rev E 2023; 107:064141. [PMID: 37464619 DOI: 10.1103/physreve.107.064141] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2023] [Accepted: 05/30/2023] [Indexed: 07/20/2023]
Abstract
We consider N Brownian motions diffusing independently on a line, starting at x_{0}>0, in the presence of an absorbing target at the origin. The walkers undergo stochastic resetting under two protocols: (A) each walker resets independently to x_{0} with rate r and (B) all walkers reset simultaneously to x_{0} with rate r. We derive an explicit analytical expression for the mean first-passage time to the origin in terms of an integral which is evaluated numerically using Mathematica. We show that, as a function of r and for fixed x_{0}, it has a minimum at an optimal value r^{*}>0 as long as N<N_{c}. Thus resetting is beneficial for the search for N<N_{c}. When N>N_{c}, the optimal value occurs at r^{*}=0 indicating that resetting hinders search processes. We obtain different values of N_{c} for protocols A and B; indeed, for N≤7 resetting is beneficial in protocol A, while for N≤6 resetting is beneficial for protocol B. Our theoretical predictions are verified in numerical Langevin simulations.
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Affiliation(s)
- Marco Biroli
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Satya N Majumdar
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
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Biroli M, Larralde H, Majumdar SN, Schehr G. Extreme Statistics and Spacing Distribution in a Brownian Gas Correlated by Resetting. PHYSICAL REVIEW LETTERS 2023; 130:207101. [PMID: 37267543 DOI: 10.1103/physrevlett.130.207101] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2022] [Revised: 03/21/2023] [Accepted: 04/14/2023] [Indexed: 06/04/2023]
Abstract
We study a one-dimensional gas of N Brownian particles that diffuse independently, but are simultaneously reset to the origin at a constant rate r. The system approaches a nonequilibrium stationary state with long-range interactions induced by the simultaneous resetting. Despite the presence of strong correlations, we show that several observables can be computed exactly, which include the global average density, the distribution of the position of the kth rightmost particle, and the spacing distribution between two successive particles. Our analytical results are confirmed by numerical simulations. We also discuss a possible experimental realization of this resetting gas using optical traps.
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Affiliation(s)
- Marco Biroli
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Hernan Larralde
- Instituto de Ciencias Fisicas, UNAM, Avenida Universidad s/n, CP 62210 Cuernavaca, Morelos, Mexico
| | - Satya N Majumdar
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
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Sasorov P, Vilenkin A, Smith NR. Probabilities of moderately atypical fluctuations of the size of a swarm of Brownian bees. Phys Rev E 2023; 107:014140. [PMID: 36797921 DOI: 10.1103/physreve.107.014140] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/15/2022] [Accepted: 01/17/2023] [Indexed: 02/03/2023]
Abstract
The "Brownian bees" model describes an ensemble of N= const independent branching Brownian particles. The conservation of N is provided by a modified branching process. When a particle branches into two particles, the particle which is farthest from the origin is eliminated simultaneously. The spatial density of the particles is governed by the solution of a free boundary problem for a reaction-diffusion equation in the limit of N≫1. At long times, the particle density approaches a spherically symmetric steady-state solution with a compact support of radius ℓ[over ¯]_{0}. However, at finite N, the radius of this support, L, fluctuates. The variance of these fluctuations appears to exhibit a logarithmic anomaly [Siboni et al., Phys. Rev. E 104, 054131 (2021)2470-004510.1103/PhysRevE.104.054131]. It is proportional to N^{-1}lnN at N→∞. We investigate here the tails of the probability density function (PDF), P(L), of the swarm radius, when the absolute value of the radius fluctuation ΔL=L-ℓ[over ¯]_{0} is sufficiently larger than the typical fluctuations' scale determined by the variance. For negative deviations the PDF can be obtained in the framework of the optimal fluctuation method. This part of the PDF displays the scaling behavior lnP∝-NΔL^{2}ln^{-1}(ΔL^{-2}), demonstrating a logarithmic anomaly at small negative ΔL. For the opposite sign of the fluctuation, ΔL>0, the PDF can be obtained with an approximation of a single particle, running away. We find that lnP∝-N^{1/2}ΔL. We consider in this paper only the case when |ΔL| is much less than the typical radius of the swarm at N≫1.
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Affiliation(s)
- Pavel Sasorov
- Institute of Physics CAS, ELI Beamlines, 182 21 Prague, Czech Republic
| | - Arkady Vilenkin
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Naftali R Smith
- Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer, 8499000, Israel
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Krapivsky PL, Vilk O, Meerson B. Competition in a system of Brownian particles: Encouraging achievers. Phys Rev E 2022; 106:034125. [PMID: 36266791 DOI: 10.1103/physreve.106.034125] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2022] [Accepted: 09/01/2022] [Indexed: 06/16/2023]
Abstract
We introduce and analytically and numerically study a simple model of interagent competition, where underachievement is strongly discouraged. We consider N≫1 particles performing independent Brownian motions on the line. Two particles are selected at random and at random times, and the particle closest to the origin is reset to it. We show that, in the limit of N→∞, the dynamics of the coarse-grained particle density field can be described by a nonlocal hydrodynamic theory which was encountered in a study of the spatial extent of epidemics in a critical regime. The hydrodynamic theory predicts relaxation of the system toward a stationary density profile of the "swarm" of particles, which exhibits a power-law decay at large distances. An interesting feature of this relaxation is a nonstationary "halo" around the stationary solution, which continues to expand in a self-similar manner. The expansion is ultimately arrested by finite-N effects at a distance of order sqrt[N] from the origin, which gives an estimate of the average radius of the swarm. The hydrodynamic theory does not capture the behavior of the particle farthest from the origin-the current leader. We suggest a simple scenario for typical fluctuations of the leader's distance from the origin and show that the mean distance continues to grow indefinitely as sqrt[t]. Finally, we extend the inter-agent competition from n=2 to an arbitrary number n of competing Brownian particles (n≪N). Our analytical predictions are supported by Monte Carlo simulations.
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Affiliation(s)
- P L Krapivsky
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
- Santa Fe Institute, Santa Fe, New Mexico 87501, USA
| | - Ohad Vilk
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
- Movement Ecology Lab, Department of Ecology, Evolution and Behavior, Alexander Silberman Institute of Life Sciences, Hebrew University of Jerusalem, Jerusalem 91904, Israel
- Minerva Center for Movement Ecology, Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Baruch Meerson
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
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