1
|
Burenev IN, Majumdar SN, Rosso A. Occupation time of a system of Brownian particles on the line with steplike initial condition. Phys Rev E 2024; 109:044150. [PMID: 38755944 DOI: 10.1103/physreve.109.044150] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/30/2023] [Accepted: 02/29/2024] [Indexed: 05/18/2024]
Abstract
We consider a system of noninteracting Brownian particles on the line with steplike initial condition and study the statistics of the occupation time on the positive half-line. We demonstrate that even at large times, the behavior of the occupation time exhibits long-lasting memory effects of the initialization. Specifically, we calculate the mean and the variance of the occupation time, demonstrating that the memory effects in the variance are determined by a generalized compressibility (or Fano factor), associated with the initial condition. In the particular case of the uncorrelated uniform initial condition we conduct a detailed study of two probability distributions of the occupation time: annealed (averaged over all possible initial configurations) and quenched (for a typical configuration). We show that at large times both the annealed and the quenched distributions admit large deviation form and we compute analytically the associated rate functions. We verify our analytical predictions via numerical simulations using importance sampling Monte Carlo strategy.
Collapse
Affiliation(s)
- Ivan N Burenev
- LPTMS, CNRS, Université Paris-Saclay, 91405 Orsay, France
| | | | - Alberto Rosso
- LPTMS, CNRS, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
2
|
Biroli M, Kulkarni M, Majumdar SN, Schehr G. Dynamically emergent correlations between particles in a switching harmonic trap. Phys Rev E 2024; 109:L032106. [PMID: 38632730 DOI: 10.1103/physreve.109.l032106] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/07/2023] [Accepted: 02/09/2024] [Indexed: 04/19/2024]
Abstract
We study a one dimensional gas of N noninteracting diffusing particles in a harmonic trap, whose stiffness switches between two values μ_{1} and μ_{2} with constant rates r_{1} and r_{2}, respectively. Despite the absence of direct interaction between the particles, we show that strong correlations between them emerge in the stationary state at long times, induced purely by the dynamics itself. We compute exactly the joint distribution of the positions of the particles in the stationary state, which allows us to compute several physical observables analytically. In particular, we show that the extreme value statistics (EVS), i.e., the distribution of the position of the rightmost particle, has a nontrivial shape in the large N limit. The scaling function characterizing this EVS has a finite support with a tunable shape (by varying the parameters). Remarkably, this scaling function turns out to be universal. First, it also describes the distribution of the position of the kth rightmost particle in a 1d trap. Moreover, the distribution of the position of the particle farthest from the center of the harmonic trap in d dimensions is also described by the same scaling function for all d≥1. Numerical simulations are in excellent agreement with our analytical predictions.
Collapse
Affiliation(s)
- Marco Biroli
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Manas Kulkarni
- ICTS, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - 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
| |
Collapse
|
3
|
Boyer D, Mercado-Vásquez G, Majumdar SN, Schehr G. Optimizing the random search of a finite-lived target by a Lévy flight. Phys Rev E 2024; 109:L022103. [PMID: 38491567 DOI: 10.1103/physreve.109.l022103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2023] [Accepted: 01/22/2024] [Indexed: 03/18/2024]
Abstract
In many random search processes of interest in chemistry, biology, or during rescue operations, an entity must find a specific target site before the latter becomes inactive, no longer available for reaction or lost. We present exact results on a minimal model system, a one-dimensional searcher performing a discrete time random walk, or Lévy flight. In contrast with the case of a permanent target, the capture probability and the conditional mean first passage time can be optimized. The optimal Lévy index takes a nontrivial value, even in the long lifetime limit, and exhibits an abrupt transition as the initial distance to the target is varied. Depending on the target lifetime, this transition is discontinuous or continuous, separated by a nonconventional tricritical point. These results pave the way to the optimization of search processes under time constraints.
Collapse
Affiliation(s)
- Denis Boyer
- Instituto de Física, Universidad Nacional Autónoma de México, Ciudad de México 04510, México
| | - Gabriel Mercado-Vásquez
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois, 60637, USA
| | - 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
| |
Collapse
|
4
|
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] [What about the content of this article? (0)] [Affiliation(s)] [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.
Collapse
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
| |
Collapse
|
5
|
Majumdar SN, Mori F, Vivo P. Nonlinear-cost random walk: Exact statistics of the distance covered for fixed budget. Phys Rev E 2023; 108:064122. [PMID: 38243551 DOI: 10.1103/physreve.108.064122] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/13/2023] [Accepted: 11/22/2023] [Indexed: 01/21/2024]
Abstract
We consider the nonlinear-cost random-walk model in discrete time introduced in Phys. Rev. Lett. 130, 237102 (2023)10.1103/PhysRevLett.130.237102, where a fee is charged for each jump of the walker. The nonlinear cost function is such that slow or short jumps incur a flat fee, while for fast or long jumps the cost is proportional to the distance covered. In this paper we compute analytically the average and variance of the distance covered in n steps when the total budget C is fixed, as well as the statistics of the number of long or short jumps in a trajectory of length n, for the exponential jump distribution. These observables exhibit a very rich and nonmonotonic scaling behavior as a function of the variable C/n, which is traced back to the makeup of a typical trajectory in terms of long or short jumps, and the resulting entropy thereof. As a by-product, we compute the asymptotic behavior of ratios of Kummer hypergeometric functions when both the first and last arguments are large. All our analytical results are corroborated by numerical simulations.
Collapse
Affiliation(s)
- Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, and Université Paris-Saclay, 91405 Orsay, France
| | - Francesco Mori
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom
| | - Pierpaolo Vivo
- Department of Mathematics, King's College London, London WC2R 2LS, United Kingdom
| |
Collapse
|
6
|
Burenev IN, Majumdar SN, Rosso A. Local time of a system of Brownian particles on the line with steplike initial condition. Phys Rev E 2023; 108:064113. [PMID: 38243455 DOI: 10.1103/physreve.108.064113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2023] [Accepted: 10/30/2023] [Indexed: 01/21/2024]
Abstract
We consider a system of noninteracting Brownian particles on a line with a steplike initial condition, and we investigate the behavior of the local time at the origin at large times. We compute the mean and the variance of the local time, and we show that the memory effects are governed by the Fano factor associated with the initial condition. For the uniform initial condition, we show that the probability distribution of the local time admits a large deviation form, and we compute the corresponding large deviation functions for the annealed and quenched averaging schemes. The two resulting large deviation functions are very different. Our analytical results are supported by extensive numerical simulations.
Collapse
Affiliation(s)
- Ivan N Burenev
- LPTMS, CNRS, Université Paris-Saclay, 91405 Orsay, France
| | | | - Alberto Rosso
- LPTMS, CNRS, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
7
|
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] [What about the content of this article? (0)] [Affiliation(s)] [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.
Collapse
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
| |
Collapse
|
8
|
Majumdar SN, Mori F, Vivo P. Cost of Diffusion: Nonlinearity and Giant Fluctuations. Phys Rev Lett 2023; 130:237102. [PMID: 37354426 DOI: 10.1103/physrevlett.130.237102] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2023] [Revised: 04/14/2023] [Accepted: 05/15/2023] [Indexed: 06/26/2023]
Abstract
We introduce a simple model of diffusive jump process where a fee is charged for each jump. The nonlinear cost function is such that slow jumps incur a flat fee, while for fast jumps the cost is proportional to the velocity of the jump. The model-inspired by the way taxi meters work-exhibits a very rich behavior. The cost for trajectories of equal length and equal duration exhibits giant fluctuations at a critical value of the scaled distance traveled. Furthermore, the full distribution of the cost until the target is reached exhibits an interesting "freezing" transition in the large-deviation regime. All the analytical results are corroborated by numerical simulations. Our results also apply to elastic systems near the depinning transition, when driven by a random force.
Collapse
Affiliation(s)
- 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 3RH, United Kingdom
| | - Pierpaolo Vivo
- Department of Mathematics, King's College London, London WC2R 2LS, United Kingdom
| |
Collapse
|
9
|
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] [What about the content of this article? (0)] [Affiliation(s)] [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.
Collapse
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
| |
Collapse
|
10
|
Flack A, Le Doussal P, Majumdar SN, Schehr G. Out-of-equilibrium dynamics of repulsive ranked diffusions: The expanding crystal. Phys Rev E 2023; 107:064105. [PMID: 37464700 DOI: 10.1103/physreve.107.064105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2023] [Accepted: 05/05/2023] [Indexed: 07/20/2023]
Abstract
We study the nonequilibrium Langevin dynamics of N particles in one dimension with Coulomb repulsive linear interactions. This is a dynamical version of the so-called jellium model (without confinement) also known as ranked diffusion. Using a mapping to the Lieb-Liniger model of quantum bosons, we obtain an exact formula for the joint distribution of the positions of the N particles at time t, all starting from the origin. A saddle-point analysis shows that the system converges at long time to a linearly expanding crystal. Properly rescaled, this dynamical state resembles the equilibrium crystal in a time-dependent effective quadratic potential. This analogy allows us to study the fluctuations around the perfect crystal, which, to leading order, are Gaussian. There are however deviations from this Gaussian behavior, which embody long-range correlations of purely dynamical origin, characterized by the higher-order cumulants of, e.g., the gaps between the particles, which we calculate exactly. We complement these results using a recent approach by one of us in terms of a noisy Burgers equation. In the large-N limit, the mean density of the gas can be obtained at any time from the solution of a deterministic viscous Burgers equation. This approach provides a quantitative description of the dense regime at shorter times. Our predictions are in good agreement with numerical simulations for finite and large N.
Collapse
Affiliation(s)
- Ana Flack
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Pierre Le Doussal
- Laboratoire de Physique de l'Ecole Normale Supérieure, CNRS, ENS and PSL Université, Sorbonne Université, and Université Paris Cité, 24 Rue Lhomond, 75005 Paris, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR No. 7589, Sorbonne Université, 4 Place Jussieu, 75252 Paris Cedex 05, France
| |
Collapse
|
11
|
Biroli M, Larralde H, Majumdar SN, Schehr G. Extreme Statistics and Spacing Distribution in a Brownian Gas Correlated by Resetting. Phys Rev Lett 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] [What about the content of this article? (0)] [Affiliation(s)] [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.
Collapse
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
| |
Collapse
|
12
|
Mori F, Majumdar SN, Schehr G. Time to reach the maximum for a stationary stochastic process. Phys Rev E 2022; 106:054110. [PMID: 36559509 DOI: 10.1103/physreve.106.054110] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2022] [Accepted: 10/11/2022] [Indexed: 11/06/2022]
Abstract
We consider a one-dimensional stationary time series of fixed duration T. We investigate the time t_{m} at which the process reaches the global maximum within the time interval [0,T]. By using a path-decomposition technique, we compute the probability density function P(t_{m}|T) of t_{m} for several processes, that are either at equilibrium (such as the Ornstein-Uhlenbeck process) or out of equilibrium (such as Brownian motion with stochastic resetting). We show that for equilibrium processes the distribution of P(t_{m}|T) is always symmetric around the midpoint t_{m}=T/2, as a consequence of the time-reversal symmetry. This property can be used to detect nonequilibrium fluctuations in stationary time series. Moreover, for a diffusive particle in a confining potential, we show that the scaled distribution P(t_{m}|T) becomes universal, i.e., independent of the details of the potential, at late times. This distribution P(t_{m}|T) becomes uniform in the "bulk" 1≪t_{m}≪T and has a nontrivial universal shape in the "edge regimes" t_{m}→0 and t_{m}→T. Some of these results have been announced in a recent letter [Europhys. Lett. 135, 30003 (2021)0295-507510.1209/0295-5075/ac19ee].
Collapse
Affiliation(s)
- Francesco Mori
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - 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 4 Place Jussieu, 75252 Paris Cedex 05, France
| |
Collapse
|
13
|
Smith NR, Le Doussal P, Majumdar SN, Schehr G. Exact position distribution of a harmonically confined run-and-tumble particle in two dimensions. Phys Rev E 2022; 106:054133. [PMID: 36559430 DOI: 10.1103/physreve.106.054133] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/14/2022] [Accepted: 10/21/2022] [Indexed: 11/16/2022]
Abstract
We consider an overdamped run-and-tumble particle in two dimensions, with self-propulsion in an orientation that stochastically rotates by 90^{∘} at a constant rate, clockwise or counterclockwise with equal probabilities. In addition, the particle is confined by an external harmonic potential of stiffness μ, and possibly diffuses. We find the exact time-dependent distribution P(x,y,t) of the particle's position, and in particular, the steady-state distribution P_{st}(x,y) that is reached in the long-time limit. We also find P(x,y,t) for a "free" particle, μ=0. We achieve this by showing that, under a proper change of coordinates, the problem decomposes into two statistically independent one-dimensional problems, whose exact solution has recently been obtained. We then extend these results in several directions, to two such run-and-tumble particles with a harmonic interaction, to analogous systems of dimension three or higher, and by allowing stochastic resetting.
Collapse
Affiliation(s)
- Naftali R Smith
- Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus 8499000, Israel
| | - Pierre Le Doussal
- Laboratoire de Physique de l'Ecole Normale Supérieure, CNRS, ENS and Université PSL, Sorbonne Université, Université de Paris, 75005 Paris, France
| | | | - Grégory Schehr
- Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
| |
Collapse
|
14
|
Tucci G, Gambassi A, Majumdar SN, Schehr G. First-passage time of run-and-tumble particles with noninstantaneous resetting. Phys Rev E 2022; 106:044127. [PMID: 36397533 DOI: 10.1103/physreve.106.044127] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/04/2022] [Accepted: 09/09/2022] [Indexed: 06/16/2023]
Abstract
We study the statistics of the first-passage time of a single run-and-tumble particle (RTP) in one spatial dimension, with or without resetting, to a fixed target located at L>0. First, we compute the first-passage time distribution of a free RTP, without resetting or in a confining potential, but averaged over the initial position drawn from an arbitrary distribution p(x). Recent experiments used a noninstantaneous resetting protocol that motivated us to study in particular the case where p(x) corresponds to the stationary non-Boltzmann distribution of an RTP in the presence of a harmonic trap. This distribution p(x) is characterized by a parameter ν>0, which depends on the microscopic parameters of the RTP dynamics. We show that the first-passage time distribution of the free RTP, drawn from this initial distribution, develops interesting singular behaviors, depending on the value of ν. We then switch on resetting, mimicked by relaxation of the RTP in the presence of a harmonic trap. Resetting leads to a finite mean first-passage time and we study this as a function of the resetting rate for different values of the parameters ν and b=L/c, where c is the position of the right edge of the initial distribution p(x). In the diffusive limit of the RTP dynamics, we find a rich phase diagram in the (b,ν) plane, with an interesting reentrance phase transition. Away from the diffusive limit, qualitatively similar rich behaviors emerge for the full RTP dynamics.
Collapse
Affiliation(s)
- Gennaro Tucci
- SISSA-International School for Advanced Studies and INFN, via Bonomea 265, I-34136 Trieste, Italy
| | - Andrea Gambassi
- SISSA-International School for Advanced Studies and INFN, via Bonomea 265, I-34136 Trieste, Italy
| | - 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, 4 Place Jussieu, 75252 Paris Cedex 05, France
| |
Collapse
|
15
|
De Bruyne B, Majumdar SN, Schehr G. Optimal Resetting Brownian Bridges via Enhanced Fluctuations. Phys Rev Lett 2022; 128:200603. [PMID: 35657896 DOI: 10.1103/physrevlett.128.200603] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/06/2022] [Revised: 04/06/2022] [Accepted: 04/25/2022] [Indexed: 06/15/2023]
Abstract
We introduce a resetting Brownian bridge as a simple model to study search processes where the total search time t_{f} is finite and the searcher returns to its starting point at t_{f}. This is simply a Brownian motion with a Poissonian resetting rate r to the origin which is constrained to start and end at the origin at time t_{f}. We unveil a surprising general mechanism that enhances fluctuations of a Brownian bridge, by introducing a small amount of resetting. This is verified for different observables, such as the mean-square displacement, the hitting probability of a fixed target and the expected maximum. This mechanism, valid for a Brownian bridge in arbitrary dimensions, leads to a finite optimal resetting rate that minimizes the time to search a fixed target. The physical reason behind an optimal resetting rate in this case is entirely different from that of resetting Brownian motions without the bridge constraint. We also derive an exact effective Langevin equation that generates numerically the trajectories of a resetting Brownian bridge in all dimensions via a completely rejection-free algorithm.
Collapse
Affiliation(s)
- Benjamin De Bruyne
- 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
| |
Collapse
|
16
|
Schimmenti VM, Majumdar SN, Rosso A. Statistical properties of avalanches via the c-record process. Phys Rev E 2021; 104:064129. [PMID: 35030910 DOI: 10.1103/physreve.104.064129] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/04/2021] [Accepted: 10/26/2021] [Indexed: 06/14/2023]
Abstract
We study the statistics of avalanches, as a response to an applied force, undergone by a particle hopping on a one-dimensional lattice where the pinning forces at each site are independent and identically distributed (i.i.d.), each drawn from a continuous f(x). The avalanches in this model correspond to the interrecord intervals in a modified record process of i.i.d. variables, defined by a single parameter c>0. This parameter characterizes the record formation via the recursive process R_{k}>R_{k-1}-c, where R_{k} denotes the value of the kth record. We show that for c>0, if f(x) decays slower than an exponential for large x, the record process is nonstationary as in the standard c=0 case. In contrast, if f(x) has a faster than exponential tail, the record process becomes stationary and the avalanche size distribution π(n) has a decay faster than 1/n^{2} for large n. The marginal case where f(x) decays exponentially for large x exhibits a phase transition from a nonstationary phase to a stationary phase as c increases through a critical value c_{crit}. Focusing on f(x)=e^{-x} (with x≥0), we show that c_{crit}=1 and for c<1, the record statistics is nonstationary. However, for c>1, the record statistics is stationary with avalanche size distribution π(n)∼n^{-1-λ(c)} for large n. Consequently, for c>1, the mean number of records up to N steps grows algebraically ∼N^{λ(c)} for large N. Remarkably, the exponent λ(c) depends continuously on c for c>1 and is given by the unique positive root of c=-ln(1-λ)/λ. We also unveil the presence of nontrivial correlations between avalanches in the stationary phase that resemble earthquake sequences.
Collapse
Affiliation(s)
| | | | - Alberto Rosso
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
| |
Collapse
|
17
|
Le Doussal P, Majumdar SN, Schehr G. Stationary nonequilibrium bound state of a pair of run and tumble particles. Phys Rev E 2021; 104:044103. [PMID: 34781527 DOI: 10.1103/physreve.104.044103] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2021] [Accepted: 09/09/2021] [Indexed: 11/07/2022]
Abstract
We study two interacting identical run-and-tumble particles (RTPs) in one dimension. Each particle is driven by a telegraphic noise and, in some cases, also subjected to a thermal white noise with a corresponding diffusion constant D. We are interested in the stationary bound state formed by the two RTPs in the presence of a mutual attractive interaction. The distribution of the relative coordinate y indeed reaches a steady state that we characterize in terms of the solution of a second-order differential equation. We obtain the explicit formula for the stationary probability P(y) of y for two examples of interaction potential V(y). The first one corresponds to V(y)∼|y|. In this case, for D=0 we find that P(y) contains a δ function part at y=0, signaling a strong clustering effect, together with a smooth exponential component. For D>0, the δ function part broadens, leading instead to weak clustering. The second example is the harmonic attraction V(y)∼y^{2} in which case, for D=0, P(y) is supported on a finite interval. We unveil an interesting relation between this two-RTP model with harmonic attraction and a three-state single-RTP model in one dimension, as well as with a four-state single-RTP model in two dimensions. We also provide a general discussion of the stationary bound state, including examples where it is not unique, e.g., when the particles cannot cross due to an additional short-range repulsion.
Collapse
Affiliation(s)
- Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France
| | | | - Grégory Schehr
- Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
| |
Collapse
|
18
|
De Bruyne B, Majumdar SN, Schehr G. Generating discrete-time constrained random walks and Lévy flights. Phys Rev E 2021; 104:024117. [PMID: 34525540 DOI: 10.1103/physreve.104.024117] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2021] [Accepted: 07/27/2021] [Indexed: 11/07/2022]
Abstract
We introduce a method to exactly generate bridge trajectories for discrete-time random walks, with arbitrary jump distributions, that are constrained to initially start at the origin and return to the origin after a fixed time. The method is based on an effective jump distribution that implicitly accounts for the bridge constraint. It is illustrated on various jump distributions and is shown to be very efficient in practice. In addition, we show how to generalize the method to other types of constrained random walks such as generalized bridges, excursions, and meanders.
Collapse
Affiliation(s)
- Benjamin De Bruyne
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - 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, 4 Place Jussieu, 75252 Paris Cedex 05, France
| |
Collapse
|
19
|
Mori F, Le Doussal P, Majumdar SN, Schehr G. Condensation transition in the late-time position of a run-and-tumble particle. Phys Rev E 2021; 103:062134. [PMID: 34271704 DOI: 10.1103/physreve.103.062134] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2021] [Accepted: 05/25/2021] [Indexed: 11/07/2022]
Abstract
We study the position distribution P(R[over ⃗],N) of a run-and-tumble particle (RTP) in arbitrary dimension d, after N runs. We assume that the constant speed v>0 of the particle during each running phase is independently drawn from a probability distribution W(v) and that the direction of the particle is chosen isotropically after each tumbling. The position distribution is clearly isotropic, P(R[over ⃗],N)→P(R,N) where R=|R[over ⃗]|. We show that, under certain conditions on d and W(v) and for large N, a condensation transition occurs at some critical value of R=R_{c}∼O(N) located in the large-deviation regime of P(R,N). For R<R_{c} (subcritical fluid phase), all runs are roughly of the same size in a typical trajectory. In contrast, an RTP trajectory with R>R_{c} is typically dominated by a "condensate," i.e., a large single run that subsumes a finite fraction of the total displacement (supercritical condensed phase). Focusing on the family of speed distributions W(v)=α(1-v/v_{0})^{α-1}/v_{0}, parametrized by α>0, we show that, for large N, P(R,N)∼exp[-Nψ_{d,α}(R/N)], and we compute exactly the rate function ψ_{d,α}(z) for any d and α. We show that the transition manifests itself as a singularity of this rate function at R=R_{c} and that its order depends continuously on d and α. We also compute the distribution of the condensate size for R>R_{c}. Finally, we study the model when the total duration T of the RTP, instead of the total number of runs, is fixed. Our analytical predictions are confirmed by numerical simulations, performed using a constrained Markov chain Monte Carlo technique, with precision ∼10^{-100}.
Collapse
Affiliation(s)
- Francesco Mori
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Pierre Le Doussal
- Laboratoire de Physique de l'Ecole Normale Supérieure, PSL University, CNRS, Sorbonne Universités, 24 rue Lhomond, 75231 Paris, 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
| |
Collapse
|
20
|
Besga B, Faisant F, Petrosyan A, Ciliberto S, Majumdar SN. Dynamical phase transition in the first-passage probability of a Brownian motion. Phys Rev E 2021; 104:L012102. [PMID: 34412290 DOI: 10.1103/physreve.104.l012102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/14/2021] [Accepted: 06/22/2021] [Indexed: 06/13/2023]
Abstract
We study the first-passage time distribution (FPTD) F(t_{f}|x_{0},L) for a freely diffusing particle starting at x_{0} in one dimension, to a target located at L, averaged over the initial position x_{0} drawn from a normalized distribution (1/σ)g(x_{0}/σ) of finite width σ. We show the averaged FPTD undergoes a sharp dynamical phase transition from a two-peak structure for b=L/σ>b_{c} to a single-peak structure for b<b_{c}. This transition is generated by the competition of two characteristic timescales σ^{2}/D and L^{2}/D, where D is the diffusion coefficient. A very good agreement is found between theoretical predictions and experimental results obtained with a Brownian bead whose diffusion is initialized by an optical trap which determines the initial distribution g(x_{0}/σ). We show that this transition is robust: It is present for all initial conditions with a finite σ, in all dimensions, and also exists for more general stochastic processes going beyond free diffusion.
Collapse
Affiliation(s)
- B Besga
- Université Lyon, ENS de Lyon, Université Claude Bernard, CNRS, Laboratoire de Physique, UMR 5672, F-69342 Lyon, France
| | - F Faisant
- Université Lyon, ENS de Lyon, Université Claude Bernard, CNRS, Laboratoire de Physique, UMR 5672, F-69342 Lyon, France
| | - A Petrosyan
- Université Lyon, ENS de Lyon, Université Claude Bernard, CNRS, Laboratoire de Physique, UMR 5672, F-69342 Lyon, France
| | - S Ciliberto
- Université Lyon, ENS de Lyon, Université Claude Bernard, CNRS, Laboratoire de Physique, UMR 5672, F-69342 Lyon, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, UMR 8626, F-91405 Orsay, France
| |
Collapse
|
21
|
Smith NR, Le Doussal P, Majumdar SN, Schehr G. Counting statistics for noninteracting fermions in a d-dimensional potential. Phys Rev E 2021; 103:L030105. [PMID: 33862753 DOI: 10.1103/physreve.103.l030105] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/07/2020] [Accepted: 03/03/2021] [Indexed: 11/07/2022]
Abstract
We develop a first-principles approach to compute the counting statistics in the ground state of N noninteracting spinless fermions in a general potential in arbitrary dimensions d (central for d>1). In a confining potential, the Fermi gas is supported over a bounded domain. In d=1, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions N_{D} in a domain D of macroscopic size in the bulk of the support. We show that the variance of N_{D} grows as N^{(d-1)/d}(A_{d}logN+B_{d}) for large N, and obtain the explicit dependence of A_{d},B_{d} on the potential and on the size of D (for a spherical domain in d>1). This generalizes the free-fermion results for microscopic domains, given in d=1 by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem D, in any dimension, supported by exact results for d=1.
Collapse
Affiliation(s)
- Naftali R Smith
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
22
|
Gradenigo G, Iubini S, Livi R, Majumdar SN. Condensation transition and ensemble inequivalence in the discrete nonlinear Schrödinger equation. Eur Phys J E Soft Matter 2021; 44:29. [PMID: 33710395 DOI: 10.1140/epje/s10189-021-00046-5] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/16/2020] [Accepted: 02/25/2021] [Indexed: 06/12/2023]
Abstract
The thermodynamics of the discrete nonlinear Schrödinger equation in the vicinity of infinite temperature is explicitly solved in the microcanonical ensemble by means of large-deviation techniques. A first-order phase transition between a thermalized phase and a condensed (localized) one occurs at the infinite-temperature line. Inequivalence between statistical ensembles characterizes the condensed phase, where the grand-canonical representation does not apply. The control over finite-size corrections of the microcanonical partition function allows to design an experimental test of delocalized negative-temperature states in lattices of cold atoms.
Collapse
Affiliation(s)
- Giacomo Gradenigo
- Gran Sasso Science Institute, Viale F. Crispi 7, 67100, L'Aquila, Italy.
| | - Stefano Iubini
- Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Via Madonna del Piano 10, 50019, Sesto Fiorentino, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, Via G. Sansone 1, 50019, Sesto Fiorentino, Italy
| | - Roberto Livi
- Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Via Madonna del Piano 10, 50019, Sesto Fiorentino, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, Via G. Sansone 1, 50019, Sesto Fiorentino, Italy
- Dipartimento di Fisica e Astronomia and CSDC, Università di Firenze, Via G. Sansone 1, 50019, Sesto Fiorentino, Italy
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405, Orsay, France
| |
Collapse
|
23
|
Majumdar SN, Mori F, Schawe H, Schehr G. Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting. Phys Rev E 2021; 103:022135. [PMID: 33736082 DOI: 10.1103/physreve.103.022135] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2020] [Accepted: 01/28/2021] [Indexed: 06/12/2023]
Abstract
We compute exactly the mean perimeter and the mean area of the convex hull of a two-dimensional isotropic Brownian motion of duration t and diffusion constant D, in the presence of resetting to the origin at a constant rate r. We show that for any t, the mean perimeter is given by 〈L(t)〉=2πsqrt[D/r]f_{1}(rt) and the mean area is given by 〈A(t)〉=2πD/rf_{2}(rt) where the scaling functions f_{1}(z) and f_{2}(z) are computed explicitly. For large t≫1/r, the mean perimeter grows extremely slowly as 〈L(t)〉∝ln(rt) with time. Likewise, the mean area also grows slowly as 〈A(t)〉∝ln^{2}(rt) for t≫1/r. Our exact results indicate that the convex hull, in the presence of resetting, approaches a circular shape at late times due to the isotropy of the Brownian motion. Numerical simulations are in perfect agreement with our analytical predictions.
Collapse
Affiliation(s)
- Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Francesco Mori
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Hendrik Schawe
- Laboratoire de Physique Théorique et Modélisation, UMR-8089 CNRS, CY Cergy Paris Université, 95510 Cergy, France
| | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
24
|
Abstract
We study a class of stochastic processes of the type d^{n}x/dt^{n}=v_{0}σ(t) where n>0 is a positive integer and σ(t)=±1 represents an active telegraphic noise that flips from one state to the other with a constant rate γ. For n=1, it reduces to the standard run-and-tumble process for active particles in one dimension. This process can be analytically continued to any n>0, including noninteger values. We compute exactly the mean-squared displacement at time t for all n>0 and show that at late times while it grows as ∼t^{2n-1} for n>1/2, it approaches a constant for n<1/2. In the marginal case n=1/2, it grows very slowly with time as ∼lnt. Thus, the process undergoes a localization transition at n=1/2. We also show that the position distribution p_{n}(x,t) remains time-dependent even at late times for n≥1/2, but approaches a stationary time-independent form for n<1/2. The tails of the position distribution at late times exhibit a large deviation form, p_{n}(x,t)∼exp[-γtΦ_{n}(x/x^{*}(t))], where x^{*}(t)=v_{0}t^{n}/Γ(n+1). We compute the rate function Φ_{n}(z) analytically for all n>0 and also numerically using importance sampling methods, finding excellent agreement between them. For three special values n=1, n=2, and n=1/2 we compute the exact cumulant-generating function of the position distribution at all times t.
Collapse
Affiliation(s)
- David S Dean
- Univ. Bordeaux and CNRS, Laboratoire Ondes et Matière d'Aquitaine (LOMA), UMR 5798, F-33400 Talence, France
| | - Satya N Majumdar
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Hendrik Schawe
- LPTM, UMR 8089, CY Cergy Paris Université, CNRS, 95000 Cergy, France
| |
Collapse
|
25
|
Sabhapandit S, Majumdar SN. Freezing Transition in the Barrier Crossing Rate of a Diffusing Particle. Phys Rev Lett 2020; 125:200601. [PMID: 33258622 DOI: 10.1103/physrevlett.125.200601] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2020] [Revised: 09/21/2020] [Accepted: 10/08/2020] [Indexed: 06/12/2023]
Abstract
We study the decay rate θ(a) that characterizes the late time exponential decay of the first-passage probability density F_{a}(t|0)∼e^{-θ(a)t} of a diffusing particle in a one dimensional confining potential U(x), starting from the origin, to a position located at a>0. For general confining potential U(x) we show that θ(a), a measure of the barrier (located at a) crossing rate, has three distinct behaviors as a function of a, depending on the tail of U(x) as x→-∞. In particular, for potentials behaving as U(x)∼|x| when x→-∞, we show that a novel freezing transition occurs at a critical value a=a_{c}, i.e., θ(a) increases monotonically as a decreases till a_{c}, and for a≤a_{c} it freezes to θ(a)=θ(a_{c}). Our results are established using a general mapping to a quantum problem and by exact solution in three representative cases, supported by numerical simulations. We show that the freezing transition occurs when in the associated quantum problem, the gap between the ground state (bound) and the continuum of scattering states vanishes.
Collapse
Affiliation(s)
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
26
|
Mori F, Le Doussal P, Majumdar SN, Schehr G. Universal properties of a run-and-tumble particle in arbitrary dimension. Phys Rev E 2020; 102:042133. [PMID: 33212668 DOI: 10.1103/physreve.102.042133] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2020] [Accepted: 08/18/2020] [Indexed: 06/11/2023]
Abstract
We consider an active run-and-tumble particle (RTP) in d dimensions, starting from the origin and evolving over a time interval [0,t]. We examine three different models for the dynamics of the RTP: the standard RTP model with instantaneous tumblings, a variant with instantaneous runs and a general model in which both the tumblings and the runs are noninstantaneous. For each of these models, we use the Sparre Andersen theorem for discrete-time random walks to compute exactly the probability that the x component does not change sign up to time t, showing that it does not depend on d. As a consequence of this result, we compute exactly other x-component properties, namely, the distribution of the time of the maximum and the record statistics, showing that they are universal, i.e., they do not depend on d. Moreover, we show that these universal results hold also if the speed v of the particle after each tumbling is random, drawn from a generic probability distribution. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 124, 090603 (2020)10.1103/PhysRevLett.124.090603].
Collapse
Affiliation(s)
- Francesco Mori
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Pierre Le Doussal
- Laboratoire de Physique de l'Ecole Normale Supérieure, PSL University, CNRS, Sorbonne Universités, 24 rue Lhomond, 75231 Paris, France
| | - Satya N Majumdar
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
27
|
Majumdar SN, Meerson B. Toward the full short-time statistics of an active Brownian particle on the plane. Phys Rev E 2020; 102:022113. [PMID: 32942466 DOI: 10.1103/physreve.102.022113] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/28/2020] [Accepted: 07/21/2020] [Indexed: 06/11/2023]
Abstract
We study the position distribution of a single active Brownian particle (ABP) on the plane. We show that this distribution has a compact support, the boundary of which is an expanding circle. We focus on a short-time regime and employ the optimal fluctuation method to study large deviations of the particle position coordinates x and y. We determine the optimal paths of the ABP, conditioned on reaching specified values of x and y, and the large deviation functions of the marginal distributions of x and of y. These marginal distributions match continuously with "near tails" of the x and y distributions of typical fluctuations, studied earlier. We also calculate the large deviation function of the joint x and y distribution P(x,y,t) in a vicinity of a special "zero-noise" point, and show that lnP(x,y,t) has a nontrivial self-similar structure as a function of x, y, and t. The joint distribution vanishes extremely fast at the expanding circle, exhibiting an essential singularity there. This singularity is inherited by the marginal x- and y-distributions. We argue that this fingerprint of the short-time dynamics remains there at all times.
Collapse
Affiliation(s)
- Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Baruch Meerson
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
| |
Collapse
|
28
|
Mori F, Majumdar SN, Schehr G. Distribution of the time between maximum and minimum of random walks. Phys Rev E 2020; 101:052111. [PMID: 32575204 DOI: 10.1103/physreve.101.052111] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2020] [Accepted: 04/17/2020] [Indexed: 06/11/2023]
Abstract
We consider a one-dimensional Brownian motion of fixed duration T. Using a path-integral technique, we compute exactly the probability distribution of the difference τ=t_{min}-t_{max} between the time t_{min} of the global minimum and the time t_{max} of the global maximum. We extend this result to a Brownian bridge, i.e., a periodic Brownian motion of period T. In both cases, we compute analytically the first few moments of τ, as well as the covariance of t_{max} and t_{min}, showing that these times are anticorrelated. We demonstrate that the distribution of τ for Brownian motion is valid for discrete-time random walks with n steps and with a finite jump variance, in the limit n→∞. In the case of Lévy flights, which have a divergent jump variance, we numerically verify that the distribution of τ differs from the Brownian case. For random walks with continuous and symmetric jumps we numerically verify that the probability of the event "τ=n" is exactly 1/(2n) for any finite n, independently of the jump distribution. Our results can be also applied to describe the distance between the maximal and minimal height of (1+1)-dimensional stationary-state Kardar-Parisi-Zhang interfaces growing over a substrate of finite size L. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 123, 200201 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.200201].
Collapse
Affiliation(s)
- Francesco Mori
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
29
|
Banerjee T, Majumdar SN, Rosso A, Schehr G. Current fluctuations in noninteracting run-and-tumble particles in one dimension. Phys Rev E 2020; 101:052101. [PMID: 32575200 DOI: 10.1103/physreve.101.052101] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2020] [Accepted: 03/30/2020] [Indexed: 06/11/2023]
Abstract
We present a general framework to study the distribution of the flux through the origin up to time t, in a noninteracting one-dimensional system of particles with a step initial condition with a fixed density ρ of particles to the left of the origin. We focus principally on two cases: (i) particles undergoing diffusive dynamics (passive case) and (ii) run-and-tumble dynamics for each particle (active case). In analogy with disordered systems, we consider the flux distribution for both the annealed and the quenched initial conditions, for passive and active particles. In the annealed case, we show that, for arbitrary particle dynamics, the flux distribution is a Poissonian with a mean μ(t) that we compute exactly in terms of the Green's function of the single-particle dynamics. For the quenched case, we show that, for the run-and-tumble dynamics, the quenched flux distribution takes an anomalous large-deviation form at large times, P_{qu}(Q,t)∼exp[-ρv_{0}γt^{2}ψ_{RTP}(Q/ρv_{0}t)], where γ is the rate of tumbling and v_{0} is the ballistic speed between two successive tumblings. In this paper, we compute the rate function ψ_{RTP}(q) and show that it is nontrivial. Our method also gives access to the probability of the rare event that, at time t, there is no particle to the right of the origin. For diffusive and run-and-tumble dynamics, we find that this probability decays with time as a stretched exponential, ∼exp(-csqrt[t]), where the constant c can be computed exactly. We verify our results for these large deviations by using an importance sampling Monte Carlo method.
Collapse
Affiliation(s)
- Tirthankar Banerjee
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
- Instituut voor Theoretische Fysica, KU Leuven, 3001 Heverlee, Belgium
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Alberto Rosso
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
30
|
Mori F, Le Doussal P, Majumdar SN, Schehr G. Universal Survival Probability for a d-Dimensional Run-and-Tumble Particle. Phys Rev Lett 2020; 124:090603. [PMID: 32202896 DOI: 10.1103/physrevlett.124.090603] [Citation(s) in RCA: 23] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2019] [Accepted: 02/05/2020] [Indexed: 06/10/2023]
Abstract
We consider an active run-and-tumble particle (RTP) in d dimensions and compute exactly the probability S(t) that the x component of the position of the RTP does not change sign up to time t. When the tumblings occur at a constant rate, we show that S(t) is independent of d for any finite time t (and not just for large t), as a consequence of the celebrated Sparre Andersen theorem for discrete-time random walks in one dimension. Moreover, we show that this universal result holds for a much wider class of RTP models in which the speed v of the particle after each tumbling is random, drawn from an arbitrary probability distribution. We further demonstrate, as a consequence, the universality of the record statistics in the RTP problem.
Collapse
Affiliation(s)
- Francesco Mori
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Pierre Le Doussal
- Laboratoire de Physique de l'Ecole Normale Supérieure, PSL University, CNRS, Sorbonne Universités, 24 rue Lhomond, 75231 Paris, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
31
|
Basu U, Majumdar SN, Rosso A, Schehr G. Long-time position distribution of an active Brownian particle in two dimensions. Phys Rev E 2019; 100:062116. [PMID: 31962395 DOI: 10.1103/physreve.100.062116] [Citation(s) in RCA: 26] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/10/2019] [Indexed: 06/10/2023]
Abstract
We study the late-time dynamics of a single active Brownian particle in two dimensions with speed v_{0} and rotation diffusion constant D_{R}. We show that at late times t≫D_{R}^{-1}, while the position probability distribution P(x,y,t) in the x-y plane approaches a Gaussian form near its peak describing the typical diffusive fluctuations, it has non-Gaussian tails describing atypical rare fluctuations when sqrt[x^{2}+y^{2}]∼v_{0}t. In this regime, the distribution admits a large deviation form, P(x,y,t)∼exp{-tD_{R}Φ[sqrt[x^{2}+y^{2}]/(v_{0}t)]}, where we compute the rate function Φ(z) analytically and also numerically using an importance sampling method. We show that the rate function Φ(z), encoding the rare fluctuations, still carries the trace of activity even at late times. Another way of detecting activity at late times is to subject the active particle to an external harmonic potential. In this case we show that the stationary distribution P_{stat}(x,y) depends explicitly on the activity parameter D_{R}^{-1} and undergoes a crossover, as D_{R} increases, from a ring shape in the strongly active limit (D_{R}→0) to a Gaussian shape in the strongly passive limit (D_{R}→∞).
Collapse
Affiliation(s)
- Urna Basu
- Raman Research Institute, Bengaluru 560080, India
| | - Satya N Majumdar
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Alberto Rosso
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
32
|
Abstract
We present an exact solution for the probability density function P(τ=t_{min}-t_{max}|T) of the time difference between the minimum and the maximum of a one-dimensional Brownian motion of duration T. We then generalize our results to a Brownian bridge, i.e., a periodic Brownian motion of period T. We demonstrate that these results can be directly applied to study the position difference between the minimal and the maximal heights of a fluctuating (1+1)-dimensional Kardar-Parisi-Zhang interface on a substrate of size L, in its stationary state. We show that the Brownian motion result is universal and, asymptotically, holds for any discrete-time random walk with a finite jump variance. We also compute this distribution numerically for Lévy flights and find that it differs from the Brownian motion result.
Collapse
Affiliation(s)
- Francesco Mori
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
33
|
Agarwal S, Dhar A, Kulkarni M, Kundu A, Majumdar SN, Mukamel D, Schehr G. Harmonically Confined Particles with Long-Range Repulsive Interactions. Phys Rev Lett 2019; 123:100603. [PMID: 31573302 DOI: 10.1103/physrevlett.123.100603] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2019] [Indexed: 06/10/2023]
Abstract
We study an interacting system of N classical particles on a line at thermal equilibrium. The particles are confined by a harmonic trap and repel each other via pairwise interaction potential that behaves as a power law ∝∑[under i≠j][over N]|x_{i}-x_{j}|^{-k} (with k>-2) of their mutual distance. This is a generalization of the well-known cases of the one-component plasma (k=-1), Dyson's log gas (k→0^{+}), and the Calogero-Moser model (k=2). Because of the competition between harmonic confinement and pairwise repulsion, the particles spread over a finite region of space for all k>-2. We compute exactly the average density profile for large N for all k>-2 and show that while it is independent of temperature for sufficiently low temperature, it has a rich and nontrivial dependence on k with distinct behavior for -2<k<1, k>1 and k=1.
Collapse
Affiliation(s)
- S Agarwal
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
- Birla Institute of Technology and Science, Pilani 333031, India
| | - A Dhar
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - M Kulkarni
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - A Kundu
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - S N Majumdar
- LPTMS, CNRS, Univ. Paris-Sud, Universite Paris-Saclay, 91405 Orsay, France
| | - D Mukamel
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
| | - G Schehr
- LPTMS, CNRS, Univ. Paris-Sud, Universite Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
34
|
Abstract
We revisit a family of integrals that delude intuition and that recently appeared in mathematical literature in connection with computer algebra package verification. We show that the remarkable properties displayed by these integrals become transparent when formulated in the language of random walks. In turn, the random walk view naturally leads to a plethora of nontrivial generalizations that are worked out. Related complex identities are also derived, without the need of explicit calculation. The crux of our treatment lies in a causality argument where a message that travels at finite speed signals the existence of a boundary.
Collapse
Affiliation(s)
- Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Emmanuel Trizac
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
35
|
Lacroix-A-Chez-Toine B, Garzón JAM, Calva CSH, Castillo IP, Kundu A, Majumdar SN, Schehr G. Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble. Phys Rev E 2019; 100:012137. [PMID: 31499884 DOI: 10.1103/physreve.100.012137] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2019] [Indexed: 06/10/2023]
Abstract
We study the Ginibre ensemble of N×N complex random matrices and compute exactly, for any finite N, the full distribution as well as all the cumulants of the number N_{r} of eigenvalues within a disk of radius r centered at the origin. In the limit of large N, when the average density of eigenvalues becomes uniform over the unit disk, we show that for 0<r<1 the fluctuations of N_{r} around its mean value 〈N_{r}〉≈Nr^{2} display three different regimes: (i) a typical Gaussian regime where the fluctuations are of order O(N^{1/4}), (ii) an intermediate regime where N_{r}-〈N_{r}〉=O(sqrt[N]), and (iii) a large deviation regime where N_{r}-〈N_{r}〉=O(N). This intermediate behavior (ii) had been overlooked in previous studies and we show here that it ensures a smooth matching between the typical and the large deviation regimes. In addition, we demonstrate that this intermediate regime controls all the (centered) cumulants of N_{r}, which are all of order O(sqrt[N]). We show that the intermediate deviation function that describes these intermediate fluctuations can be computed explicitly and we demonstrate that it is universal, i.e., it holds for a large class of complex random matrices. Our analytical results are corroborated by precise "importance sampling" Monte Carlo simulations.
Collapse
Affiliation(s)
| | - Jeyson Andrés Monroy Garzón
- Departamento de Física Cuántica y Fotónica, Instituto de Física, UNAM, P.O. Box 20-364, 01000 Mexico Distrito Federal, Mexico
| | | | - Isaac Pérez Castillo
- Departamento de Física Cuántica y Fotónica, Instituto de Física, UNAM, P.O. Box 20-364, 01000 Mexico Distrito Federal, Mexico
- London Mathematical Laboratory, 18 Margravine Gardens, London W6 8RH, United Kingdom
| | - Anupam Kundu
- International Centre for Theoretical Sciences, TIFR, Bangalore 560089, India
| | - Satya N Majumdar
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
36
|
Abstract
We study active particles performing independent run-and-tumble motion on an infinite line with velocities v_{0}σ(t), where σ(t)=±1 is a dichotomous telegraphic noise with constant flipping rate γ. We first consider a single particle in the presence of an absorbing wall at x=0 and calculate the probability that it has survived up to time t and is at position x at time t. We then consider two particles with independent telegraphic noises and compute exactly the probability that they do not cross up to time t. In contrast with the case of passive (Brownian) particles this problem of two run-and-tumble particles (RTPs) cannot be reduced to a single RTP with an absorbing wall. Nevertheless, we are able to compute exactly the probability of no crossing of two independent RTPs up to time t and find that it decays at large time as t^{-1/2} with an amplitude that depends on the initial condition. We show that this amplitude vanishes if one extrapolates the starting distance between the two particles to a negative value-similarly to the Milne extrapolation length in neutron scattering.
Collapse
Affiliation(s)
- Pierre Le Doussal
- Laboratoire de Physique de l'Ecole Normale Supérieure, PSL University, CNRS, Sorbonne Universités, 24 rue Lhomond, 75231 Paris, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
37
|
Abstract
Daily precipitation time series are composed of null entries corresponding to dry days and nonzero entries that describe the rainfall amounts on wet days. Assuming that wet days follow a Bernoulli process with success probability p, we show that the presence of dry days induces negative correlations between record-breaking precipitation events. The resulting nonmonotonic behavior of the Fano factor of the record counting process is recovered in empirical data. We derive the full probability distribution P(R,n) of the number of records R_{n} up to time n, and show that for large n, it converges to a Poisson distribution with parameter ln(pn). We also study in detail the joint limit p→0, n→∞, which yields a random record model in continuous time t=pn.
Collapse
Affiliation(s)
- Satya N Majumdar
- Université Paris-Sud, CNRS, LPTMS, UMR 8626, 91405 Orsay, France
| | | | - Joachim Krug
- Institute for Biological Physics, University of Cologne, 50937 Köln, Germany
| |
Collapse
|
38
|
Dhar A, Kundu A, Majumdar SN, Sabhapandit S, Schehr G. Run-and-tumble particle in one-dimensional confining potentials: Steady-state, relaxation, and first-passage properties. Phys Rev E 2019; 99:032132. [PMID: 30999430 DOI: 10.1103/physreve.99.032132] [Citation(s) in RCA: 60] [Impact Index Per Article: 12.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2018] [Indexed: 06/09/2023]
Abstract
We study the dynamics of a one-dimensional run-and-tumble particle subjected to confining potentials of the type V(x)=α|x|^{p}, with p>0. The noise that drives the particle dynamics is telegraphic and alternates between ±1 values. We show that the stationary probability density P(x) has a rich behavior in the (p,α) plane. For p>1, the distribution has a finite support in [x_{-},x_{+}] and there is a critical line α_{c}(p) that separates an activelike phase for α>α_{c}(p) where P(x) diverges at x_{±}, from a passivelike phase for α<α_{c}(p) where P(x) vanishes at x_{±}. For p<1, the stationary density P(x) collapses to a delta function at the origin, P(x)=δ(x). In the marginal case p=1, we show that, for α<α_{c}, the stationary density P(x) is a symmetric exponential, while for α>α_{c}, it again is a delta function P(x)=δ(x). For the harmonic case p=2, we obtain exactly the full time-dependent distribution P(x,t), which allows us to study how the system relaxes to its stationary state. In addition, for this p=2 case, we also study analytically the full distribution of the first-passage time to the origin. Numerical simulations are in complete agreement with our analytical predictions.
Collapse
Affiliation(s)
- Abhishek Dhar
- International Centre for Theoretical Sciences, TIFR, Bangalore 560089, India
| | - Anupam Kundu
- International Centre for Theoretical Sciences, TIFR, Bangalore 560089, India
| | - Satya N Majumdar
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | | | - Grégory Schehr
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
39
|
Le Doussal P, Majumdar SN, Schehr G. Multicritical Edge Statistics for the Momenta of Fermions in Nonharmonic Traps. Phys Rev Lett 2018; 121:030603. [PMID: 30085768 DOI: 10.1103/physrevlett.121.030603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2018] [Revised: 05/24/2018] [Indexed: 06/08/2023]
Abstract
We compute the joint statistics of the momenta p_{i} of N noninteracting fermions in a trap, near the Fermi edge, with a particular focus on the largest one p_{max}. For a 1D harmonic trap, momenta and positions play a symmetric role, and hence the joint statistics of momenta are identical to that of the positions. In particular, p_{max}, as x_{max}, is distributed according to the Tracy-Widom distribution. Here we show that novel "momentum edge statistics" emerge when the curvature of the potential vanishes, i.e., for "flat traps" near their minimum, with V(x)∼x^{2n} and n>1. These are based on generalizations of the Airy kernel that we obtain explicitly. The fluctuations of p_{max} are governed by new universal distributions determined from the nth member of the second Painlevé hierarchy of nonlinear differential equations, with connections to multicritical random matrix models. Finite temperature extensions and possible experimental signatures in cold atoms are discussed.
Collapse
Affiliation(s)
- Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
40
|
Abstract
A global picture of a random particle movement is given by the convex hull of the visited points. We obtained numerically the probability distributions of the volume and surface of the convex hulls of a selection of three types of self-avoiding random walks, namely, the classical self-avoiding walk, the smart-kinetic self-avoiding walk, and the loop-erased random walk. To obtain a comprehensive description of the measured random quantities, we applied sophisticated large-deviation techniques, which allowed us to obtain the distributions over a large range of support down to probabilities far smaller than P=10^{-100}. We give an approximate closed form of the so-called large-deviation rate function Φ which generalizes above the upper critical dimension to the previously studied case of the standard random walk. Further, we show correlations between the two observables also in the limits of atypical large or small values.
Collapse
Affiliation(s)
- Hendrik Schawe
- Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany and LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Alexander K Hartmann
- Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany and LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
41
|
Abstract
The distribution of the hypervolume V and surface ∂V of convex hulls of (multiple) random walks in higher dimensions are determined numerically, especially containing probabilities far smaller than P=10^{-1000} to estimate large deviation properties. For arbitrary dimensions and large walk lengths T, we suggest a scaling behavior of the distribution with the length of the walk T similar to the two-dimensional case and behavior of the distributions in the tails. We underpin both with numerical data in d=3 and d=4 dimensions. Further, we confirm the analytically known means of those distributions and calculate their variances for large T.
Collapse
Affiliation(s)
- Hendrik Schawe
- Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany and LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Alexander K Hartmann
- Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany and LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
42
|
Falcón-Cortés A, Boyer D, Giuggioli L, Majumdar SN. Localization Transition Induced by Learning in Random Searches. Phys Rev Lett 2017; 119:140603. [PMID: 29053283 DOI: 10.1103/physrevlett.119.140603] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/27/2017] [Indexed: 06/07/2023]
Abstract
We solve an adaptive search model where a random walker or Lévy flight stochastically resets to previously visited sites on a d-dimensional lattice containing one trapping site. Because of reinforcement, a phase transition occurs when the resetting rate crosses a threshold above which nondiffusive stationary states emerge, localized around the inhomogeneity. The threshold depends on the trapping strength and on the walker's return probability in the memoryless case. The transition belongs to the same class as the self-consistent theory of Anderson localization. These results show that similarly to many living organisms and unlike the well-studied Markovian walks, non-Markov movement processes can allow agents to learn about their environment and promise to bring adaptive solutions in search tasks.
Collapse
Affiliation(s)
- Andrea Falcón-Cortés
- Instituto de Física, Universidad Nacional Autónoma de México, Ciudad de México 04510, Mexico
| | - Denis Boyer
- Instituto de Física, Universidad Nacional Autónoma de México, Ciudad de México 04510, Mexico
| | - Luca Giuggioli
- Bristol Centre for Complexity Sciences, Department of Engineering Mathematics and School of Biological Sciences, University of Bristol, Bristol BS8 1UB, United Kingdom
| | - Satya N Majumdar
- Université Paris-Sud, CNRS, LPTMS, UMR 8626, Orsay F-91405, France
| |
Collapse
|
43
|
Grela J, Majumdar SN, Schehr G. Kinetic Energy of a Trapped Fermi Gas at Finite Temperature. Phys Rev Lett 2017; 119:130601. [PMID: 29341702 DOI: 10.1103/physrevlett.119.130601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/06/2017] [Indexed: 06/07/2023]
Abstract
We study the statistics of the kinetic (or, equivalently, potential) energy for N noninteracting fermions in a 1d harmonic trap of frequency ω at finite temperature T. Remarkably, we find an exact solution for the full distribution of the kinetic energy, at any temperature T and for any N, using a nontrivial mapping to an integrable Calogero-Moser-Sutherland model. As a function of temperature T and for large N, we identify (i) a quantum regime, for T∼ℏω, where quantum fluctuations dominate and (ii) a thermal regime, for T∼Nℏω, governed by thermal fluctuations. We show how the mean and the variance as well as the large deviation function associated with the distribution of the kinetic energy cross over from the quantum to the thermal regime as T increases.
Collapse
Affiliation(s)
- Jacek Grela
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
- M. Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Centre, Jagiellonian University, PL-30-059 Cracow, Poland
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
44
|
Dhar A, Kundu A, Majumdar SN, Sabhapandit S, Schehr G. Exact Extremal Statistics in the Classical 1D Coulomb Gas. Phys Rev Lett 2017; 119:060601. [PMID: 28949607 DOI: 10.1103/physrevlett.119.060601] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/28/2017] [Indexed: 06/07/2023]
Abstract
We consider a one-dimensional classical Coulomb gas of N-like charges in a harmonic potential-also known as the one-dimensional one-component plasma. We compute, analytically, the probability distribution of the position x_{max} of the rightmost charge in the limit of large N. We show that the typical fluctuations of x_{max} around its mean are described by a nontrivial scaling function, with asymmetric tails. This distribution is different from the Tracy-Widom distribution of x_{max} for Dyson's log gas. We also compute the large deviation functions of x_{max} explicitly and show that the system exhibits a third-order phase transition, as in the log gas. Our theoretical predictions are verified numerically.
Collapse
Affiliation(s)
- Abhishek Dhar
- International Centre for Theoretical Sciences, TIFR, Bangalore 560089, India
| | - Anupam Kundu
- International Centre for Theoretical Sciences, TIFR, Bangalore 560089, India
| | - Satya N Majumdar
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | | | - Grégory Schehr
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
45
|
Majumdar SN, Sabhapandit S, Schehr G. Exact distributions of cover times for N independent random walkers in one dimension. Phys Rev E 2017; 94:062131. [PMID: 28085303 DOI: 10.1103/physreve.94.062131] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/21/2016] [Indexed: 11/07/2022]
Abstract
We study the probability density function (PDF) of the cover time t_{c} of a finite interval of size L by N independent one-dimensional Brownian motions, each with diffusion constant D. The cover time t_{c} is the minimum time needed such that each point of the entire interval is visited by at least one of the N walkers. We derive exact results for the full PDF of t_{c} for arbitrary N≥1 for both reflecting and periodic boundary conditions. The PDFs depend explicitly on N and on the boundary conditions. In the limit of large N, we show that t_{c} approaches its average value of 〈t_{c}〉≈L^{2}/(16DlnN) with fluctuations vanishing as 1/(lnN)^{2}. We also compute the centered and scaled limiting distributions for large N for both boundary conditions and show that they are given by nontrivial N independent scaling functions.
Collapse
Affiliation(s)
- Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | | | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
46
|
Abstract
We study the polygons governing the convex hull of a point set created by the steps of n independent two-dimensional random walkers. Each such walk consists of T discrete time steps, where x and y increments are independent and identically distributed Gaussian. We analyze area A and perimeter L of the convex hulls. We obtain probability densities for these two quantities over a large range of the support by using a large-deviation approach allowing us to study densities below 10^{-900}. We find that the densities exhibit in the limit T→∞ a time-independent scaling behavior as a function of A/T and L/sqrt[T], respectively. As in the case of one walker (n=1), the densities follow Gaussian distributions for L and sqrt[A], respectively. We also obtained the rate functions for the area and perimeter, rescaled with the scaling behavior of their maximum possible values, and found limiting functions for T→∞, revealing that the densities follow the large-deviation principle. These rate functions can be described by a power law for n→∞ as found in the n=1 case. We also investigated the behavior of the averages as a function of the number of walks n and found good agreement with the predicted behavior.
Collapse
Affiliation(s)
- Timo Dewenter
- Institut für Physik, Universität Oldenburg, D-26111 Oldenburg, Germany
| | - Gunnar Claussen
- Institut für Physik, Universität Oldenburg, D-26111 Oldenburg, Germany.,Fachbereich Ingenieurwissenschaften, Jade Hochschule Wilhelmshaven/Oldenburg/Elsfleth, D-26389 Wilhelmshaven, Germany
| | | | - Satya N Majumdar
- Laboratoire de Physique Théorique et Modèles Statistiques (UMR 8626 du CNRS), Université de Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France
| |
Collapse
|
47
|
Marino R, Majumdar SN, Schehr G, Vivo P. Number statistics for β-ensembles of random matrices: Applications to trapped fermions at zero temperature. Phys Rev E 2016; 94:032115. [PMID: 27739840 DOI: 10.1103/physreve.94.032115] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2016] [Indexed: 11/07/2022]
Abstract
Let P_{β}^{(V)}(N_{I}) be the probability that a N×Nβ-ensemble of random matrices with confining potential V(x) has N_{I} eigenvalues inside an interval I=[a,b] on the real line. We introduce a general formalism, based on the Coulomb gas technique and the resolvent method, to compute analytically P_{β}^{(V)}(N_{I}) for large N. We show that this probability scales for large N as P_{β}^{(V)}(N_{I})≈exp[-βN^{2}ψ^{(V)}(N_{I}/N)], where β is the Dyson index of the ensemble. The rate function ψ^{(V)}(k_{I}), independent of β, is computed in terms of single integrals that can be easily evaluated numerically. The general formalism is then applied to the classical β-Gaussian (I=[-L,L]), β-Wishart (I=[1,L]), and β-Cauchy (I=[-L,L]) ensembles. Expanding the rate function around its minimum, we find that generically the number variance var(N_{I}) exhibits a nonmonotonic behavior as a function of the size of the interval, with a maximum that can be precisely characterized. These analytical results, corroborated by numerical simulations, provide the full counting statistics of many systems where random matrix models apply. In particular, we present results for the full counting statistics of zero-temperature one-dimensional spinless fermions in a harmonic trap.
Collapse
Affiliation(s)
- Ricardo Marino
- Department of Physics of Complex Systems, Weizmann Institute of Science, 76100 Rehovot, Israel
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Pierpaolo Vivo
- King's College London, Department of Mathematics, Strand, London WC2R 2LS, United Kingdom
| |
Collapse
|
48
|
Le Doussal P, Majumdar SN, Rosso A, Schehr G. Exact Short-Time Height Distribution in the One-Dimensional Kardar-Parisi-Zhang Equation and Edge Fermions at High Temperature. Phys Rev Lett 2016; 117:070403. [PMID: 27563940 DOI: 10.1103/physrevlett.117.070403] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2016] [Indexed: 06/06/2023]
Abstract
We consider the early time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions in curved (or droplet) geometry. We show that for short time t, the probability distribution P(H,t) of the height H at a given point x takes the scaling form P(H,t)∼exp[-Φ_{drop}(H)/sqrt[t]] where the rate function Φ_{drop}(H) is computed exactly for all H. While it is Gaussian in the center, i.e., for small H, the probability distribution function has highly asymmetric non-Gaussian tails that we characterize in detail. This function Φ_{drop}(H) is surprisingly reminiscent of the large deviation function describing the stationary fluctuations of finite-size models belonging to the KPZ universality class. Thanks to a recently discovered connection between the KPZ equation and free fermions, our results have interesting implications for the fluctuations of the rightmost fermion in a harmonic trap at high temperature and the full counting statistics at the edge.
Collapse
Affiliation(s)
- Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France
| | - Satya N Majumdar
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Alberto Rosso
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
49
|
Godrèche C, Majumdar SN, Schehr G. Exact Statistics of Record Increments of Random Walks and Lévy Flights. Phys Rev Lett 2016; 117:010601. [PMID: 27419552 DOI: 10.1103/physrevlett.117.010601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/05/2016] [Indexed: 06/06/2023]
Abstract
We study the statistics of increments in record values in a time series {x_{0}=0,x_{1},x_{2},…,x_{n}} generated by the positions of a random walk (discrete time, continuous space) of duration n steps. For arbitrary jump length distribution, including Lévy flights, we show that the distribution of the record increment becomes stationary, i.e., independent of n for large n, and compute it explicitly for a wide class of jump distributions. In addition, we compute exactly the probability Q(n) that the record increments decrease monotonically up to step n. Remarkably, Q(n) is universal (i.e., independent of the jump distribution) for each n, decaying as Q(n)∼A/sqrt[n] for large n, with a universal amplitude A=e/sqrt[π]=1.53362….
Collapse
Affiliation(s)
- Claude Godrèche
- Institut de Physique Théorique, Université Paris-Saclay, CEA and CNRS, 91191 Gif-sur-Yvette, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
50
|
Abstract
We study extreme-value statistics for spatially extended models exhibiting mixed-order phase transitions (MOT). These are phase transitions that exhibit features common to both first-order (discontinuity of the order parameter) and second-order (diverging correlation length) transitions. We consider here the truncated inverse distance squared Ising model, which is a prototypical model exhibiting MOT, and study analytically the extreme-value statistics of the domain lengths The lengths of the domains are identically distributed random variables except for the global constraint that their sum equals the total system size L. In addition, the number of such domains is also a fluctuating variable, and not fixed. In the paramagnetic phase, we show that the distribution of the largest domain length l_{max} converges, in the large L limit, to a Gumbel distribution. However, at the critical point (for a certain range of parameters) and in the ferromagnetic phase, we show that the fluctuations of l_{max} are governed by novel distributions, which we compute exactly. Our main analytical results are verified by numerical simulations.
Collapse
Affiliation(s)
- Amir Bar
- Department of Complex Systems, Weizmann Institute, Rehovot, Israel
| | - Satya N Majumdar
- Université Paris-Sud, CNRS, LPTMS, UMR 8626, Orsay F-91405, France
| | - Grégory Schehr
- Université Paris-Sud, CNRS, LPTMS, UMR 8626, Orsay F-91405, France
| | - David Mukamel
- Department of Complex Systems, Weizmann Institute, Rehovot, Israel
| |
Collapse
|