1
|
Dupont T, Giordano S, Cleri F, Blossey R. Short-time expansion of one-dimensional Fokker-Planck equations with heterogeneous diffusion. Phys Rev E 2024; 109:064106. [PMID: 39020937 DOI: 10.1103/physreve.109.064106] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/04/2024] [Accepted: 05/07/2024] [Indexed: 07/20/2024]
Abstract
We formulate a short-time expansion for one-dimensional Fokker-Planck equations with spatially dependent diffusion coefficients, derived from stochastic processes with Gaussian white noise, for general values of the discretization parameter 0≤α≤1 of the stochastic integral. The kernel of the Fokker-Planck equation (the propagator) can be expressed as a product of a singular and a regular term. While the singular term can be given in closed form, the regular term can be computed from a Taylor expansion whose coefficients obey simple ordinary differential equations. We illustrate the application of our approach with examples taken from statistical physics and biophysics. Furthermore, we show how our formalism allows us to define a class of stochastic equations which can be treated exactly. The convergence of the expansion cannot be guaranteed independently from the discretization parameter α.
Collapse
Affiliation(s)
| | | | - Fabrizio Cleri
- University of Lille, Institut d'Électronique, de Microélectronique et de Nanotechnologie (IEMN CNRS UMR8520) and Departement de Physique, F-59652 Villeneuve d'Ascq, France
| | | |
Collapse
|
2
|
Lima HS, Tsallis C, Nobre FD. First-Principle Validation of Fourier's Law: One-Dimensional Classical Inertial Heisenberg Model. ENTROPY (BASEL, SWITZERLAND) 2023; 26:25. [PMID: 38248151 PMCID: PMC11154294 DOI: 10.3390/e26010025] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/21/2023] [Revised: 12/17/2023] [Accepted: 12/22/2023] [Indexed: 01/23/2024]
Abstract
The thermal conductance of a one-dimensional classical inertial Heisenberg model of linear size L is computed, considering the first and last particles in thermal contact with heat baths at higher and lower temperatures, Th and Tl (Th>Tl), respectively. These particles at the extremities of the chain are subjected to standard Langevin dynamics, whereas all remaining rotators (i=2,⋯,L-1) interact by means of nearest-neighbor ferromagnetic couplings and evolve in time following their own equations of motion, being investigated numerically through molecular-dynamics numerical simulations. Fourier's law for the heat flux is verified numerically, with the thermal conductivity becoming independent of the lattice size in the limit L→∞, scaling with the temperature, as κ(T)∼T-2.25, where T=(Th+Tl)/2. Moreover, the thermal conductance, σ(L,T)≡κ(T)/L, is well-fitted by a function, which is typical of nonextensive statistical mechanics, according to σ(L,T)=Aexpq(-Bxη), where A and B are constants, x=L0.475T, q=2.28±0.04, and η=2.88±0.04.
Collapse
Affiliation(s)
- Henrique Santos Lima
- Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, RJ, Brazil; (C.T.); (F.D.N.)
- National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, RJ, Brazil
| | - Constantino Tsallis
- Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, RJ, Brazil; (C.T.); (F.D.N.)
- National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, RJ, Brazil
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
- Complexity Science Hub Vienna, Josefstädter Strasse 39, 1080 Vienna, Austria
| | - Fernando Dantas Nobre
- Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, RJ, Brazil; (C.T.); (F.D.N.)
- National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, RJ, Brazil
| |
Collapse
|
3
|
Giordano S, Cleri F, Blossey R. Infinite ergodicity in generalized geometric Brownian motions with nonlinear drift. Phys Rev E 2023; 107:044111. [PMID: 37198762 DOI: 10.1103/physreve.107.044111] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/05/2022] [Accepted: 03/25/2023] [Indexed: 05/19/2023]
Abstract
Geometric Brownian motion is an exemplary stochastic processes obeying multiplicative noise, with widespread applications in several fields, e.g., in finance, in physics, and biology. The definition of the process depends crucially on the interpretation of the stochastic integrals which involves the discretization parameter α with 0≤α≤1, giving rise to the well-known special cases α=0 (Itô), α=1/2 (Fisk-Stratonovich), and α=1 (Hänggi-Klimontovich or anti-Itô). In this paper we study the asymptotic limits of the probability distribution functions of geometric Brownian motion and some related generalizations. We establish the conditions for the existence of normalizable asymptotic distributions depending on the discretization parameter α. Using the infinite ergodicity approach, recently applied to stochastic processes with multiplicative noise by E. Barkai and collaborators, we show how meaningful asymptotic results can be formulated in a transparent way.
Collapse
Affiliation(s)
- Stefano Giordano
- University of Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, UMR 8520 - IEMN - Institut d'Électronique, de Microélectronique et de Nanotechnologie, F-59000 Lille, France
| | - Fabrizio Cleri
- University of Lille, Institut d'Électronique, de Microélectronique et de Nanotechnologie (IEMN CNRS UMR8520) and Departement de Physique, F-59652 Villeneuve d'Ascq, France
| | - Ralf Blossey
- University of Lille, Unité de Glycobiologie Structurale et Fonctionnelle (UGSF), CNRS UMR8576, F-59000 Lille, France
| |
Collapse
|
4
|
Defaveri L, Olivares C, Anteneodo C. Heat flux in chains of nonlocally coupled harmonic oscillators: Mean-field limit. Phys Rev E 2022; 105:054149. [PMID: 35706286 DOI: 10.1103/physreve.105.054149] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/09/2021] [Accepted: 05/04/2022] [Indexed: 06/15/2023]
Abstract
We consider one-dimensional systems of all-to-all harmonically coupled particles with arbitrary masses, subject to two Langevin thermal baths. The couplings correspond to the mean-field limit of long-range interactions. Additionally, the particles can be subject to a harmonic on-site potential to break momentum conservation. Using the nonequilibrium Green's operator formalism, we calculate the transmittance, the heat flow, and local temperatures for arbitrary configurations of masses. For identical masses, we show analytically that the heat flux decays with the system size N as 1/N regardless of the conservation or not of the momentum and of the introduction or not of a Kac factor. These results describe, in good agreement, the thermal behavior of systems with small heterogeneity in the masses.
Collapse
Affiliation(s)
- Lucianno Defaveri
- Department of Physics, PUC-Rio, Rio de Janeiro, 22451-900 Rio de Janeiro, Brazil
| | - Carlos Olivares
- Department of Physics, PUC-Rio, Rio de Janeiro, 22451-900 Rio de Janeiro, Brazil
- Laboratoire de physique de l'École normale supérieure (PSL University), CNRS, Sorbonne Université, and Université de Paris, 75005 Paris, France
| | - Celia Anteneodo
- Department of Physics, PUC-Rio, Rio de Janeiro, 22451-900 Rio de Janeiro, Brazil
- Laboratoire de physique de l'École normale supérieure (PSL University), CNRS, Sorbonne Université, and Université de Paris, 75005 Paris, France
- Institute of Science and Technology for Complex Systems, INCT-SC, Brazil
| |
Collapse
|
5
|
de Oliveira MJ. Heat transport along a chain of coupled quantum harmonic oscillators. Phys Rev E 2017; 95:042113. [PMID: 28505787 DOI: 10.1103/physreve.95.042113] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2017] [Indexed: 11/07/2022]
Abstract
I study the heat transport properties of a chain of coupled quantum harmonic oscillators in contact at its ends with two heat reservoirs at distinct temperatures. My approach is based on the use of an evolution equation for the density operator which is a canonical quantization of the classical Fokker-Planck-Kramers equation. I set up the evolution equation for the covariances and obtain the stationary covariances at the stationary states from which I determine the thermal conductance in closed form when the interparticle interaction is small. The conductance is finite in the thermodynamic limit implying an infinite thermal conductivity.
Collapse
Affiliation(s)
- Mário J de Oliveira
- Instituto de Física, Universidade de São Paulo, Rua do Matão, 1371, 05508-090 São Paulo, São Paulo, Brazil
| |
Collapse
|
6
|
Olivares C, Anteneodo C. Role of the range of the interactions in thermal conduction. Phys Rev E 2016; 94:042117. [PMID: 27841624 DOI: 10.1103/physreve.94.042117] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2016] [Indexed: 06/06/2023]
Abstract
We investigate thermal transport along a one-dimensional lattice of classical inertial rotators, with attractive couplings that decrease with distance as r^{-α} (α≥0), subject at its ends to Brownian heat reservoirs at different temperatures with average value T. By means of numerical integration of the equations of motion, we show the effects of the range of the interactions in the temperature profile and energy transport and determine the domain of validity of Fourier's law in this context. We find that Fourier's law, as signaled by a finite κ in the thermodynamic limit, holds only for sufficiently short-range interactions, with α>α_{c}(T). For α<α_{c}(T), a kind of insulator behavior emerges at any T.
Collapse
Affiliation(s)
| | - Celia Anteneodo
- Department of Physics, PUC-Rio, Rio de Janeiro, Brazil
- Institute of Science and Technology for Complex Systems, Rio de Janeiro, Brazil
| |
Collapse
|
7
|
Guimarães PH, Landi GT, de Oliveira MJ. Nonequilibrium quantum chains under multisite Lindblad baths. Phys Rev E 2016; 94:032139. [PMID: 27739825 DOI: 10.1103/physreve.94.032139] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/14/2016] [Indexed: 06/06/2023]
Abstract
We study a quantum XX chain coupled to two heat reservoirs that act on multiple sites and are kept at different temperatures and chemical potentials. The baths are described by Lindblad dissipators, which are constructed by direct coupling to the fermionic normal modes of the chain. Using a perturbative method, we are able to find analytical formulas for all steady-state properties of the system. We compute both the particle or magnetization current and the energy current, both of which are found to have the structure of Landauer's formula. We also obtain exact formulas for the Onsager coefficients. All properties are found to differ substantially from those of a single-site bath. In particular, we find a strong dependence on the intensity of the bath couplings. In the weak-coupling regime, we show that the Onsager reciprocal relations are satisfied.
Collapse
Affiliation(s)
- Pedro H Guimarães
- Instituto de Física da Universidade de São Paulo, 05314-970 São Paulo, Brazil
| | | | - Mario J de Oliveira
- Instituto de Física da Universidade de São Paulo, 05314-970 São Paulo, Brazil
| |
Collapse
|
8
|
Guimarães PH, Landi GT, de Oliveira MJ. Thermal rectification in anharmonic chains under an energy-conserving noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062120. [PMID: 26764645 DOI: 10.1103/physreve.92.062120] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2015] [Indexed: 06/05/2023]
Abstract
Systems in which the heat flux depends on the direction of the flow are said to present thermal rectification. This effect has attracted much theoretical and experimental interest in recent years. However, in most theoretical models the effect is found to vanish in the thermodynamic limit, in disagreement with experiment. The purpose of this paper is to show that the rectification may be restored by including an energy-conserving noise which randomly flips the velocity of the particles with a certain rate λ. It is shown that as long as λ is nonzero, the rectification remains finite in the thermodynamic limit. This is illustrated in a classical harmonic chain subject to a quartic pinning potential (the Φ(4) model) and coupled to heat baths by Langevin equations.
Collapse
Affiliation(s)
- Pedro H Guimarães
- Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05314-970 São Paulo, Brazil
| | | | - Mário J de Oliveira
- Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05314-970 São Paulo, Brazil
| |
Collapse
|
9
|
Popkov V, Schadschneider A, Schmidt J, Schütz GM. Fibonacci family of dynamical universality classes. Proc Natl Acad Sci U S A 2015; 112:12645-50. [PMID: 26424449 PMCID: PMC4611608 DOI: 10.1073/pnas.1512261112] [Citation(s) in RCA: 28] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Universality is a well-established central concept of equilibrium physics. However, in systems far away from equilibrium, a deeper understanding of its underlying principles is still lacking. Up to now, a few classes have been identified. Besides the diffusive universality class with dynamical exponent [Formula: see text], another prominent example is the superdiffusive Kardar-Parisi-Zhang (KPZ) class with [Formula: see text]. It appears, e.g., in low-dimensional dynamical phenomena far from thermal equilibrium that exhibit some conservation law. Here we show that both classes are only part of an infinite discrete family of nonequilibrium universality classes. Remarkably, their dynamical exponents [Formula: see text] are given by ratios of neighboring Fibonacci numbers, starting with either [Formula: see text] (if a KPZ mode exist) or [Formula: see text] (if a diffusive mode is present). If neither a diffusive nor a KPZ mode is present, all dynamical modes have the Golden Mean [Formula: see text] as dynamical exponent. The universal scaling functions of these Fibonacci modes are asymmetric Lévy distributions that are completely fixed by the macroscopic current density relation and compressibility matrix of the system and hence accessible to experimental measurement.
Collapse
Affiliation(s)
- Vladislav Popkov
- Institut für Theoretische Physik, Universität zu Köln, 50937 Cologne, Germany; Centro Interdipartimentale per lo Studio di Dinamiche Complesse, Università di Firenze, 50019 Sesto Fiorentino, Italy
| | | | - Johannes Schmidt
- Institut für Theoretische Physik, Universität zu Köln, 50937 Cologne, Germany
| | - Gunter M Schütz
- Theoretical Soft Matter and Biophysics, Institute of Complex Systems II, Forschungszentrum Jülich, 52425 Jülich, Germany
| |
Collapse
|
10
|
Nicacio F, Ferraro A, Imparato A, Paternostro M, Semião FL. Thermal transport in out-of-equilibrium quantum harmonic chains. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042116. [PMID: 25974448 DOI: 10.1103/physreve.91.042116] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2014] [Indexed: 06/04/2023]
Abstract
We address the problem of heat transport in a chain of coupled quantum harmonic oscillators, exposed to the influences of local environments of various nature, stressing the effects that the specific nature of the environment has on the phenomenology of the transport process. We study in detail the behavior of thermodynamically relevant quantities such as heat currents and mean energies of the oscillators, establishing rigorous analytical conditions for the existence of a steady state, whose features we analyze carefully. In particular, we assess the conditions that should be faced to recover trends reminiscent of the classical Fourier law of heat conduction and highlight how such a possibility depends on the environment linked to our system.
Collapse
Affiliation(s)
- F Nicacio
- Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Santo André, 09210-170 São Paulo, Brazil
| | - A Ferraro
- Centre for Theoretical Atomic, Molecular, and Optical Physics, School of Mathematics and Physics, Queen's University, Belfast BT7 1NN, United Kingdom
| | - A Imparato
- Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, Building 1520, DK-8000 Aarhus C, Denmark
| | - M Paternostro
- Centre for Theoretical Atomic, Molecular, and Optical Physics, School of Mathematics and Physics, Queen's University, Belfast BT7 1NN, United Kingdom
| | - F L Semião
- Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Santo André, 09210-170 São Paulo, Brazil
| |
Collapse
|
11
|
Tomé T, de Oliveira MJ. Stochastic approach to equilibrium and nonequilibrium thermodynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042140. [PMID: 25974471 DOI: 10.1103/physreve.91.042140] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/05/2015] [Indexed: 06/04/2023]
Abstract
We develop the stochastic approach to thermodynamics based on stochastic dynamics, which can be discrete (master equation) and continuous (Fokker-Planck equation), and on two assumptions concerning entropy. The first is the definition of entropy itself and the second the definition of entropy production rate, which is non-negative and vanishes in thermodynamic equilibrium. Based on these assumptions, we study interacting systems with many degrees of freedom in equilibrium or out of thermodynamic equilibrium and how the macroscopic laws are derived from the stochastic dynamics. These studies include the quasiequilibrium processes; the convexity of the equilibrium surface; the monotonic time behavior of thermodynamic potentials, including entropy; the bilinear form of the entropy production rate; the Onsager coefficients and reciprocal relations; and the nonequilibrium steady states of chemical reactions.
Collapse
Affiliation(s)
- Tânia Tomé
- Instituto de Física, Universidade de São Paulo, Caixa Postal 66318 05314-970 São Paulo, São Paulo, Brazil
| | - Mário J de Oliveira
- Instituto de Física, Universidade de São Paulo, Caixa Postal 66318 05314-970 São Paulo, São Paulo, Brazil
| |
Collapse
|
12
|
Landi GT, Novais E, de Oliveira MJ, Karevski D. Flux rectification in the quantum XXZ chain. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042142. [PMID: 25375473 DOI: 10.1103/physreve.90.042142] [Citation(s) in RCA: 39] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/10/2014] [Indexed: 06/04/2023]
Abstract
Thermal rectification is the phenomenon by which the flux of heat depends on the direction of the flow. It has attracted much interest in recent years due to the possibility of devising thermal diodes. In this paper, we consider the rectification phenomenon in the quantum XXZ chain subject to an inhomogeneous field. The chain is driven out of equilibrium by the contact at its boundaries with two different reservoirs, leading to a constant flow of magnetization from one bath to the other. The nonunitary dynamics of this system, which is modeled by a Lindblad master equation, is treated exactly for small sizes and numerically for larger ones. The functional dependence of the rectification coefficient on the model parameters (anisotropy, field amplitude, and out of equilibrium driving strength) is investigated in full detail. Close to the XX point and at small inhomogeneity and low driving, we have found an explicit expression for the rectification coefficient that is valid at all system sizes. In particular, it shows that the phenomenon of rectification persists even in the thermodynamic limit. Finally, we prove that in the case of the XX chain, there is no rectification.
Collapse
Affiliation(s)
| | - E Novais
- Universidade Federal do ABC, 09210-580 Santo André, Brazil
| | | | - Dragi Karevski
- Université de Lorraine, Institut Jean Lamour, Vandoeuvre lès Nancy F-54506, France and CNRS, UMR 7198, IJL, Groupe de Physique Statistique, Nancy, France
| |
Collapse
|