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Quintana M, Berger A. Verification of scaling behavior near dynamic phase transitions for nonantisymmetric field sequences. Phys Rev E 2024; 109:054112. [PMID: 38907487 DOI: 10.1103/physreve.109.054112] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2024] [Accepted: 04/11/2024] [Indexed: 06/24/2024]
Abstract
We investigate the scaling behavior of the magnetic dynamic order parameter Q in the vicinity of the dynamic phase transition (DPT) in the presence of temporal field sequences H(t) that are periodic with period P but lack half-wave antisymmetry. We verify by means of mean-field calculations that the scaling of Q is preserved in the vicinity of the second-order phase transition if one defines a suitable generalized conjugate field H^{*} that reestablishes the proper time-reversal symmetry. For the purpose of our quantitative data analysis, we employ the dynamic equivalent of the Arrott-Noakes equation of state, which allows for a simultaneous scaling analysis of the period P and the conjugate-field H^{*} dependence of Q. By doing so, we demonstrate that both the scaling behavior and universality are preserved, even if the dynamics is driven by a more general applied field sequence that lacks antisymmetry.
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Affiliation(s)
- M Quintana
- CIC nanoGUNE BRTA, E-20018 Donostia-San Sebastián, Spain
| | - A Berger
- CIC nanoGUNE BRTA, E-20018 Donostia-San Sebastián, Spain
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2
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Defaveri L, Barkai E, Kessler DA. Stretched-exponential relaxation in weakly confined Brownian systems through large deviation theory. Phys Rev E 2024; 109:L022102. [PMID: 38491584 DOI: 10.1103/physreve.109.l022102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2023] [Accepted: 12/15/2023] [Indexed: 03/18/2024]
Abstract
Stretched-exponential relaxation is a widely observed phenomenon found in ordered ferromagnets as well as glassy systems. One modeling approach connects this behavior to a droplet dynamics described by an effective Langevin equation for the droplet radius with an r^{2/3} potential. Here, we study a Brownian particle under the influence of a general confining, albeit weak, potential field that grows with distance as a sublinear power law. We find that for this memoryless model, observables display stretched-exponential relaxation. The probability density function of the system is studied using a rate-function ansatz. We obtain analytically the stretched-exponential exponent along with an anomalous power-law scaling of length with time. The rate function exhibits a point of nonanalyticity, indicating a dynamical phase transition. In particular, the rate function is double valued both to the left and right of this point, leading to four different rate functions, depending on the choice of initial conditions and symmetry.
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Affiliation(s)
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - David A Kessler
- Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
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Wei Q, Wang W, Zhou H, Metzler R, Chechkin A. Time-fractional Caputo derivative versus other integrodifferential operators in generalized Fokker-Planck and generalized Langevin equations. Phys Rev E 2023; 108:024125. [PMID: 37723675 DOI: 10.1103/physreve.108.024125] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2023] [Accepted: 07/21/2023] [Indexed: 09/20/2023]
Abstract
Fractional diffusion and Fokker-Planck equations are widely used tools to describe anomalous diffusion in a large variety of complex systems. The equivalent formulations in terms of Caputo or Riemann-Liouville fractional derivatives can be derived as continuum limits of continuous-time random walks and are associated with the Mittag-Leffler relaxation of Fourier modes, interpolating between a short-time stretched exponential and a long-time inverse power-law scaling. More recently, a number of other integrodifferential operators have been proposed, including the Caputo-Fabrizio and Atangana-Baleanu forms. Moreover, the conformable derivative has been introduced. We study here the dynamics of the associated generalized Fokker-Planck equations from the perspective of the moments, the time-averaged mean-squared displacements, and the autocovariance functions. We also study generalized Langevin equations based on these generalized operators. The differences between the Fokker-Planck and Langevin equations with different integrodifferential operators are discussed and compared with the dynamic behavior of established models of scaled Brownian motion and fractional Brownian motion. We demonstrate that the integrodifferential operators with exponential and Mittag-Leffler kernels are not suitable to be introduced to Fokker-Planck and Langevin equations for the physically relevant diffusion scenarios discussed in our paper. The conformable and Caputo Langevin equations are unveiled to share similar properties with scaled and fractional Brownian motion, respectively.
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Affiliation(s)
- Qing Wei
- School of Mechanics and Civil Engineering, China University of Mining and Technology, Beijing 100083, People's Republic of China
- University of Potsdam, Institute of Physics & Astronomy, 14476 Potsdam-Golm, Germany
| | - Wei Wang
- University of Potsdam, Institute of Physics & Astronomy, 14476 Potsdam-Golm, Germany
| | - Hongwei Zhou
- School of Energy and Mining Engineering, China University of Mining and Technology, Beijing 100083, People's Republic of China
| | - Ralf Metzler
- University of Potsdam, Institute of Physics & Astronomy, 14476 Potsdam-Golm, Germany
- Asia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea
| | - Aleksei Chechkin
- University of Potsdam, Institute of Physics & Astronomy, 14476 Potsdam-Golm, Germany
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland
- Akhiezer Institute for Theoretical Physics National Science Center, Kharkiv Institute of Physics and Technology, Akademichna 1, Kharkiv 61108, Ukraine
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Teza G, Yaacoby R, Raz O. Relaxation Shortcuts through Boundary Coupling. PHYSICAL REVIEW LETTERS 2023; 131:017101. [PMID: 37478423 DOI: 10.1103/physrevlett.131.017101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2021] [Revised: 07/06/2022] [Accepted: 05/17/2023] [Indexed: 07/23/2023]
Abstract
When a hot system cools down faster than an equivalent cold one, it exhibits the Mpemba effect (ME). This counterintuitive phenomenon was observed in several systems including water, magnetic alloys, and polymers. In most experiments the system is coupled to the bath through its boundaries, but all theories so far assumed bulk coupling. Here we build a general framework to characterize anomalous relaxations through boundary coupling, and present two emblematic setups: a diffusing particle and an Ising antiferromagnet. In the latter, we show that the ME can survive even arbitrarily weak couplings.
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Affiliation(s)
- Gianluca Teza
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
| | - Ran Yaacoby
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
| | - Oren Raz
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
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Stella AL, Chechkin A, Teza G. Anomalous Dynamical Scaling Determines Universal Critical Singularities. PHYSICAL REVIEW LETTERS 2023; 130:207104. [PMID: 37267558 DOI: 10.1103/physrevlett.130.207104] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2022] [Accepted: 04/19/2023] [Indexed: 06/04/2023]
Abstract
Anomalous diffusion phenomena occur on length scales spanning from intracellular to astrophysical ranges. A specific form of decay at a large argument of the probability density function of rescaled displacement (scaling function) is derived and shown to imply universal singularities in the normalized cumulant generator. Exact calculations for continuous time random walks provide paradigmatic examples connected with singularities of second order phase transitions. In the biased case scaling is restricted to displacements in the drift direction and singularities have no equilibrium analogue.
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Affiliation(s)
- Attilio L Stella
- Department of Physics and Astronomy, University of Padova, Via Marzolo 8, I-35131 Padova, Italy and INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy
| | - Aleksei Chechkin
- Institute of Physics and Astronomy, University of Potsdam, D-14476 Potsdam-Golm, Germany, Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, University of Science and Technology, Wyspianskiego 27, 50-370 Wrocław, Poland, and Akhiezer Institute for Theoretical Physics National Science Center ''Kharkov Institute of Physics and Technology,'' 61108, Kharkiv, Ukraine
| | - Gianluca Teza
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
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Teza G, Yaacoby R, Raz O. Eigenvalue Crossing as a Phase Transition in Relaxation Dynamics. PHYSICAL REVIEW LETTERS 2023; 130:207103. [PMID: 37267560 DOI: 10.1103/physrevlett.130.207103] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/14/2022] [Accepted: 04/07/2023] [Indexed: 06/04/2023]
Abstract
When a system's parameter is abruptly changed, a relaxation toward the new equilibrium of the system follows. We show that a crossing between the second and third eigenvalues of the relaxation operator results in a singularity in the dynamics analogous to a first-order equilibrium phase transition. While dynamical phase transitions are intrinsically hard to detect in nature, here we show how this kind of transition can be observed in an experimentally feasible four-state colloidal system. Finally, analytical proof of survival in the thermodynamic limit of a many body (1D Ising) model is provided.
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Affiliation(s)
- Gianluca Teza
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
| | - Ran Yaacoby
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
| | - Oren Raz
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
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