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Sun Y, Wang L. Heat transport in an angular-momentum-conserving lattice. Phys Rev E 2024; 109:034118. [PMID: 38632744 DOI: 10.1103/physreve.109.034118] [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: 02/23/2024] [Indexed: 04/19/2024]
Abstract
It is expected that the energy-diffusion propagator in a one-dimensional nonlinear lattice with three conserved quantities: energy, momentum, and stretch, consists of a central heat mode and two sound modes. The heat mode follows a Lévy distribution. Consequently, the heat diffusion is super, i.e., the second moment of the diffusion propagator diverges as t^{β} with β>1; and the heat conduction is anomalous, i.e., the heat conductivity is size dependent and diverges with size N by N^{α}, with α>0. In this paper, we study a one-dimensional lattice with two-dimensional transverse motions, in which the total angular momentum also conserves. More importantly, the diffusion of this conserved quantity is ballistic. Surprisingly, the above pictures and the values of the mentioned power exponents keep unchanged. The universality of the scalings is then further extended. On the other hand, the detailed strengths of heat transports are largely enhanced. Such a counterintuitive finding can be explained by the change of the phonon mean-free path of the lattices.
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Affiliation(s)
- Yachao Sun
- Department of Physics, Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices, and Key Laboratory of Quantum State Construction and Manipulation (Ministry of Education), Renmin University of China, Beijing 100872, People's Republic of China
| | - Lei Wang
- Department of Physics, Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices, and Key Laboratory of Quantum State Construction and Manipulation (Ministry of Education), Renmin University of China, Beijing 100872, People's Republic of China
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2
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Guo Y, Sun Y, Wang L. Energy diffusion in two-dimensional momentum-conserving nonlinear lattices: Lévy walk and renormalized phonon. Phys Rev E 2023; 107:014109. [PMID: 36797934 DOI: 10.1103/physreve.107.014109] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/28/2022] [Accepted: 12/21/2022] [Indexed: 01/12/2023]
Abstract
The energy diffusion process in a few two-dimensional Fermi-Pasta-Ulam-type lattices is numerically simulated via the equilibrium local energy spatiotemporal correlation. Just as the nonlinear fluctuating hydrodynamic theory suggested, the diffusion propagator consists of a bell-shaped central heat mode and a sound mode extending with a constant speed. The profiles of the heat and sound modes satisfy the scaling properties from a random-walk-with-velocity-fluctuation process very well. An effective phonon approach is proposed, which expects the frequencies of renormalized phonons as well as the sound speed with quite good accuracy. Since many existing analytical and numerical studies indicate that heat conduction in such two-dimensional momentum-conserving lattices is divergent and the thermal conductivity κ increases logarithmically with lattice length, it is expected that the mean-square displacement of energy diffusion grows as tlnt. Discrepancies, however, are noticeably observed.
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Affiliation(s)
- Yanjiang Guo
- Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices, Renmin University of China, Beijing 100872, People's Republic of China
| | - Yachao Sun
- Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices, Renmin University of China, Beijing 100872, People's Republic of China
| | - Lei Wang
- Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices, Renmin University of China, Beijing 100872, People's Republic of China
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Ming Y, Hu H, Li HM, Ding ZJ, Ren J. Universal Kardar-Parisi-Zhang transient diffusion in nonequilibrium anharmonic chains. Phys Rev E 2023; 107:014204. [PMID: 36797957 DOI: 10.1103/physreve.107.014204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/10/2021] [Accepted: 12/10/2022] [Indexed: 01/05/2023]
Abstract
The well known nonlinear fluctuating hydrodynamics theory has grouped diffusions in anharmonic chains into two universality classes: one is the Kardar-Parisi-Zhang (KPZ) class for chains with either asymmetric potential or nonzero static pressure and the other is the Gaussian class for chains with symmetric potential at zero static pressure, such as Fermi-Pasta-Ulam-Tsingou (FPUT)-β chains. However, little is known of the nonequilibrium transient diffusion in anharmonic chains. Here, we reveal that the KPZ class is the only universality class for nonequilibrium transient diffusion, manifested as the KPZ scaling of the side peaks of momentum correlation (corresponding to the sound modes correlation), which was completely unexpected in equilibrium FPUT-β chains. The underlying mechanism is that the nonequilibrium soliton dynamics cause nonzero transient pressure so that the sound modes satisfy approximately the noisy Burgers equation, in which the collisions of solitons was proved to yield the KPZ dynamic exponent of the soliton dispersion. Therefore, the unexpected KPZ universality class is obtained in the nonequilibrium transient diffusion in FPUT-β chains and the corresponding carriers of nonequilibrium transient diffusion are attributed to solitons.
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Affiliation(s)
- Yi Ming
- School of Physics and Optoelectronics Engineering, Anhui University, Hefei, Anhui 230601, China
| | - Hao Hu
- School of Physics and Optoelectronics Engineering, Anhui University, Hefei, Anhui 230601, China
| | - Hui-Min Li
- Supercomputing Center, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Ze-Jun Ding
- Department of Physics and Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jie Ren
- Center for Phononics and Thermal Energy Science, China-EU Joint Center for Nanophononics, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
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Sun Y, Wang L. Correlation functions and their universal connection during an extremely slow equilibration process. Phys Rev E 2022; 105:054114. [PMID: 35706236 DOI: 10.1103/physreve.105.054114] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2021] [Accepted: 04/25/2022] [Indexed: 06/15/2023]
Abstract
We study the equilibration process of a one-dimensional lattice with transverse motions and external magnetic field. Starting from certain initial states, the system commonly reaches a metastable transient state shortly and then stays there for an extremely long time before it finally arrives in the ergodic equilibrium state. The relaxation time T_{eq} diverges even much more rapidly than exponential, which, compared with the widely reported power-law or even exponential divergence in many other systems, implies much higher stability of the transient state. Two correlation functions, the spatiotemporal correlation of the local energy and the autocorrelation of global heat current, are studied in both the metastable transient and the final equilibrium states. It is revealed that the correlations behave entirely differently in the two different states. In the former case they suggest normal heat diffusion and normal heat conduction; whereas in the latter one they indicate super heat diffusion and anomalous heat conduction. More importantly, we confirm that a general relation which connects the two correlations keeps valid not only in the equilibrium state but in the metastable transient state as well. The universality of the connection is, thus, extended.
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Affiliation(s)
- Yachao Sun
- Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices, Renmin University of China, Beijing 100872, People's Republic of China
| | - Lei Wang
- Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices, Renmin University of China, Beijing 100872, People's Republic of China
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Xu L, Wang L. Response and correlation functions of nonlinear systems in equilibrium states. Phys Rev E 2018; 96:052139. [PMID: 29347673 DOI: 10.1103/physreve.96.052139] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2017] [Indexed: 11/07/2022]
Abstract
In this paper, we study systematically a serial of correlation functions in some one-dimensional nonlinear lattices. Due to the energy conservation law, they are implicitly interdependent. Various transport coefficients are thus also connected. In the studies of the autocorrelations of local energy density and of local heat current, a general relation between diverging heat conduction and super heat diffusion has been proposed recently. We clarify that such a relation is valid only in systems without temperature pressure. In those with temperature pressure, a constant but nontrivial term appears. This term explains a previously observed fact that heat diffusion in such systems is always ballistic but heat conduction can diverge very slowly. Such a result not only disproves the existence of any general relation between diverging heat conduction and super heat diffusion, but it also breaks the long-term presumption that ballistic heat conduction and diffusion always coexist.
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Affiliation(s)
- Lubo Xu
- Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-Nano Devices, Renmin University of China, Beijing 100872, People's Republic of China
| | - Lei Wang
- Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-Nano Devices, Renmin University of China, Beijing 100872, People's Republic of China
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Xu L, Wang L. Resonance phonon approach to phonon relaxation time and mean free path in one-dimensional nonlinear lattices. Phys Rev E 2017; 95:042138. [PMID: 28505770 DOI: 10.1103/physreve.95.042138] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2017] [Indexed: 11/07/2022]
Abstract
We extend a previously proposed resonance phonon approach that is based on the linear response theory. By studying the complex response function in depth, we work out the phonon relaxation time besides the oscillating frequency of the phonons in a few one-dimensional nonlinear lattices. The results in the large wave-number-k regime agree with the expectations of the effective phonon theory. However, in the small-k limit they follow different scaling laws. The phonon mean free path can also be calculated indirectly. It coincides well with that derived from the anharmonic phonon approach. A power-law divergent heat conduction, i.e., the heat conductivity κ depends on lattice length N by κ∼N^{β} with β>0, then is supported for the momentum-conserving lattices. Furthermore, this approach can be applied to diatomic lattices. So obtained relaxation time quantitatively agrees with that from the effective phonon theory. As for the mean free path, the resonance phonon approach can detect both the acoustic and the optical branches, whereas the anharmonic phonon approach can only detect a combined branch, i.e., the acoustic branch for small k and the optical branch for large k.
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Affiliation(s)
- Lubo Xu
- Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-Nano Devices, Renmin University of China, Beijing 100872, People's Republic of China
| | - Lei Wang
- Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-Nano Devices, Renmin University of China, Beijing 100872, People's Republic of China
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Zhang C, Kang W, Wang J. Thermal conductance of one-dimensional materials calculated with typical lattice models. Phys Rev E 2016; 94:052131. [PMID: 27967010 DOI: 10.1103/physreve.94.052131] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/18/2016] [Indexed: 11/07/2022]
Abstract
We show through calculations on typical lattice models that thermal conductance σ can well describe the near-equilibrium thermal transport property of one-dimensional materials of finite length, which presents a situation often met in the application of nanoscale devices. The σ generally contains contributions from the material itself and those from the thermal reservoirs. The intrinsic σ of the material, i.e., the one with the fewest external influences, can be efficiently calculated with the help of the "blackbody"-like nonreflective thermal reservoir, either through the nonequilibrium method or through the Green-Kubo-type formula. σ thus calculated would be helpful to guide the design of thermal management and heat control in nanoscale devices.
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Affiliation(s)
- Chunyi Zhang
- HEDPS, Center for Applied Physics and Technology, College of Engineering, Peking University, Beijing 100871, China
| | - Wei Kang
- HEDPS, Center for Applied Physics and Technology, College of Engineering, Peking University, Beijing 100871, China
| | - Jianxiang Wang
- HEDPS, Center for Applied Physics and Technology, College of Engineering, Peking University, Beijing 100871, China.,State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China
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Xu L, Wang L. Dispersion and absorption in one-dimensional nonlinear lattices: A resonance phonon approach. Phys Rev E 2016; 94:030101. [PMID: 27739752 DOI: 10.1103/physreve.94.030101] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2016] [Indexed: 11/07/2022]
Abstract
Based on the linear response theory, we propose a resonance phonon (r-ph) approach to study the renormalized phonons in a few one-dimensional nonlinear lattices. Compared with the existing anharmonic phonon (a-ph) approach, the dispersion relations derived from this approach agree with the expectations of the effective phonon (e-ph) theory much better. The application is also largely extended, i.e., it is applicable in many extreme situations, e.g., high frequency, high temperature, etc., where the existing one can hardly work. Furthermore, two separated phonon branches (one acoustic and one optical) with a clear gap in between can be observed by the r-ph approach in a diatomic anharmonic lattice. While only one combined branch can be detected in the same lattice with both the a-ph approach and the e-ph theory.
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Affiliation(s)
- Lubo Xu
- Department of Physics and Beijing Key Laboratory of Optoelectronic Functional Materials and Micro-Nano Devices, Renmin University of China, Beijing 100872, People's Republic of China
| | - Lei Wang
- Department of Physics and Beijing Key Laboratory of Optoelectronic Functional Materials and Micro-Nano Devices, Renmin University of China, Beijing 100872, People's Republic of China
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Gao Z, Li N, Li B. Heat conduction and energy diffusion in momentum-conserving one-dimensional full-lattice ding-a-ling model. Phys Rev E 2016; 93:022102. [PMID: 26986283 DOI: 10.1103/physreve.93.022102] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/13/2015] [Indexed: 06/05/2023]
Abstract
The ding-a-ling model is a kind of half lattice and half hard-point-gas (HPG) model. The original ding-a-ling model proposed by Casati et al. does not conserve total momentum and has been found to exhibit normal heat conduction behavior. Recently, a modified ding-a-ling model which conserves total momentum has been studied and normal heat conduction has also been claimed. In this work, we propose a full-lattice ding-a-ling model without hard point collisions where total momentum is also conserved. We investigate the heat conduction and energy diffusion of this full-lattice ding-a-ling model with three different nonlinear inter-particle potential forms. For symmetrical potential lattices, the thermal conductivities diverges with lattice length and their energy diffusions are superdiffusive signaturing anomalous heat conduction. For asymmetrical potential lattices, although the thermal conductivity seems to converge as the length increases, the energy diffusion is definitely deviating from normal diffusion behavior indicating anomalous heat conduction as well. No normal heat conduction behavior can be found for the full-lattice ding-a-ling model.
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Affiliation(s)
- Zhibin Gao
- Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University, 200092 Shanghai, People's Republic of China
| | - Nianbei Li
- Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University, 200092 Shanghai, People's Republic of China
| | - Baowen Li
- Department of Mechanical Engineering, University of Colorado Boulder, Colorado 80309, USA
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Zhang Z, Tang C, Tong P. Dynamical thermalization of Frenkel-Kontorova model in the thermodynamic limit. Phys Rev E 2016; 93:022216. [PMID: 26986340 DOI: 10.1103/physreve.93.022216] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2015] [Indexed: 06/05/2023]
Abstract
We study numerically the process of dynamical thermalization in the Frenkel-Kontorova (FK) model with weak nonlinearity. The total energy has initially equidistributed among some of the lowest frequency linear modes. It is found that the energy transfers continuously to the high-frequency modes and finally evolves towards energy equipartition in the FK model. However, the metastable state, which was found in Fermi-Pasta-Ulam (FPU) model and φ(4) model in a relatively short time scale, is not found in the FK model. We further perform a very accurate systematic study of the equipartition time T(eq) as functions of the particle number N, the nonlinear parameter β, and the energy density ɛ. In the thermodynamic limit, the dependence of T(eq) on β and ɛ is found to display a power law behavior: T(eq)∝β(a)ɛ(b). The exponents a and b are numerically found to be approximately -2.0 and 1.43. This scaling law is also quite different from those of the FPU-β model and φ(4) model.
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Affiliation(s)
- Zhenjun Zhang
- College of Science, Hohai University, Nanjing, Jiangsu 210098, People's Republic of China
| | - Chunmei Tang
- College of Science, Hohai University, Nanjing, Jiangsu 210098, People's Republic of China
| | - Peiqing Tong
- School of Physics Science and Technology, Nanjing Normal University, Nanjing, Jiangsu 210023, People's Republic of China
- Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing, Jiangsu 210023, People's Republic of China
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Zhao H, Wang L. Deviation from the Maxwell-Cattaneo law: Role of asymmetric interparticle interactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:042136. [PMID: 26565197 DOI: 10.1103/physreve.92.042136] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/31/2015] [Indexed: 06/05/2023]
Abstract
Nonstationary heat conduction in a few one-dimensional nonlinear lattices is studied numerically based on the Maxwell-Cattaneo (MC) law. We simulate the relaxation process and calculate the magnitudes of the temperature oscillation A(T)(t) and the local heat current oscillation A(j)(t). A phase difference between A(T)(t) and A(j)(t) is observed, which not only verifies the existence of the time lag τ in the MC law but also provides a better way of determining the critical wavelength L(*) that separates between oscillatory and diffusive relaxation modes. However, clear deviations from the MC law are observed. Not only do the decay exponents differ from the theoretical expectations, but, more importantly, suboscillation in the diffusive regime, which is not expected by the MC law, is found in the lattices with asymmetric interactions as well. These findings imply that higher-order effects must be considered in order to well describe the nonstationary heat conduction process in these systems.
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Affiliation(s)
- Huizhu Zhao
- Department of Physics and Beijing Key Laboratory of Optoelectronic Functional Materials and Micro-nano Devices, Renmin University of China, Beijing 100872, People's Republic of China
| | - Lei Wang
- Department of Physics and Beijing Key Laboratory of Optoelectronic Functional Materials and Micro-nano Devices, Renmin University of China, Beijing 100872, People's Republic of China
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