Anomalies of Lévy-based thermal transport from the Lévy-Fokker-Planck equation

Shu-Nan Li; Bing-Yang Cao; Shu-Nan Li; Bing-Yang Cao

doi:10.3934/math.2021402

AIMS Mathematics

2021, Volume 6, Issue 7: 6868-6881. doi: 10.3934/math.2021402

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Anomalies of Lévy-based thermal transport from the Lévy-Fokker-Planck equation

Shu-Nan Li ,
Bing-Yang Cao ^,

Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

Received: 29 January 2021 Accepted: 25 March 2021 Published: 22 April 2021
MSC : 34A08, 80A05, 80M60

Lévy-type behaviors are widely involved in anomalous thermal transport, yet generic investigations based on the mathematical descriptions of the confined Lévy flights are still lacking. In the frameworks of classical irreversible thermodynamics and Boltzmann-Gibbs statistical mechanics, the Lévy-Fokker-Planck equation is connected to near-equilibrium thermal transport. In this work, we show that thermal transport dominated by the confined Lévy flights will be paired with an anomaly, namely that the local effective thermal conductivity is nonlocal. It is demonstrated that the near-equilibrium assumption is not unconditionally valid, which relies on several thermodynamic restrictions expressed by the probability density function (PDF). It is illustrated that the Lévy-Fokker-Planck equation based on the Caputo operator will give rise to two signatures of anomalous thermal transport, the power-law size-dependence of the global effective thermal conductivity and nonlinear boundary asymptotics of the stationary temperature profile. These anomalies are interrelated with each other, and their quantitative relations can be considered as criteria for Lévy-based thermal transport.
- Lévy-Fokker-Planck equation,
- fractional-order derivative,
- anomalous thermal transport,
- irreversible thermodynamics,
- effective thermal conductivity
Citation: Shu-Nan Li, Bing-Yang Cao. Anomalies of Lévy-based thermal transport from the Lévy-Fokker-Planck equation[J]. AIMS Mathematics, 2021, 6(7): 6868-6881. doi: 10.3934/math.2021402

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Abstract

Lévy-type behaviors are widely involved in anomalous thermal transport, yet generic investigations based on the mathematical descriptions of the confined Lévy flights are still lacking. In the frameworks of classical irreversible thermodynamics and Boltzmann-Gibbs statistical mechanics, the Lévy-Fokker-Planck equation is connected to near-equilibrium thermal transport. In this work, we show that thermal transport dominated by the confined Lévy flights will be paired with an anomaly, namely that the local effective thermal conductivity is nonlocal. It is demonstrated that the near-equilibrium assumption is not unconditionally valid, which relies on several thermodynamic restrictions expressed by the probability density function (PDF). It is illustrated that the Lévy-Fokker-Planck equation based on the Caputo operator will give rise to two signatures of anomalous thermal transport, the power-law size-dependence of the global effective thermal conductivity and nonlinear boundary asymptotics of the stationary temperature profile. These anomalies are interrelated with each other, and their quantitative relations can be considered as criteria for Lévy-based thermal transport.

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