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FOURIER'S LAW: A CHALLENGE TO THEORISTS
Abstract
We present a selective overview of the current state of our knowledge (more precisely of our ignorance) regarding the derivation of Fourier’s Law, J(r) = −κ∇T(r); J the heat flux, T the temperature and κ, the heat conductivity. This law is empirically well tested for both fluids and crystals, when the temperature varies slowly on the microscopic scale, with κ an intrinsic property which depends only on the system’s equilibrium parameters, such as the local temperature and density. There is however at present no rigorous mathematical derivation of Fourier’s law and ipso facto of Kubo’s formula for κ, involving integrals over equilibrium time correlations, for any system (or model) with a deterministic, e.g. Hamiltonian, microscopic evolution. 1
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The connection between the steady states of systems evolving under the invented thermostated dynamics and those obtained from more realistic models for which the stationary ̄ρ is not singular is still not entirely clear; the same is true for the correct identification of entropy in such systems.
“The connection between the steady states of systems evolving under the invented thermostated dynamics and those obtained from more realistic models for which the stationary ¯ ρ is not singular, is still not entirely clear. The same is true for the correct identification of entropy in such systems [17-23].”
From On the (Boltzmann) entropy of non-equilibrium systems (2004) · cites this paper as [23]
The σ in the thermostated dynamics is frequently interpreted as an 'entropy production' and in some cases has a form similar to the hydrodynamic entropy production in a stationary open system with currents.
“This σ is frequently interpreted as an 'entropy production' and indeed has, in some cases, a form similar to the hydrodynamic entropy production in a stationary open system in which there are currents.”
From On the (Boltzmann) entropy of non-equilibrium systems (2004) · cites this paper as [20]
The connection between the steady states of systems evolving under the invented thermostated dynamics and those obtained from more realistic models for which the stationary ̄ρ is not singular is still not entirely clear; the same is true for the correct identification of entropy in such systems.
“The connection between the steady states of systems evolving under the invented thermostated dynamics and those obtained from more realistic models for which the stationary ¯ ρ is not singular, is still not entirely clear. The same is true for the correct identification of entropy in such systems [17-23].”
From On the (Boltzmann) entropy of non-equilibrium systems (2004) · cites this paper as [20]
For a closed system evolving under deterministic non-Hamiltonian thermostated dynamics, starting with an initial density ρ(X,0) uniform (or absolutely continuous) with respect to the appropriate Lebesgue measure, the dynamics leads to ρ(X,t)→ ̄ρ as t→∞, with ̄ρ singular with respect to Lebesgue measure (generally an SRB measure), and S_G(ρ)→-∞ decreasing with time.
“Starting with an initial density ρ ( X, 0), uniform (or absolutely continuous) with respect to the appropriate Lebesgue measure, the dynamics leads to ρ ( X,t ) → ¯ ρ , as t →∞ , with ¯ ρ singular with respect to Lebesgue measure, (¯ ρ is generally an SRB measure), and S G ( ρ ) →-∞ decreasing with time”
From On the (Boltzmann) entropy of non-equilibrium systems (2004) · cites this paper as [20]
For open systems in contact with thermal reservoirs, the time evolution of the microstate X is no longer given by a Hamiltonian since the system is not isolated, and S_G(ρ_t) need no longer be constant in time.
“The situation is different when one considers open systems, e.g. systems in contact with thermal reservoirs [19, 20]. The time evolution of the microstate X of such a system is then no longer given by a Hamiltonian since the system is not isolated and S G ( ρ t ) need no longer be constant in time.”
From On the (Boltzmann) entropy of non-equilibrium systems (2004) · cites this paper as [20]
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On the (Boltzmann) entropy of non-equilibrium systems
Goldstein, S., & Lebowitz, J. L. (2004) · Physica D: Nonlinear Phenomena · doi:10.1016/j.physd.2004.01.008 · Cited by 240
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Metadata
- Authors
- BONETTO, F., LEBOWITZ, J. L., & REY-BELLET, L.
- Affiliations
- Rutgers, The State University of New Jersey (US)
- Venue
- Mathematical Physics 2000
- Published
- 2000-05
- Pages
- 128-150
- OpenAlex
- W1597941147
- Cited by
- 303
- References
- 0
- Citations by year
- 2026: 2 · 2025: 5 · 2024: 6 · 2023: 11 · 2022: 7 · 2021: 10 · 2020: 15 · 2019: 22 · 2018: 11 · 2017: 8 · 2016: 26 · 2015: 17 · 2014: 17 · 2013: 14 · 2012: 18
- FWCI
- 3.88
- Type
- book chapter
- Language
- en
- Open access
- closed
- Locations
- IMPERIAL COLLEGE PRESS eBooks (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Physical Sciences › Materials Science › Materials Chemistry › Thermal properties of materials
- Physical Sciences › Mathematics › Applied Mathematics › Gas Dynamics and Kinetic Theory
- Keywords
- Fourier's law, heat conduction, thermal conductivity, heat flux, Kubo formula, temporal correlation, Hamiltonian dynamics, non-equilibrium thermodynamics, crystals, fluids
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