Fourier's Law confirmed for a class of small quantum systems
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The neglect of the off-diagonal entries in Eq. (52) directly yields a Lindblad QME of the same type previously used for investigations on heat transport in small quantum systems [14], which hitherto lacked a microscopic derivation.
“Moreover we would like to point out that the neglect of the off-diagonal entries in Eq. (52) directly yields a Lindblad QME which is of the same type that has previously served for investigations on heat transport in small quantum systems [14] but hitherto lacked a microscopic derivation.”
From Modeling heat transport through completely positive maps (2007) · cites this paper as [14]
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Cites
Harrison, W. (2000) · doi:10.1142/9789812386199_others05 · Cited by 1,261 · cited 2×
“In the classical phonon theory of heat conduction in insulators there are some expectations on this dependence based on experimental results [15].”
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Strong evidence of normal heat conduction in a one-dimensional quantum system
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“There are systems showing a normal heat conductance [12,11] and others where the conductivity diverges [5] (vanishing temperature gradient).”
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Self-citationSimple one-dimensional model of heat conduction which obeys Fourier's law.
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“hel et al.: Fourier’s Law confirmed for a class of small quantum systems 2 Model We consider an open quantum system, a chain of N subsystems with n levels each, with nearest neighbor interactions (cf. [6,8]). The hamiltonian H of the system thus reads H = XN µ=1 H(loc)(µ) + λ I NX−1 µ=1 H(int) µ,µ+1. (1) Here, the term…”
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“However, the approach of open-quantum-system has only quite recently been applied to the non-equilibrium problems of condensed matter physics [8, 9, 10, 11, 12, 13, 14, 15], such as the controversial quantum transport problem in one-dimension [16, 17, 18].”
MethodDiffusive and Subdiffusive Spin Transport in the Ergodic Phase of a Many-Body Localizable System
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“The second approach goes via direct simulation of out-of-equilibrium system coupled to the baths, usually using a master equation [30, 31, 32, 33, 34, 35, 36, 37] solving it by either numerical integration, diagonalization, or Monte Carlo wave-function approach.”
MethodModeling heat transport through completely positive maps
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Periodically refreshed baths to simulate open quantum many-body dynamics
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Local temperatures out of equilibrium
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“It has been found that the temperature oscillations in nanowires [140–142] strongly violate the Fourier’s law which predicts a linear temperature profile along a one-dimensional nanosystem [337, 422–424].”
BackgroundLimited validity of the Kubo formula for thermal conduction in modular quantum systems
Gemmer, J., Steinigeweg, R., & Michel, M. (2006) · Physical Review B · doi:10.1103/physrevb.73.104302 · Cited by 41 · see citing works
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Self-citationHeat transport in quantum spin chains
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Reichental, I., Klempner, A., Kafri, Y., & Podolsky, D. K. (2018) · Physical review. B./Physical review. B · doi:10.1103/physrevb.97.134301 · Cited by 34 · see citing works
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Self-citationMICROSCOPIC QUANTUM MECHANICAL FOUNDATION OF FOURIER'S LAW
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“The majority of coupling types within the chain, the Heisenberg coupling and the random next neighbor interaction, indeed show this normal behavior in the weak coupling limit (14; 21).”
BackgroundResultInfluentialSelf-citationHeat exchange mediated by a quantum system
Panasyuk, G. Y., Levin, G. A., & Yerkes, K. L. (2012) · Physical Review E · doi:10.1103/physreve.86.021116 · Cited by 19 · see citing works
Heat Conductivity in Small Quantum Systems: Kubo Formula in Liouville SpacePreprint
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Global and local relaxation of a spin chain under exact Schrödinger and master-equation dynamics
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Self-citationIsing chain: Thermal conductivity and first-principle validation of Fourier’s law
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“Nowadays, various low-dimensional systems have been experimentally [3, 4] and theoretically [5, 6, 7, 8, 9, 10, 11, 12, 13] investigated and some of them, even in one-dimension, obey that important relation [16, 17, 15, 14].”
BackgroundNon-equilibrium stationary states from the equation of motion of open systems
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“This bath coupling could be realized by standard Lindblad operators [12,14], well known from the theory of open systems in quantum optics.”
BackgroundMethodSelf-citationHeitmann, T., Richter, J., Jin, F., et al. (2023) · Physical review. B./Physical review. B · doi:10.1103/physrevb.108.l201119 · Cited by 6 · see citing works
“Then, the profile and currsnt of this steady state yield information on the transport behavior [42–45].”
BackgroundSelf-citationTransport in anisotropic model systems analyzed by a correlated projection superoperator technique
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“From the theoretical point of view, many authors have investigated Fourier’s law in a wide diversity of models [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 37, 35, 36], like a Lorentz gas [15], biological [22] and small quantum systems [21], chains of coupled harmonic [23], or anhar…”
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“This model has been extensively investigated in the context of non-equilibrium statistical mechanics, with a focus on understanding magnetization and heat transport [65, 128– 132].”
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“In an open situation, the system of actual interest can be coupled at its boundaries to two reservoirs at different temperatures or chemical potentials, such that transport is induced and a nonequilibrium steady state is usually established in the long-time limit [22–24].”
Kang, S. M., Jung, Y. J., & Lee, J. H. (2014) · Journal of Life Science · doi:10.5352/jls.2014.24.4.437 · Cited by 1 · see citing works
“를 equation 1 [27]에 따라 퓨리에 변환하여 적외선 흡수 스펙 트럼을 얻은 뒤 이를 분석하는 비분산형 분광기로, 유기 화합 물을 구성하는 작용기의 특성 흡수대를 이용하여 미지 시료의 동정, 정량, 구조 해석 등에 흔히 이용된다.”
Open System Approach to Transport1
Jochen Gemmer, B., Mathias Michel, A., & Mahler, G. (2009) · Lecture notes in physics · doi:10.1007/978-3-540-70510-9_21
Self-citationQuantum mechanics and classical computing
Mahler, G. (2006) · Informatik - Forschung und Entwicklung · doi:10.1007/s00450-006-0015-8
Self-citationQuantum typicality approach to energy flow between two spin-chain domains at different temperatures
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“Then, in the long-time limit, a nonequilibrium steady state usually emerges with a characteristic density profile and a constant current [41–43].”
Fourier’s law breakdown for the planar-rotor chain with long-range interactions
Lima, H. S., Tsallis, C., Eroglu, D., et al. (2026) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2026.131314
- Authors
- Michel, M., Hartmann, M., Gemmer, J., & Mahler, G.
- Affiliations
- University of Stuttgart (DE)
- Corresponding author
- Michel, M.
- Venue
- The European Physical Journal B - Condensed Matter
- Published
- 2003-08-01
- Volume
- 34
- Issue
- 3
- Pages
- 325-330
- ISSN
- 1434-6028 (ISSN-L)
- 1434-6036
- OpenAlex
- W2087841179
- Cited by
- 80
- References
- 15
- Citations by year
- 2026: 2 · 2025: 3 · 2024: 1 · 2023: 5 · 2022: 1 · 2021: 4 · 2019: 2 · 2018: 1 · 2016: 2 · 2014: 1 · 2013: 1 · 2012: 4
- FWCI
- 4.28
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- www.springer.com/tdm (tdm)
- Locations
- The European Physical Journal B (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Springer Link (Chiba Institute of Technology) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and electron transport phenomena
- Keywords
- Fourier's law, heat conduction, Lindblad dynamics, spin chains, energy transport, linear transport, small quantum systems, interaction strength, open quantum systems
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