Fourier’s law for a boundary-thermostated anharmonic chain
For $N \ge 2$ take scalar positions and momenta $(q_i, p_i)_{i=1}^N$, fixed endpoints $q_0 = q_{N+1} = 0$, and the Hamiltonian $$H_N = \sum_{i=1}^N \left[ \frac{p_i^2}{2} + U(q_i) \right] + \sum_{i=0}^{N} V(q_{i+1} - q_i), \qquad U(r) = V(r) = \frac{r^2}{2} + \frac{r^4}{4}.$$ In the interior the dynamics is given by Hamilton's equations. At sites $1$ and $N$ add $-p_1\,dt + \sqrt{4}\,dW_L$ and $-p_N\,dt + \sqrt{2}\,dW_R$ respectively to $dp_i$, with independent Brownian motions $W_L$, $W_R$, so that the bath temperatures are $2$ and $1$; there is no noise or other forcing in the bulk. Let $\mu_N$ be the invariant probability measure. For an interior bond put $j_i = -\frac12 (p_i + p_{i+1}) V'(q_{i+1} - q_i)$; by stationarity $J_N = \int j_i \, d\mu_N$ does not depend on $i$.
Bonetto, Lebowitz and Rey-Bellet define the conductivity as the stationary current times the length divided by the temperature difference, in the limit of vanishing temperature difference and then of infinite length, and ask for a proof, for a Hamiltonian microscopic dynamics, that it exists and is positive and finite. For the harmonic chain, Rieder, Lebowitz and Lieb found that the stationary current is proportional to the temperature difference rather than to the temperature gradient. Bonetto, Lebowitz and Lukkarinen proved Fourier's law for a harmonic crystal with self-consistent stochastic reservoirs at every site, and Canestrari, Liverani and Olla derived the heat equation for a deterministic chain driven by an additional chaotic force. Carmona proved existence, uniqueness and exponential convergence to the invariant measure for chains with Langevin baths at both ends whose interaction grows at least as fast as the pinning.
Problem. Does $\lim_{N \to \infty} N J_N$ exist and lie in $(0, \infty)$?
Source: Stated by AIM — Open Applied Problems, as an explicit instance of the derivation of Fourier's law from a Hamiltonian microscopic dynamics posed by Bonetto, Lebowitz and Rey-Bellet (2000).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, anharmonic chains, heat conduction, nonequilibrium steady states
References
@incollection{bonetto2000,
author = {Bonetto, F. and Lebowitz, J. L. and Rey-Bellet, L.},
title = {Fourier's law: a challenge to theorists},
booktitle = {Mathematical Physics 2000},
editor = {Fokas, Athanassios and Grigoryan, Alexander and Kibble, Tom and Zegarlinski, Boguslaw},
publisher = {Imperial College Press},
address = {London},
year = {2000},
pages = {128--150},
doi = {10.1142/9781848160224_0008},
eprint = {math-ph/0002052},
archivePrefix = {arXiv},
}
@article{rieder1967,
author = {Rieder, Z. and Lebowitz, J. L. and Lieb, E.},
title = {Properties of a harmonic crystal in a stationary nonequilibrium state},
journal = {J. Math. Phys.},
volume = {8},
number = {5},
year = {1967},
pages = {1073--1078},
doi = {10.1063/1.1705319},
}
@article{bonetto2004,
author = {Bonetto, F. and Lebowitz, J. L. and Lukkarinen, J.},
title = {Fourier's law for a harmonic crystal with self-consistent stochastic reservoirs},
journal = {J. Stat. Phys.},
volume = {116},
number = {1-4},
year = {2004},
pages = {783--813},
doi = {10.1023/B:JOSS.0000037232.14365.10},
eprint = {math-ph/0307035},
archivePrefix = {arXiv},
}
@article{canestrari2026,
author = {Canestrari, Giovanni and Liverani, Carlangelo and Olla, Stefano},
title = {Heat equation from a deterministic dynamics},
journal = {Invent. Math.},
volume = {245},
number = {3},
year = {2026},
pages = {1437--1504},
doi = {10.1007/s00222-026-01429-1},
eprint = {2310.13338},
archivePrefix = {arXiv},
}
@article{carmona2007,
author = {Carmona, Philippe},
title = {Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths},
journal = {Stochastic Process. Appl.},
volume = {117},
number = {8},
year = {2007},
pages = {1076--1092},
doi = {10.1016/j.spa.2006.12.003},
eprint = {math/0611689},
archivePrefix = {arXiv},
}
@misc{aim-anharmonic-chain-fourier-law,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Fourier’s law for a boundary-thermostated anharmonic chain}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/228-anharmonic-chain-fourier-law.md},
note = {Entry 228 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00573, "Fourier’s law for a boundary-thermostated anharmonic chain", clopen.solutions.