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Green-Kubo formula for weakly coupled systems with noise

Bernardin, Cédric; Huveneers, François; Lebowitz, Joel L.; Liverani, Carlangelo; Olla, Stefano (2015), Green-Kubo formula for weakly coupled systems with noise, Communications in Mathematical Physics, 334, 3, p. 1377-1412. http://dx.doi.org/10.1007/s00220-014-2206-7

Type
Article accepté pour publication ou publié
Date
2015
Journal name
Communications in Mathematical Physics
Volume
334
Number
3
Publisher
Springer
Pages
1377-1412
Publication identifier
http://dx.doi.org/10.1007/s00220-014-2206-7
Metadata
Show full item record
Author(s)
Bernardin, Cédric
Laboratoire Jean Alexandre Dieudonné [JAD]
Huveneers, François
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Lebowitz, Joel L.
Center for Mathematical Sciences Research
Liverani, Carlangelo
Dipartimento di Matematica [Roma II] [DIPMAT]
Olla, Stefano cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
We consider an infinite system of cells coupled into a chain by a smooth nearest neighbour potential $\varepsilon V$. The uncoupled system (cells) evolve according to Hamiltonian dynamics perturbed stochastically with an energy conserving noise of strength $\zeta$. We study the Green-Kubo (GK) formula $\kappa(\varepsilon,\zeta)$ for the heat conductivity of this system which exists and is finite for $\zeta >0$, by formally expanding $\kappa(\varepsilon,\zeta)$ in a power series in $\varepsilon$, $\kappa(\varepsilon,\zeta) = \sum_{n\ge 2} \varepsilon^n \kappa_n(\zeta)$. We show that $\kappa_2(\zeta)$ is the same as the conductivity obtained in the weak coupling (van Hove) limit where time is rescaled as $\varepsilon^{-2}t$. $\kappa_2(\zeta)$ is conjectured to approach as $\zeta \to 0$ a value proportional to that obtained for the weak coupling limit of the purely Hamiltonian chain. We also show that the $\kappa_2(\zeta)$ from the GK formula, is the same as the one obtained from the flux of an open system in contact with Langevin reservoirs. Finally we show that the limit $\zeta\to 0$ of $\kappa_2(\zeta)$ is finite for the pinned anharmonic oscillators due to phase mixing caused by the non-resonating frequencies of the neighbouring cells. This limit is bounded for coupled rotors and vanishes for harmonic chain with random pinning.
Subjects / Keywords
thermal conductivity; small noise; coupling expansion; Green-Kubo formula

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