## Non-Equilibrium Statistical Mechanics of Strongly Anharmonic Chains of Oscillators (2000)

Venue: | Commun. Math. Phys |

Citations: | 51 - 14 self |

### BibTeX

@ARTICLE{Eckmann00non-equilibriumstatistical,

author = {J.-P. Eckmann and M. Hairer},

title = {Non-Equilibrium Statistical Mechanics of Strongly Anharmonic Chains of Oscillators},

journal = {Commun. Math. Phys},

year = {2000},

volume = {212},

pages = {105--164}

}

### OpenURL

### Abstract

We study the model of a strongly non-linear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, Rey-Bellet [EPR99a, EPR99b] to potentials with essentially arbitrary growth at infinity. This extension is possible by introducing a stronger version of Hormander's theorem for Kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains. Introduction In this paper, we study the statistical mechanics of a highly non-linear chain of oscillators coupled to two heat reservoirs which are at (arbitrary) different temperatures. We show that such systems have, under suitable conditions, a unique stationary state, in which heat flows from the hotter reservoir to the cooler one. These results are an extension of the same statements obtained by Eckmann, Pillet and ReyBellet in [EPR99a, EPR99b] where it was ass...

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Citation Context ... describing a chain of particles with nearest-neighbor interaction (Figure 1). This chain is linearly coupled to heat baths B i represented by free fields at temperatures T i . We proceed then, as in =-=[EPR99a], to a red-=-uction to a stochastic differential equation, see (1.2). Associated with it is an "effective energy" G, described in (1.6), which is equal to H S with some quadratic terms from the heat bath... |

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Citation Context ...d result. Finally, we want to show the strict positivity and the uniqueness of the invariant measure. The proof of this result will only be sketched, as it simply retraces the proof of Theorem 3.6 in =-=[EPR99b]-=-. Proposition 7.6. The density h of the invariant measuresis a strictly positive function. Moreover, the invariant measure is unique. Sketch of proof. The idea is to show that the control system assoc... |

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Citation Context ...ome physical reason behind this. If a stationary state exists, this means that even if the chain is in a state of very high energy, the mean time to reach a region with low energy is finite (see e.g. =-=[Has80]-=-). But if m < n, the relative strength of the coupling versus the one-body potential goes to zero at high energy. The consequence is that there is almost no energy transmitted between particles. Since... |

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Citation Context ...ome physical reason behind this. If a stationary state exists, this means that even if the chain is in a state of very high energy, the mean time to reach a region with low energy is finite (see e.g. =-=[Has80]-=-). But if m ! n, the relative strength of the coupling versus the one-body potential goes to zero at high energy. The consequence is that there is almost no energy transmitted between particles. Since... |