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How Hot Can a Heat Bath Get?

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Abstract

We study a model of two interacting Hamiltonian particles subject to a common potential in contact with two Langevin heat reservoirs: one at finite and one at infinite temperature. This is a toy model for ‘extreme’ non-equilibrium statistical mechanics. We provide a full picture of the long-time behaviour of such a system, including the existence/non-existence of a non-equilibrium steady state, the precise tail behaviour of the energy in such a state, as well as the speed of convergence toward the steady state.

Despite its apparent simplicity, this model exhibits a surprisingly rich variety of long time behaviours, depending on the parameter regime: if the surrounding potential is ‘too stiff’, then no stationary state can exist. In the softer regimes, the tails of the energy in the stationary state can be either algebraic, fractional exponential, or exponential. Correspondingly, the speed of convergence to the stationary state can be either algebraic, stretched exponential, or exponential. Regarding both types of claims, we obtain matching upper and lower bounds.

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Authors and Affiliations

  1. Mathematics Institute, The University of Warwick, Coventry, CV4 7AL, United Kingdom

    Martin Hairer

  2. Courant Institute, New York University, New York, NY, 10012, USA

    Martin Hairer

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  1. Martin Hairer
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Correspondence to Martin Hairer.

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Communicated by A. Kupiainen

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Hairer, M. How Hot Can a Heat Bath Get?. Commun. Math. Phys. 292, 131–177 (2009). https://doi.org/10.1007/s00220-009-0857-6

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  • DOI: https://doi.org/10.1007/s00220-009-0857-6

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