### Abstract

We study the dynamics of covariances in a chain of harmonic oscillators with conservative noise in contact with two stochastic Langevin heat baths. The noise amounts to random collisions between nearest-neighbour oscillators that exchange their momenta. In a recent paper (Lepri et al 2009 J. Phys. A: Math. Theor. 42 025001), we have studied the stationary state of this system with fixed boundary conditions, finding analytical exact expressions for the temperature profile and the heat current in the thermodynamic (continuum) limit. In this paper, we extend the analysis to the evolution of the covariance matrix and to generic boundary conditions. Our main purpose is to construct a hydrodynamic description of the relaxation to the stationary state, starting from the exact equations governing the evolution of the correlation matrix. We identify and adiabatically eliminate the fast variables, arriving at a continuity equation for the temperature profile T(y, t), complemented by an ordinary equation that accounts for the evolution in the bulk. Altogether, we find that the evolution of T(y, t) is the result of fractional diffusion.

Original language | English |
---|---|

Article number | 065002 |

Number of pages | 22 |

Journal | Journal of Physics. A, Mathematical and theoretical |

Volume | 43 |

Issue number | 6 |

DOIs | |

Publication status | Published - 12 Feb 2010 |

### Keywords

- conduction
- lattices

### Cite this

*Journal of Physics. A, Mathematical and theoretical*,

*43*(6), [065002]. https://doi.org/10.1088/1751-8113/43/6/065002

**Nonequilibrium dynamics of a stochastic model of anomalous heat transport.** / Lepri, Stefano; Mejia-Monasterio, Carlos; Politi, Antonio.

Research output: Contribution to journal › Article

*Journal of Physics. A, Mathematical and theoretical*, vol. 43, no. 6, 065002. https://doi.org/10.1088/1751-8113/43/6/065002

}

TY - JOUR

T1 - Nonequilibrium dynamics of a stochastic model of anomalous heat transport

AU - Lepri, Stefano

AU - Mejia-Monasterio, Carlos

AU - Politi, Antonio

PY - 2010/2/12

Y1 - 2010/2/12

N2 - We study the dynamics of covariances in a chain of harmonic oscillators with conservative noise in contact with two stochastic Langevin heat baths. The noise amounts to random collisions between nearest-neighbour oscillators that exchange their momenta. In a recent paper (Lepri et al 2009 J. Phys. A: Math. Theor. 42 025001), we have studied the stationary state of this system with fixed boundary conditions, finding analytical exact expressions for the temperature profile and the heat current in the thermodynamic (continuum) limit. In this paper, we extend the analysis to the evolution of the covariance matrix and to generic boundary conditions. Our main purpose is to construct a hydrodynamic description of the relaxation to the stationary state, starting from the exact equations governing the evolution of the correlation matrix. We identify and adiabatically eliminate the fast variables, arriving at a continuity equation for the temperature profile T(y, t), complemented by an ordinary equation that accounts for the evolution in the bulk. Altogether, we find that the evolution of T(y, t) is the result of fractional diffusion.

AB - We study the dynamics of covariances in a chain of harmonic oscillators with conservative noise in contact with two stochastic Langevin heat baths. The noise amounts to random collisions between nearest-neighbour oscillators that exchange their momenta. In a recent paper (Lepri et al 2009 J. Phys. A: Math. Theor. 42 025001), we have studied the stationary state of this system with fixed boundary conditions, finding analytical exact expressions for the temperature profile and the heat current in the thermodynamic (continuum) limit. In this paper, we extend the analysis to the evolution of the covariance matrix and to generic boundary conditions. Our main purpose is to construct a hydrodynamic description of the relaxation to the stationary state, starting from the exact equations governing the evolution of the correlation matrix. We identify and adiabatically eliminate the fast variables, arriving at a continuity equation for the temperature profile T(y, t), complemented by an ordinary equation that accounts for the evolution in the bulk. Altogether, we find that the evolution of T(y, t) is the result of fractional diffusion.

KW - conduction

KW - lattices

U2 - 10.1088/1751-8113/43/6/065002

DO - 10.1088/1751-8113/43/6/065002

M3 - Article

VL - 43

JO - Journal of Physics. A, Mathematical and theoretical

JF - Journal of Physics. A, Mathematical and theoretical

SN - 1751-8113

IS - 6

M1 - 065002

ER -