Heat Conduction Networks: Disposition of Heat Baths and Invariant Measure

TL;DR

This paper studies heat conduction networks with oscillators and heat baths, establishing conditions for the existence and uniqueness of their stationary states, especially focusing on the arrangement of heat baths and interaction strength.

Contribution

It generalizes previous results to complex networks, providing sharp conditions for invariant measure existence and uniqueness based on heat bath placement and interaction strength.

Findings

Condition for unique invariant measure derived using LaSalle's principle.

Sharpness of the condition demonstrated for linear oscillators.

Strong interactions between particles ensure the existence of invariant measure.

Abstract

We consider a model of heat conduction networks consisting of oscillators in contact with heat baths at different temperatures. Our aim is to generalize the results concerning the existence and uniqueness of the stationnary state already obtained when the network is reduced to a chain of particles. Using Lasalle's principle, we establish a condition on the disposition of the heat baths among the network that ensures the uniqueness of the invariant measure. We will show that this condition is sharp when the oscillators are linear. Moreover, when the interaction between the particles is stronger than the pinning, we prove that this condition implies the existence of the invariant measure.