Heat Conduction Networks: Disposition of Heat Baths and Invariant Measure

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. 1. Definitions and Results 1.1. The motivations. We consider an arbitrary graph. At each vertex of this graph, there is a particle interacting with the substrate and with its neighbours. Among these particles, some are linked to heat baths; an Ornstein-Uhlenbeck process models this interaction. Given this graph, we establish conditions on the disposition of the heat baths that entails existence and uniqueness of the invariant measure. When the graph is reduced to a chain, each extremal particle is connected to a heat bath. This model has been studied in [EPRB99b, EPRB99a, EH00, RBT02]. The uniqueness of the invariant measure is obtained using controllability properties. This property is deeply connected to the geometry of the chain: the behaviour of the extremal particle entails the behaviour of its neighbour and so on. . . The existence of the invariant measure when the interaction is stronger than the pinning has also been obtained. These results were used in [Car07] to solve some variations of this model developed in [BO05] on the one side and [LS04] on the other. To avoid the particular geometry of the chain, we work with general networks. These heat conduction networks have been introduced in [MNV03] and [RB03]. Let us notice that an Ornstein-Uhlenbeck process is the sum of a damping term and an excitation term. To understand the effect of each of these quantities, we will not suppose that the heat baths have non-negative temperatures. A recent work of [BLLO08] uses this kind of result to prove the existence of a self-consistent temperature profile. We will see that their results are closely related to the geometry of the network they consider. First we introduce the model. Then we will state our main results linking existence and uniqueness of the invariant measure to the disposition of the heat baths. Intuitively, the existence and the uniqueness of the invariant measure is related to the disposition of the damped particles, i.e. the particles interacting with Date: February 3, 2009. 2000 Mathematics Subject Classification. Primary 82C22 ; Secondary 60K35, 60H10.