1219 0022-4715/03/0600-1219/0 © 2003 Plenum Publishing Corporation Journal of Statistical Physics, Vol. 111, Nos. 5/6, June 2003 (© 2003) Heat Conduction Networks Christian Maes, 1 Karel Netoc ˇny ´, 2 and Michel Verschuere 3 1 Instituut voor Theoretische Fysica K.U. Leuven, B-3001 Leuven, Belgium; e-mail: christian. maes@fys.kuleuven.ac.be 2 E-mail: karel.netocny@fys.kuleuven.ac.be 3 Aspirant FWO Vlaanderen; e-mail: michel.verschuere@fys.kuleuven.ac.be Received April 16, 2002; accepted November 21, 2002 We study networks of interacting oscillators, driven at the boundary by heat baths at possibly different temperatures. A set of first elementary questions are discussed concerning the uniqueness of a stationary possibly Gibbsian density and the nature of the entropy production and the local heat currents. We also derive a Carnot efficiency relation for the work that can be extracted from the heat engine. KEY WORDS: Heat current; entropy production; nonequilibrium state. 1. THE MODEL AND RESULTS Consider a finite connected graph G=(V, ’ ) with vertex set V. Two vertices (=sites) i ] j ¥ V are called nearest neighbors if there is an edge between them: i ’ j. Every site i ¥ V carries a momentum and position coordinate (p i ,q i ) ¥ R 2 . Generalizations to higher dimensional coordinate vectors are straightforward. We select a non-empty subset “V … V, called boundary sites, that, below, will be imagined connected to thermal baths at possibly different temperatures. States ( p, q) are elements ((p i ,q i ), i ¥ V) ¥ R 2 |V | and r will denote a probability density (with respect to dp dq= < i dp i dq i ) on it. The coupling between the degrees of freedom is modeled by the Hamiltonian H( p, q)= C i ¥ V p 2 i 2 +U(q) (1.1)