Institute for Mathematical Physics Entropy Production in Open Volume{preserving Systems Entropy Production in Open Volume-preserving Systems

We describe a mechanism leading to positive entropy production in volume-preserving systems under nonequilibrium conditions. We consider volume-preserving systems sustaining a diiusion process like the multibaker map or the Lorentz gas. A continuous ux of particles is imposed across the system resulting in a stationary gradient of concentration. In the limit where such ux boundary conditions are imposed at arbitrarily separated boundaries for a xed gradient, the invariant measure becomes singular. For instance, in the multibaker map, the limit invariant measure has a cumulative function given in terms of the nondiierentiable Takagi function. Because of this singularity of the invariant measure, the entropy must be deened as an-entropy instead of the Gibbs entropy which would require the existence of a regular measure with a density. The-entropy production is then shown to be asymptoti-cally positive and, moreover, given by the entropy production expected from irreversible thermodynamics.