Thermodynamic behavior of an area-preserving multibaker map with energy

Abstract.

A multibaker map with “kinetic energy” is proposed which incorporates an external field. The map is volume-preserving, time-reversal symmetric and conserves total energy. In an appropriate macroscopic limit, the particle distribution is shown to obey a Smoluchowski-type equation. For the cases without any external field and with a constant external force, the nonequilibrium stationary states are constructed by solving the evolution equation of the partially integrated distribution functions. These states are described by singular functions such as incomplete Takagi functions and Lebesgue's singular functions. In an appropriate macroscopic limit, the mass flows for the stationary states are shown to be identical to the ones expected from the Smoluchowski equation and a “heat flow” proportional to the local energy gradient appears. The Gaspard–Gilbert–Dorfman entropy production is calculated for the stationary states and is shown to be positive. Particularly, for the case with a constant external force, when the energy distribution is independent of the spatial distribution, the entropy production reduces to the one consistent with classical thermodynamics. The result shows that there exists a volume-preserving driven multibaker map whose entropy production is consistent with classical thermodynamics.