Thermodynamic behavior of an area-preserving multibaker map with energy

A multibaker map with "kinetic energy'' is proposed which incorporates an e xternal field. The map is volume-preserving, time-reversal symmetric and co nserves total energy. In an appropriate macroscopic limit, the particle dis tribution is shown to obey a Smoluchowski-type equation. For the cases with out any external field and with a constant external force, the nonequilibri um stationary states are constructed by solving the evolution equation of t he partially integrated distribution functions. These states are described by singular functions such as incomplete Takagi functions and Lebesgue's si ngular functions. In an appropriate macroscopic limit, the mass flows for t he stationary states are shown to be identical to the ones expected from th e Smoluchuwski equation and a "heat flow'' proportional to the local energy gradient appears. The Gaspard-Gilbert-Dorfman entropy production is calcul ated for the stationary states and is shown to be positive. Particularly, f or 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 th ere exists a volume-preserving driven multibaker map whose entropy producti on is consistent with classical thermodynamics.