Entropy Currents for Reversible Processes in a System of Differential   equations. -- The Case of Latticized Classical Field Theory --
  We consider a very complicated system of some latticized differential equations that is considered as equations of motion for a field theory. We define macro state restrictions for such a system analogous to thermodynamical states of a system in statistical mechanics. For the case in which we have assumed adiabaticity in a generalized way which is equivalent to reversible processes. It is shown that we can define various entropy currents, not only one. It is indeed surprising that, for a two dimensional example of lattice field theory, we get three different entropy currents, all conserved under the adiabaticity condition. 
Holger B. Nielsen and Masao Ninomiya