Remarks on NonHamiltonian Statistical Mechanics: Lyapunov Exponents and   Phase-Space Dimensionality Loss
  The dissipation associated with nonequilibrium flow processes is reflected by the formation of strange attractor distributions in phase space. The information dimension of these attractors is less than that of the equilibrium phase space, corresponding to the extreme rarity of nonequilibrium states. Here we take advantage of a simple model for heat conduction to demonstrate that the nonequilibrium dimensionality loss can definitely exceed the number of phase-space dimensions required to thermostat an otherwise Hamiltonian system. 
Wm. G. Hoover, H. A. Posch, K. Aoki, D. Kusnezov