D ec 2 00 1 Steady state thermodynamics for heat conduction

Following the proposal of steady state thermodynamics (SST) by Oono and Paniconi, we develop a phenomenological theory for steady nonequilibrium states in systems with heat conduction. We find that there is essentially a unique consistent thermodynamics, and make concrete predictions, i.e, the existence of a new osmotic pressure and a shift in the coexistence temperature. These predictions allow one to test for the quantitative validity of SST by comparing them with experiments. Construction of a statistical mechanics that apply to nonequilibrium states has been a challenging open problem in theoretical physics [1]. But so far it is not known how the desired probability measures for nonequilibrium states look like, or even whether the measures can be written in compact forms [2]. Recalling the history that the conventional thermodynamics was an essential guide when Boltzmann, Gibbs, and others constructed equilibrium statistical mechanics, it may be a good idea to start from the level of thermodynamics. The standard theory of nonequilibrium thermodynamics [3] is based on local equilibrium hypothesis, which roughly asserts that each small part of a nonequilibrium state can be regarded as a copy of a suitable equilibrium state. But such a description seems insufficient for general nonequilibrium states. Consider, for example, a system with a steady heat flow. It is true that the quantities like the temperature and the density become essentially constant within a sufficiently small portion of the system. But no matter how small the portion is, there always exists a heat flux passing through it and hence the local state is not isotropic. This suggests that the local state cannot be identical to an equilibrium state (which is isotropic), but should be described rather as a local steady state. Among existing attempts in nonequilibrium thermodynamics to go beyond local equilibrium treatments [4], the steady state thermodynamics (SST) proposed by Oono and Paniconi [5] seems to be most sophisticated and promising. The basic strategy of [5], in our own interpretation, is to i) look for a thermodynamics which describes a steady state as a whole, ii) clarify operational procedures for determining thermodynamic quantities, and iii) respect the general mathematical structure of thermodynamics. In the present Letter, we apply this strategy to a concrete problem of heat conduction (in a fluid), and show that theoretical consistency leads one to an essentially unique thermodynamics. We then make some concrete predictions which may be confirmed quantitatively in experiments. Extensions to other systems and discussions of related microscopic results will appear in [6].