HYDRODYNAMIC MODES AS SINGULAR EIGENSTATES OF THE LIOUVILLIAN DYNAMICS - DETERMINISTIC DIFFUSION

Hydrodynamic modes of diffusion and the corresponding nonequilibrium s teady states are studied as an eigenvalue problem for the Liouvillian dynamics of spatially extended suspension flows which are special cont inuous-time dynamical systems including billiards defined on the basis of a mapping. The infinite spatial extension is taken into account by spatial Fourier transforms which decompose the observables and probab ility densities into sectors corresponding to the different values of the wave number The Frobenius-Perron operator ruling the time evolutio n in each wave number sector is reduced to a Frobenius-Perron operator associated with the mapping of the suspension flow. In this theory, t he dispersion relation of diffusion is given as a Pollicott-Ruelle res onance of the Frobenius-Perron operator and the corresponding eigensta tes are studied. Formulas are derived for the diffusion and the Burnet t coefficients in terms of the mapping of the suspension flow. Nonequi librium steady states are constructed on the basis of the eigenstates and are given by mathematical distributions without density functions, also referred to as singular measures. The nonequilibrium steady stat es are shown to obey Fick's law and to be related to Zubarev's local i ntegrals of motion. The theory is applied to the regular Lorentz gas w ith a finite horizon. Generalizations to the nonequilibrium steady sta tes associated with the other transport processes are also obtained.