Hydrodynamic modes as singular eigenstates of the Liouvillian dynamics: Deterministic diffusion

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Abstract
Hydrodynamic modes of diffusion and the corresponding nonequilibrium steady states are studied as an eigenvalue problem for the Liouvillian dynamics of spatially extended suspension flows which are special continuous-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 probability densities into sectors corresponding to the different values of the wave number. The Frobenius-Perron operator ruling the time evolution in each wave number sector is reduced to a Frobenius-Perron operator associated with the mapping of the suspension flow. In this theory, the dispersion relation of diffusion is given as a Pollicott-Ruelle resonance of the Frobenius-Perron operator and the corresponding eigenstates are studied. Formulas are derived for the diffusion and the Burnett coefficients in terms of the mapping of the suspension flow. Nonequilibrium 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 states are shown to obey Fick's law and to be related to Zubarev's local integrals of motion. The theory is applied to the regular Lorentz gas with a finite horizon. Generalizations to the nonequilibrium steady states associated with the other transport processes are also obtained.

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