Abstract
We presented a general approach for obtaining the generalized transport equations with fractional derivatives by using the Liouville equation with fractional derivatives for a system of classical particles and Zubarev's nonequilibrium statistical operator (NSO) method within Gibbs statistics. The new non-Markovian diffusion equations of ions in spatially heterogeneous environment with fractal structure and generalized Cattaneo-Maxwell diffusion equation with taking into account the space-time nonlocality are obtained. Dispersion relations are found for the Cattaneo-Maxwell diffusion equation with taking into account the space-time nonlocality in fractional derivatives. The frequency spectrum, phase and group velocities are calculated. It is shown that it has a wave behaviour with discontinuities, which are also manifested in the behaviour of the phase velocity.
Highlights
Studies of nonequilibrium processes with spatio-temporal nonlocality are relevant in the statistical physics of soft matter.One of the important problems in the theory of nonequilibrium processes of interacting particles is the calculation of memory functions in transport equations in a wide region of spatiotemporal dependence, including the region of anomalous behaviour, in particular, sub, superdiffusions, which are experimentally realized in condensed systems.Mathematical modelling of diffusion transfer processes in porous and complex nano-structured systems requires the use of transfer equations with significant spatial inhomogeneity and temporal memory
In our works [4, 6,7,8,9,10,11,12,13,14,15], a statistical approach to obtain generalized spatio-temporal nonlocal transfer equations was developed by using the Zubarev nonequilibrium statistical operator (NSO) method [16,17,18] and the Liouville equation with fractional derivatives [19, 20]
We present a statistical approach to construct generalized transfer equations using the NSO method and the Liouville equation with fractional derivatives
Summary
Studies of nonequilibrium processes with spatio-temporal nonlocality are relevant in the statistical physics of soft matter. In our works [4, 6,7,8,9,10,11,12,13,14,15], a statistical approach to obtain generalized spatio-temporal nonlocal transfer equations was developed by using the Zubarev nonequilibrium statistical operator (NSO) method [16,17,18] and the Liouville equation with fractional derivatives [19, 20]. We present a statistical approach to construct generalized transfer equations using the NSO method and the Liouville equation with fractional derivatives. After choosing parameters of the reduced description, taking into account the projections, we present the nonequilibrium particle function ρ(xN ; t) (as a solution of the Liouville equation) in the general form t.
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