I. INTRODUCTION

The fractional derivatives and integrals [1]- [4] are widely used to study anomalous diffusion in porous media, in disordered systems, in plasma physics, in turbulent, kinetic, and reaction-diffusion processes, etc. [5], [6]. In Ref. [5], [6], we discussed various approaches to obtaining the transport equations with fractional derivatives. It is important to note that, for the first time, in Refs. [7]- [10], Nigmatullin received diffusion equation with the fractional time derivatives for the mean spin density [7], the mean polarization [8], and the charge carrier concentration [9]. In Ref. [10], justification of equations with fractional derivatives is given, and the time irreversible Liouville equation with the fractional time derivative is provided. In our recent work [5], by using NSO method [11], [12] and the maximum entropy principle for the Renyi entropy, we obtained the generalized (non-Markovian) diffusion equation with fractional derivatives. The use of the Liouville equation with fractional derivatives proposed by Tarasov in Refs. [13], [14] is an important and fundamental step for obtaining this equation. By using NSO method and the maximum entropy principle for the Renyi entropy, we found a solution of the Liouville equation with fractional derivatives at a selected set of observed variables. We chose nonequilibrium average values of particle density as a parameter of reduced description, and then we received the generalized (non-Markovian) diffusion equation with fractional derivatives. In the next section by using Ref. [5], new non-Markovian diffusion equations for particles in a spatially heterogeneous environment with fractal structure are obtained. Different models of frequency-dependent memory functions are considered, and the diffusion equations with fractality of space-time are obtained.

II. LIOUVILLE EQUATION WITH FRACTIONAL DERIVATIVES FOR SYSTEM OF CLASSICAL PARTICLES

We use the Liouville equation with fractional derivatives obtained by Tarasov in Refs. [14] for a nonequilibrium particle function )

; ( t x N ρ of a classical system ( ) ( ) 0, = ) ; ( ) ; ( ) ; ( 1 = 1 = j N p N j j N r N j N F t x D v t x D t x t j j     ρ ρ ρ α α ∑ ∑ + + ∂ ∂ (1)

where se space with a fractional differential volume element [13],

N N x x x , , = 1  , } , { =

[15]

N x d x d V d α α α  1 = . Here, ) ( = 0 0 0 t p Mr m

, M is the mass of particle, 0 r is a characteristic scale in the configura- tion space, 0 p is a characteristic momentum, and 0

t is a cha- racteristic time, α d is a fractional differential [15], , ) )( ( = ) ( 2 1 = α α α j x N j dx x f D x f d j ∑ where dz z x z f n x f D n n x x − + − − Γ ∫ 1 ) ( 0 ) ( ) ( ) ( 1 = ) ( α α α (2)

is the Caputo fractional derivative, [1], [2], [16], [17] n n < <

1 α − , n n n dz z f d z f ) ( = ) ( ) (

with the properties 0 = 1

α j x D and 0 = l x x D j α , ) ( l j ≠ . j v 1 = t x D p r H D D p r H D t x iL N p r r p N j N j j j j ρ ρ α α α α α         − ∑ (4) where ) , ( p r H  

is a Hamiltonian of a system with fractional derivatives [13]. A solution of the Liouville equation (3) will be found with Zubarev`s NSO method [11]. After choosing parameters of the reduced description, taking into account projections we present the nonequilibrium particle function

′ ′ ′ − ′ − − ′ ∞ − ∫ ρ ρ ρ α ε (5)

where

      ′ ′ − − ′ ∫ ′ + t d iL t P t t T rel t t α )) ( (1 exp = ) , (

is the evolution operator in time containing the projection, exp + is ordered exponential, 0 + → ε after taking the thermodynamic limit, ) (t P rel ′ is the generalized Kawasaki-Gunton projection ope- rator depended on a structure of the relevant statistical operator (distribution function), )

; ( t x N rel ′ ρ

. By using Zubarev's NSO method [11], [12] and approach, ( )

t x N rel ′ ;

ρ will be found from the extremum of the Renyi entropy at fixed values of observed values ).

; ( ) , (1, ) , (1, = ) ( ˆt x P N T N I x P N n t n ρ α α   〉 〈 (6) ) , (1, ˆN I  α has the following form for a system of N par- ticles , ) ( , (1), = ) , (1, ˆN I I N I α α α   ) ( ) ( = ) ( ˆj j p I r I j I   α α α

and defines operation of integration . ) (

| | = ) ( ), ( ) ( = ) ( ) ( ˆdx x x d x d x f x f x I α µ µ α α α α Γ ∫ ∞ ∞ − (7) The operator ) ( , (1), = ) , (1, ˆN T T N T   defines the opera- tion . ) ) , , ) , , ( ( = ) ( ) ( ˆ2 1     j j j j j j x x f x x f x f x T + ′ + − ′

Accordingly, the average value, which is calculated with the relevant distribution function, is defined as

). ; ( ) )( , (1, ) , (1,= ) ( , t x N T N I N rel t rel ρ α α     〉 〈

According to Ref. [12], from the extremum of the Renyi entropy functional

) ( ) ( ) , (1, ) , (1, ) ; ( ) ( ) ( ) , (1, ) , (1, ˆ)) ( )( , (1, ) , (1, ln 1 1 ) ( t x P N T N I t x F x d t N T N I t N T N I q L n n n q R ρ µ ρ γ ρ ρ α α α α ′ − ′ − ′ − = ′ ∫ ∑       at fixed values of observed values t n x P α 〉 〈 ) ( ˆ and the condition of normalization 1 = ) ( ) , (1, ) , (1, ˆt N T N I ρ α ′   , the relevant distribution function takes the form , ) ; ( ) ; ( ) ( 1 1 ) ( 1 = ) ( 1 1 −                 − − − ∫ ∑ q n n n R rel t x P t x F x d H q q t Z t δ µ β ρ α (8) where ) (t Z R

is the partition function of the Renyi distribution, which is determined from the normalization condition and has the form

. ) ; ( ) ; ( ) ( 1 1 ) , (1, ) , (1, = ) ( 1 1 −                 − − − × ∫ ∑ q n n n R t x P t x F x d H q q N T N I t Z δ µ β α α  (9)

The Lagrangian multiplier γ is determined by the normaliza- tion condition. The parameters ) ; ( t x F n are determined from the self-consistency conditions .

) ( = ) ( ˆ, t rel n t n x P x P α α 〉 〈 〉 〈(10)

It is important to note that the relevant distribution function corresponded to the Gibbs entropy follows from (8) at 1 = q [5]. In the general case of the parameters

t n x P α 〉 〈 ) ( ˆ

of the reduced description of nonequilibrium processes according to (5)

′ ′ ′ ′ ′ + − ′ ∞ − ∫ ∫ ∑ β ρ µ ρ ρ ε α (11) where , ) ( ) ; ( ) (1) ; ( = ) ; ( 1 * t n n n q q n n x P t x F x d t x F t x F α α µ ′ + ′ ′ ∫ ∑ − ( ) ) ( ) ( ) ( 1 = ) ; ( 1 1 x P iL t t P t x I n q n α ψ − − ′ (12)

are the generalized flows, ) (t P is the Mori projection operator [5], and the function

) (t ψ has the following structure ). ( ) ; ( ) ( 1 = ) ( 1 x P t x F x d t n n n q q  α µ ψ ∫ ∑ − −′ ′ ′ ′ ′ ′ + 〉 〈 〉 〈 ∂ ∂ ′ − ′ ∞ − ′ ′ ∫ ∫ ∑ β ϕ µ ε α α α α(13) where (

)

) ; ( ) ; ( ) , ( ) ( ˆ) , (1, ) , (1,′ ′ ′ ′ × ′ ′ ′ ′ ρ ϕ α α  (14)

are the generalized transport kernels (the memory functions), which describe dissipative processes in the system. To demonstrate the structure of the transport equations ( 13) and the transport kernels ( 14), we will consider, for example, diffusion processes. In the next section, we obtain generalized transport equations with fractional derivatives and consider a concrete example of diffusion processes of the particle in non-homogeneous media.

III. GENERALIZED DIFFUSION EQUATIONS WITH FRACTIONAL DERIVATIVES

One of main parameters of the reduced description to describe the diffusion processes of the particles in non-homogeneous media with fractal structure is the nonequilibrium density of the particle numbers,

t n x P α 〉 〈 ) ( ˆ: t r n t r n α 〉 〈 ) ( = ) ; (   , where ) ( = ) ( ˆ1 = j N j r r r n    − ∑ δ′ ∂ ′ ∂ ⋅ ⋅ ∂ ∂ ∂ ∂ − ′ ∞ − ∫ ∫ α α ε α α α α βν µ        (15)

where

t rel q r v t t T r v t t r r D , ) ' ( ) , ( ) ( = ) , ; ' , ( α 〉 ′ 〈 ′       (16)

is the generalized coefficient diffusion of the particles within the Renyi statistics. Averaging in Eq. ( 16) is performed with the power-law Renyi distribution,

( ) , ) ( ) ; ( ) ( 1 1 ) ( 1 = ) ( 1 1 * −         − − − ∫ q R rel r n t r r d H q q t Z t    ν µ β ρ α (17)

where

(

)

1 1 * ) ( ) ; ( ) ( 1 1 ) , (1, ) , (1, = ) ( −         − − − × ∫ q R r n t r r d H q q N T N I t Z      ν µ β α α (18)

is the partition function of the relevant distribution function, H is a Hamiltonian of the system, q is the Renyi parameter

( 1 < 0 < q ). Parameter ) ; ( t r  ν is the chemical potential of the particles, which is determined from the self-consistency condition, . ) ( = ) ( ˆ, t rel t r n r n α α   (19) T k B 1/ = β ( B k is the Boltzmann constant), T is the equilib- rium value of temperature, ) ( = ) ( ˆ1 = j j N j r r v r v      − ∑ δ    − ′ − ≈ ′ δ δ

. And by excluding the parameter ) ; ' ( * t r ′  ν via the self-consistency condition, we obtain the diffusion equation with fractional derivatives from Eq. ( 15)

). ; ' ( = ) ( ˆ* 2 t r r D r n t q b t ′ ∂ ∂ ∂ ∂ ∑   ν α α α (20)

The generalized diffusion equation takes into account spatial fractality of the system and memory effects in the generalized coefficient of diffusion

) , ; ' , ( t t r r D q ′  

within the Renyi statistics. Obviously, spatial fractality of system influences on transport processes of the particles that can show up as multifractal time with characteristic relaxation times. It is known that the nonequilibrium correlation functions

) , ; ' , ( t t r r D q ′  

can not be exactly calculated, therefore the some approximations based on physical reasons are used. In the time interval t ÷ −∞ , ion transport processes in spatially non-homogeneous system can be characterized by a set of relaxation times that are associated with the nature of interaction between the particles and particles of media with fractal structure. To show the multifractal time in the generalized diffusion equation, we use the following approach for the generalized coefficient of particle diffusion ), , ( )

, ( = ) , ; , ( r r D t t W t t r r D q q     ′ ′ ′ ′(21)

where ) , ( t t W ′ can be defined as the time memory function. In view of this, Eq. ( 15) can be represented as

, ) ; ( ) , ( = ) ( ˆ) ( t d t r t t W e r n t t t t t ′ ′ Ψ ′ ∂ ∂ − ′ ∞ − ∫   ε α (22) where ). ; ( ) , ( ) ( = ) ; ( ' * ' t r r r r D r r d t r q ′ ∂ ∂ ⋅ ′ ⋅ ∂ ∂ ′ ′ Ψ ∫        βν µ α α α α α (23)

Further we apply the Fourier transform to Eq. ( 22), and as a result we get in frequency representation

). ;

( ) ( = ) ; ( ω ω ω ω r W r n i   Ψ (24)

We can represent the frequency dependence of the memory function in the following form 1, < 0 , 1

) ( = ) ( 1 ≤ + − ξ ωτ ω ω ξ i i W (25)

where the introduced relaxation time a τ characterizes the particles transport processes in the system. Then Eq. ( 24) can be represented as

t r r D D t r n t t r n t q t     βν τ α α ξ ∂ ∂ ∂ ∂ + ∂ ∂ − (31) At 1 = q

, we get the Cattaneo-type diffusion equation with fractality of space-time and the constant coefficients of the diffusion within the Gibbs statistics, ), ;

( = ) ; ( ) ; ( 2 2 1 0 2 2 t r r D D t r n t t r n t b t     ν τ α α ξ ∂ ∂ ∂ ∂ + ∂ ∂ ∑ − (32)

It should be noted that if we put 1 = α in Eqs. (31), (32), i.e. we neglect spatial fractality, we get the Cattaneo-type diffusion equations, which were obtained in Ref. [18],

). ;

( = ) ; ( ) ; ( 2 2 1 0 2 2 t r r D D t r n t t r n t t     ν τ ξ ∂ ∂ ∂ ∂ + ∂ ∂ −(

IV. CONCLUSION

We presented the general approach for obtaining the generalized transport equations with the fractional derivatives by using the Liouville equation with the fractional derivatives [14] for a system of classical particles and Zubarev's NSO method within the Renyi statistics [5]. In this approach, the new non-Markov equations of diffusion of the particles in a spatially non-homogeneous medium with a fractal structure are obtained.

By using approaches for the memory functions and fractional calculus [1]- [5], the generalized Cattaneo-type diffusion equations with taking into account fractality of space-time are obtained. It is considered the different models for the frequency dependent memory functions, which lead to the known diffusion equations with the fractality of space-time and their generalizations.