75 Generalized Transport Equation with Fractality of Space-Time. Zubarev’s NSO Method Petro Kostrobij1, Bogdan Markovych1, Olexandra Viznovych1, Mykhailo Tokarchuk1,2 1. Lviv Polytechnic National University, 12 S. Bandera str., 79013, Lviv, Ukraine, email: bohdan.m.markovych@lpnu.ua 2. Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii str., 79011, Lviv, Ukraine  Abstract: We presented a general approach for obtai- ∂ (  ) ( ) N N ning the generalized transport equations with fractional ρ ( x N ; t ) + ∑Drα ρ ( x N ; t )v j + ∑Dαp ρ ( x N ; t ) F j = 0, (1) ∂t j =1 j j =1 j derivatives by using the Liouville equation with fractional   derivatives for a system of classical particles and the Zu- where x = x1 , , x N , x j = {r j , p j } are dimensionless gene- N barev nonequilibrium statistical operator (NSO) method  ralized coordinates, r j = (r j1 , , r jm ) , and generalized mo- within Renyi statistics. Generalized Cattaneo-type diffu-  sion equations with taking into account fractality of spa- mentum, p j = ( p j1 , , p jm ) , [14] of j th particle in the pha- ce-time are obtained. se space with a fractional differential volume element [13], Keywords: fractional derivative, diffusion equation. [15] d αV = d α x1  d α x N . Here, m = Mr0 ( p0t0 ) , M is the I. INTRODUCTION mass of particle, r0 is a characteristic scale in the configura- The fractional derivatives and integrals [1]–[4] are widely tion space, p0 is a characteristic momentum, and t0 is a cha- used to study anomalous diffusion in porous media, in disor- racteristic time, d α is a fractional differential [15], dered systems, in plasma physics, in turbulent, kinetic, and 2N reaction-diffusion processes, etc. [5], [6]. In Ref. [5], [6], we discussed various approaches to obtaining the transport equ- d α f ( x) = ∑D f ( x)(dx ) , j =1 α xj j α ations with fractional derivatives. It is important to note that, where for the first time, in Refs. [7]–[10], Nigmatullin received dif- 1 x f (n) ( z) fusion equation with the fractional time derivatives for the mean spin density [7], the mean polarization [8], and the Dxα f ( x) = ∫ Γ(n − α ) 0 ( x − z )α +1− n dz (2) charge carrier concentration [9]. In Ref. [10], justification of is the Caputo fractional derivative, [1], [2], [16], [17] equations with fractional derivatives is given, and the time ir- n − 1 < α < n , f ( n ) ( z ) = d n f ( z ) dz n with the properties reversible Liouville equation with the fractional time derivati-  ve is provided. In our recent work [5], by using NSO method Dxαj 1 = 0 and Dxαj xl = 0 , ( j ≠ l ) . v j are the fields of veloci-   [11], [12] and the maximum entropy principle for the Renyi ty, F j is the force field acting on j th particle. If F j does not entropy, we obtained the generalized (non-Markovian) diffu-    sion equation with fractional derivatives. The use of the Liou- depend on p j , v j does not depend on r j , and the Helmholtz ville equation with fractional derivatives proposed by Tarasov conditions, we get the Liouville equation in the form in Refs. [13], [14] is an important and fundamental step for ∂ ρ ( x N ; t ) + iLα ρ ( x N ; t ) = 0, (3) obtaining this equation. By using NSO method and the maxi- ∂t mum entropy principle for the Renyi entropy, we found a so- where iLα is the Liouville operator with the fractional deri- lution of the Liouville equation with fractional derivatives at vatives, a selected set of observed variables. We chose nonequilibri- [     ] N um average values of particle density as a parameter of re- iLα ρ ( x N ; t ) = ∑ Dαp H (r , p ) Drα − Drα H (r , p ) Dαp ρ ( x N ; t ). (4) duced description, and then we received the generalized (non- j =1 j j j j Markovian) diffusion equation with fractional derivatives. In   where H (r , p ) is a Hamiltonian of a system with fractional the next section by using Ref. [5], new non-Markovian diffu- sion equations for particles in a spatially heterogeneous envi- derivatives [13]. A solution of the Liouville equation (3) will ronment with fractal structure are obtained. Different models be found with Zubarev`s NSO method [11]. After choosing of frequency-dependent memory functions are considered, parameters of the reduced description, taking into account projections we present the nonequilibrium particle function ( ) and the diffusion equations with fractality of space-time are obtained. ρ x N ; t (as a solution of the Liouville equation) in the gene- ral form II. LIOUVILLE EQUATION WITH FRACTIONAL DERIVATIVES FOR SYSTEM OF CLASSICAL PARTICLES ( ) ρ ( x N ; t ) = ρ rel x N ; t (5) ∫ t We use the Liouville equation with fractional derivatives − eε (t ′−t )T (t , t ′)(1 − Prel (t ′))iLα ρ rel ( x N ; t ′)dt ′, −∞ obtained by Tarasov in Refs. [14] for a nonequilibrium partic- le function ρ ( x N ; t ) of a classical system ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 76 where T (t , t ′) = exp+ − ∫ (1 − Prel (t ′))iLα dt ′ is the evolution t It is important to note that the relevant distribution function  t ′  corresponded to the Gibbs entropy follows from (8) at q = 1 operator in time containing the projection, exp+ is ordered [5]. In the general case of the parameters 〈 Pˆ ( x)〉 t of the re- n α exponential, ε → +0 after taking the thermodynamic limit, duced description of nonequilibrium processes according to Prel (t ′) is the generalized Kawasaki-Gunton projection ope- (5) and (8), we get NSO in the form rator depended on a structure of the relevant statistical opera- ρ (t ) = ρ rel (t ) tor (distribution function), ρ rel ( x N ; t ′) . By using Zubarev’s t (11) NSO method [11], [12] and approach, ρ rel x ; t ′ will be ( N ) + ∑∫ ∫ dµα ( x) eε (t ′−t )T (t , t ′) I n ( x; t ′) ρ rel (t ′) βFn* ( x; t ′)dt ′, n −∞ found from the extremum of the Renyi entropy at fixed valu- Fn ( x; t ′) where F ( x; t ′) = * , es of observed values 〈 Pˆn ( x)〉 αt , taking into account the nor- ∑∫dµ ( x) F ( x; t ′) P ( x) n q −1 t 1+ q α n n α malization condition 〈1〉 αt , rel = 1 , where the nonequilibrium n average values are found respectively [5], I n ( x; t ′) = (1 − P(t ) ) 1q ψ −1 (t )iLα Pˆn ( x) (12) 〈 Pˆn ( x)〉 αt = Iˆα (1, , N )Tˆ (1, , N ) Pˆn ρ ( x N ; t ). (6) are the generalized flows, P(t ) is the Mori projection opera- ˆI α (1, , N ) has the following form for a system of N par- tor [5], and the function ψ (t ) has the following structure   ticles Iˆα (1, , N ) = Iˆα (1), , Iˆα ( N ), Iˆα ( j ) = Iˆα (r j ) Iˆα ( p j ) ψ (t ) = 1 − qq−1 ∑ ∫dµα ( x) Fn ( x; t) Pn ( x). n and defines operation of integration By using the nonequilibrium statistical operator (11), we ∞ | x |α get the generalized transport equation for the parameters Iˆα ( x) f ( x) = ∫ −∞ f ( x)dµα ( x), dµα ( x) = Γ(α ) dx. (7) 〈 Pˆn ( x)〉 αt of the reduced description, The operator Tˆ (1, , N ) = Tˆ (1), , Tˆ ( N ) defines the opera- ∂ ˆ 〈 Pn ( x)〉 αt = 〈iLα Pˆn ( x)〉 αt , rel tion Tˆ ( x ) f ( x ) = 1 ( f ( , x′ − x ,) + f ( , x′ + x ,)). ∂t j j 2 j j j j (13) t Accordingly, the average value, which is calculated with the + ∑ ∫dµα ( x′) ∫ eε (t ′−t )ϕ P P ( x, x′; t , t ′) βFn*′ ( x′; t ′)dt ′, n n′ relevant distribution function, is defined as n′ −∞ 〈 ()〉 αt ,rel = Iˆα (1, , N )Tˆ (1, , N )() ρ rel ( x N ; t ). where According to Ref. [12], from the extremum of the Renyi ϕ P P ( x, x′; t , t ′) = Iˆα (1, , N )Tˆ (1, , N ) n n′ ( ) entropy functional (14) 1 × iLα Pˆn ( x)T (t , t ′) I n′ ( x′; t ′) ρ rel ( x N ; t ′) LR ( ρ ′) = ln Iˆα (1, , N )Tˆ (1, , N )( ρ ′(t )) q 1− q are the generalized transport kernels (the memory functions), which describe dissipative processes in the system. To de- − γIˆα (1, , N )Tˆ (1, , N ) ρ ′(t ) monstrate the structure of the transport equations (13) and the − ∑∫dµ ( x) F ( x; t ) Iˆ (1,, N )Tˆ (1,, N ) Pˆ ( x) ρ ′(t ) n α n α n transport kernels (14), we will consider, for example, diffu- sion processes. In the next section, we obtain generalized transport equations with fractional derivatives and consider a at fixed values of observed values 〈 Pˆn ( x)〉 αt and the condition concrete example of diffusion processes of the particle in of normalization Iˆα (1, , N )Tˆ (1, , N ) ρ ′(t ) = 1 , the relevant non-homogeneous media. distribution function takes the form 1 III. GENERALIZED DIFFUSION EQUATIONS WITH 1  q −1   q −1 FRACTIONAL DERIVATIVES ρ rel (t ) = 1 − Z R (t )  q  n ∑∫ β  H − dµα ( x) Fn ( x; t )δPˆn ( x; t )   , (8) One of main parameters of the reduced description to des- cribe the diffusion processes of the particles in non-homo- where Z R (t ) is the partition function of the Renyi distribu- geneous media with fractal structure is the nonequilibrium tion, which is determined from the normalization condition   and has the form density of the particle numbers, 〈 Pˆn ( x)〉 αt : n(r ; t ) = 〈 nˆ (r )〉 αt ,    where nˆ (r ) = ∑ j =1δ (r − r j ) is the microscopic density of the N Z R (t ) = Iˆα (1, , N )Tˆ (1, , N ) 1 particles. The corresponding generalized diffusion equation  q −1   q −1 (9)  × 1 − β  H − ∑∫ dµα ( x) Fn ( x; t )δPˆn ( x; t )  . for n(r ; t ) can be obtained on base of Eqs. (8), (11), (13),  q  n   t  ∂ nˆ (r ) α ∂ α  t  ∂ α βν * (r ' ; t ′) The Lagrangian multiplier γ is determined by the normaliza- ∂t ∂r −∞ ∫ ∫ =  α ⋅ dµα (r ' ) eε (t ′−t ) Dq (r , r ' ; t , t ' ) ⋅ ∂r  'α dt ′, (15) tion condition. The parameters Fn ( x; t ) are determined from the self-consistency conditions where       〈 Pˆn ( x)〉 αt = 〈 Pˆn ( x)〉 αt , rel . (10) Dq (r , r ' ; t , t ′) = 〈vˆ (r )T (t , t ′)vˆ (r ' )〉 αt ,rel (16) ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 77 is the generalized coefficient diffusion of the particles within ∂  t t  the Renyi statistics. Averaging in Eq. (16) is performed with ∂t −∞ ∫ nˆ (r ) α = eε (t ′−t )W (t , t ′)Ψ (r ; t ′)dt ′, (22) the power-law Renyi distribution, 1 where ρ rel (t ) = 1  q −1 1 − Z R (t )  (   )   q −1 β H − ∫dµα (r )ν * (r ; t )nˆ (r )  , (17)  ∫  ∂α   ∂α  Ψ (r ; t ′) = dµα (r ′)  α ⋅ D q (r , r ′) ⋅  'α βν * (r ' ; t ′). (23) q  ∂r ∂r where Further we apply the Fourier transform to Eq. (22), and as Z (t ) = Iˆα (1, , N )Tˆ (1, , N ) R a result we get in frequency representation   1 iωn(r ; ω ) = W (ω )Ψ (r ; ω ). (24) ( ∫ )  q −1 (18)     q −1 We can represent the frequency dependence of the memory × 1 − β H − dµα (r )ν * (r ; t )nˆ (r )   q  function in the following form is the partition function of the relevant distribution function, (iω )1−ξ W (ω ) = , 0 < ξ ≤ 1, (25) H is a Hamiltonian of the system, q is the Renyi parameter 1 + iωτ ( 0 < q < 1 ). where the introduced relaxation time τ a characterizes the  Parameter ν (r ; t ) is the chemical potential of the particles, particles transport processes in the system. Then Eq. (24) can which is determined from the self-consistency condition, be represented as    t  t (1 + iωτ )iωn(r ; ω ) = (iω )1−ξ Ψ (r ; ω ). (26) nˆ (r ) = nˆ (r ) α . α , rel (19) Further we use the Fourier transform to fractional derivati- β = 1/k BT ( k B is the Boltzmann constant), T is the equilib- ves of functions,   rium value of temperature, vˆ (r ) = N  j =1 j   ∑ v δ (r − r ) is the mic- j ( ) L 0 Dt1−ξ f (t ); iω = (iω )1−ξ L( f (t ); iω ). (27) By using it, the inverse transformation of Eq. (26) to time re- roscopic flux density of the particles. At q = 1 , the generali- presentation gives the Cattaneo-type generalized diffusion zed diffusion equation within the Renyi statistics goes into equation with taking into account spatial fractality, the generalized diffusion equation within the Gibbs statistics ∂2  ∂   ∂1−ξ  with fractional derivatives. If q = 1 and α = 1 , we obtain the τ 2 n(r ; t ) + n(r ; t ) = 0 Dt1−ξ Ψ (r ; t ) = 1−ξ Ψ (r ; t ), (28) ∂t ∂t ∂t generalized diffusion equation within the Gibbs statistics. In which is the new Cattaneo-type generalized equation within the Markov approximation, the generalized coefficient of the Renyi statistics with multifractal time and spatial frac- diffusion in time and space has the form     tality. At q = 1 from Eq. (29), we get the Cattaneo-type gene- ′ ′ Dq (r , r ' ; t , t ) ≈ Dqδ (t − t )δ (r − r ' ) . And by excluding the pa-  ralized equation within the Gibbs statistics with multifractal rameter ν * (r ' ; t ′) via the self-consistency condition, we ob- time and spatial fractality, tain the diffusion equation with fractional derivatives from ∂2  ∂   ∂α   Eq. (15) τ 2 n(r ; t ) + n(r ; t ) = 0 Dt1−ξ ∫dµα (r ′)  α ⋅ D(r , r ′) ∂t ∂t ∂r ∂  t ∂ 2α  (29) ∂t ∑ nˆ (r ) α = Dq 2α ν * (r ' ; t ′). ∂ r (20) ∂α  ⋅  'α βν (r ' ; t ), b ∂r The generalized diffusion equation takes into account spa- Eqs. (28), (29) contain significant spatial non-homogeneity tial fractality of the system and memory effects in the genera-     in D q (r , r ′) . If we neglect spatial non-homogeneity, lized coefficient of diffusion Dq (r , r ' ; t , t ′) within the Renyi     D q (r , r ′) = D qδ (r − r ′), (30) statistics. Obviously, spatial fractality of system influences on transport processes of the particles that can show up as mul- we get the Cattaneo-type diffusion equation with fractality of tifractal time with characteristic relaxation times. It is known space-time and the constant coefficients of the diffusion   that the nonequilibrium correlation functions Dq (r , r ' ; t , t ′) within the Renyi statistics, ∂2  ∂  ∂ 2α  can not be exactly calculated, therefore the some approxima- τ 2 n(r ; t ) + n(r ; t ) = 0 Dt1−ξ D q  2α βν * (r ; t ), (31) tions based on physical reasons are used. In the time interval ∂t ∂t ∂r −∞ ÷ t , ion transport processes in spatially non-homogeneous At q = 1 , we get the Cattaneo-type diffusion equation with system can be characterized by a set of relaxation times that fractality of space-time and the constant coefficients of the are associated with the nature of interaction between the par- diffusion within the Gibbs statistics, ticles and particles of media with fractal structure. To show ∂2  ∂  ∂ 2α  the multifractal time in the generalized diffusion equation, we τ 2 n(r ; t ) + n(r ; t ) = 0 Dt1−ξ ∑ D  2α ν (r ; t ), (32) ∂t ∂t b ∂r use the following approach for the generalized coefficient of particle diffusion It should be noted that if we put α = 1 in Eqs. (31), (32), i.e.     we neglect spatial fractality, we get the Cattaneo-type diffusi- Dq (r , r ′; t , t ′) = W (t , t ′) D q (r , r ′), (21) on equations, which were obtained in Ref. [18], where W (t , t ′) can be defined as the time memory function. ∂2  ∂  ∂2  In view of this, Eq. (15) can be represented as τ 2 n(r ; t ) + n(r ; t ) = 0 Dt1−ξ D  2 ν (r ; t ). (33) ∂t ∂t ∂r ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 78 At τ = 0 , we get an important particular case — the gene- Applications. Gordon and Breach Science Publishers, ralized diffusion equation of particles with taking into ac- 1st ed., 1993. count fractality of space-time, [3] I. Podlubny and V. T. E. Kenneth, Fractional ∂   ∂α   ∂α  Differential Equations: An Introduction to Fractional ∂t ∫ n(r ; t ) = 0 Dt1−ξ dµα (r ′) α ⋅ D q (r , r ′) ⋅  'α βν * (r ' ; t ), (34) ∂r ∂r Derivatives, Fractional Differential Equations, to and by neglecting spatial non-homogeneity of the diffusion Methods of Their Solution and Some of Their   Applications. Academic Press, 1st ed., 1998. coefficients D q (r , r ′) , we also get the diffusion equation [4] V. V. 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