Generalized Transport Equation with Fractality of Space-Time. Zubarev’s NSO Method
Petro Kostrobij
1, Bogdan Markovych
1, Olexandra Viznovych
1, Mykhailo Tokarchuk
1,21. 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- ning the generalized transport equations with fractional derivatives by using the Liouville equation with fractional derivatives for a system of classical particles and the Zu- barev nonequilibrium statistical operator (NSO) method within Renyi statistics. Generalized Cattaneo-type diffu- sion equations with taking into account fractality of spa- ce-time are obtained.
Keywords: fractional derivative, diffusion equation.
I. I
NTRODUCTIONThe fractional derivatives and integrals [1]–[4] are widely used to study anomalous diffusion in porous media, in disor- dered 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 equ- ations with fractional derivatives. It is important to note that, for the first time, in Refs. [7]–[10], Nigmatullin received dif- fusion 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 ir- reversible Liouville equation with the fractional time derivati- ve 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) diffu- sion equation with fractional derivatives. The use of the Liou- ville 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 maxi- mum entropy principle for the Renyi entropy, we found a so- lution of the Liouville equation with fractional derivatives at a selected set of observed variables. We chose nonequilibri- um average values of particle density as a parameter of re- duced 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 diffu- sion equations for particles in a spatially heterogeneous envi- ronment 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. L
IOUVILLEE
QUATION WITHF
RACTIONALD
ERIVATIVES FORS
YSTEM OFC
LASSICALP
ARTICLES We use the Liouville equation with fractional derivatives obtained by Tarasov in Refs. [14] for a nonequilibrium partic- le function ρ(xN;t) of a classical system(
( ; )) (
( ; ))
=0,)
; (
1
= 1
=
j N p N
j j N r N
j
N t D x t v D x t F
t x j j
ρ ρ
ρ +
∑
α +∑
α∂
∂ (1)
where xN=x1,,xN, xj ={rj,pj} are dimensionless gene- ralized coordinates, rj=(rj1,,rjm), and generalized mo- mentum, pj=(pj1,,pjm), [14] of jth particle in the pha- se space with a fractional differential volume element [13], [15] dαV =dαx1dαxN. Here, m=Mr0 (p0t0), M is the mass of particle, r0 is a characteristic scale in the configura- tion space, p0 is a characteristic momentum, and t0 is a cha- racteristic time, dαis a fractional differential [15],
, ) )(
(
= ) (
2
1
=
α α
α
j x
N
j
dx x f D x f
d
∑
jwhere
z dz x
z f x n
f
D n
x n
x Γ −
∫
0 ( −()) +1−) ( )
(
= 1 )
( α
α
α (2)
is the Caputo fractional derivative, [1], [2], [16], [17]
n
n−1<α < , f(n)(z)=dnf(z) dzn with the properties 0
=
α1
xj
D and Dxαjxl =0, (j≠l). vj are the fields of veloci- ty, Fj
is the force field acting on jth particle. If Fj
does not depend on pj, vj does not depend on rj, and the Helmholtz conditions, we get the Liouville equation in the form
0,
= )
; ( )
;
(x t iL x t t
N
N ρ
ρ + α
∂
∂ (3)
where iLα is the Liouville operator with the fractional deri- vatives,
[
( , ) ( , )]
( ; ).= )
; (
1
=
t x D p r H D D p r H D t
x
iL p r r p N
N
j N
j j
j
j ρ
ρ α α α α
α
∑
− (4)where H(r,p) 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
( )
xN;tρ (as a solution of the Liouville equation) in the gene- ral form
( )
, )
; ( )) ( )(1 , (
;
= )
; (
)
( T tt P t iL x t dt
e t x t x
N rel rel
t t t
N rel N
′
′
− ′
− ′− ′
∞
∫
− ρρ ρ
ε α (5)
75
ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic
where
− − ′ ′
′ +
∫
′ P t iL dtt t
T t rel
t(1 ( )) α
=exp ) ,
( is the evolution
operator in time containing the projection, exp+ is ordered exponential, ε→+0 after taking the thermodynamic limit,
) (t
Prel ′ is the generalized Kawasaki-Gunton projection ope- rator depended on a structure of the relevant statistical opera- tor (distribution function), ρrel(xN;t′). By using Zubarev’s NSO method [11], [12] and approach, ρrel
( )
xN;t′ will be found from the extremum of the Renyi entropy at fixed valu- es of observed values 〈Pˆn(x)〉tα, taking into account the nor- malization condition 1 , =1t αrel
〉
〈 , where the nonequilibrium average values are found respectively [5],
).
; ˆ ( ) , ˆ(1, ) , ˆ (1,
= )
ˆ(x I N T N P x t
Pn 〉αt α nρ N
〈 (6)
) , ˆ (1, N
Iα has the following form for a system of N par- ticles Iˆα(1,,N)=Iˆα(1),,Iˆα(N), ˆ ( )= ˆ ( )ˆ ( )
j
j I p
r I j
Iα α α and defines operation of integration
) . (
|
= | ) ( ), ( ) (
= ) ( )
ˆ ( x dx
x d x d x f x f x
Iα µα µα αα
∫
−∞∞ Γ (7)The operator Tˆ(1,,N)=Tˆ(1),,Tˆ(N) defines the opera-
tion ˆ( ) ( )= ( ( , , ) ( , , )).
2
1 j j j j
j
j f x f x x f x x
x
T ′− + ′+
Accordingly, the average value, which is calculated with the relevant distribution function, is defined as
).
; ( ) )(
, ˆ(1, ) , ˆ (1,
= )
(〉tα,rel Iα N T N ρrel xN t
〈
According to Ref. [12], from the extremum of the Renyi entropy functional
) ( ) ˆ( ) , ˆ(1, ) , ˆ (1, )
; ( ) (
) ( ) , ˆ(1, ) , ˆ (1,
)) ( )(
, ˆ(1, ) , ˆ (1, 1 ln
) 1 (
t x P N T
N I
t x F x d
t N T N I
t N T N q I
L
n n
n
q R
ρ µ
ρ γ
ρ ρ
α α α
α
′
−
− ′
− ′
′ =
∑ ∫
at fixed values of observed values 〈Pˆn(x)〉tα and the condition of normalization Iˆα(1,,N)Tˆ(1,,N)ρ′(t)=1, the relevant distribution function takes the form
, )
; ˆ( )
; ( ) 1 (
) 1 (
= 1 ) (
1 1
−
−
− −
∑ ∫
n n qn R
rel H d xF xt P xt
q q t
t Z β µ δ
ρ α (8)
where ZR(t) is the partition function of the Renyi distribu- tion, which is determined from the normalization condition and has the form
. )
; ˆ( )
; ( ) 1 (
1
) , ˆ(1, ) , ˆ (1,
= ) (
1 1
−
−
− −
×
∑ ∫
n n qn R
t x P t x F x d q H
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 Fn(x;t) are determined from the self-consistency conditions
. ) ˆ(
= ) 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 q=1 [5]. In the general case of the parameters 〈Pˆn(x)〉αt of the re- duced description of nonequilibrium processes according to (5) and (8), we get NSO in the form
, )
; ( ) ( )
; ( ) , ( )
( ) (
= ) (
* )
( T t t I xt t F xt dt
e x d
t t
n rel n
t t t
n rel
′
′
′
′
+ ′− ′
∞
−
∫
∑ ∫
µ ρ βρ ρ
α ε
(11)
where ,
) ( )
; ( ) ( 1
)
;
= ( )
;
( 1
*
t n n
n q q
n
n d x F xt P x
t x t F
x F
α α
µ ′
+
′ ′
∑ ∫
−
(
1 ())
() ˆ( )= )
;
(xt Pt 1 1 t iL P x
In ′ − qψ− α n (12) are the generalized flows, P(t) is the Mori projection opera- tor [5], and the function ψ(t) has the following structure
).
( )
; ( ) ( 1
= )
(t 1 d x Fn xt Pn x
n q
q µα
ψ − −
∑ ∫
By using the nonequilibrium statistical operator (11), we get the generalized transport equation for the parameters
t
n x
P 〉α
〈ˆ( ) of the reduced description,
, )
; ( ) ,
; , ( )
( ) ˆ (
= ) ˆ (
* )
( ,
t d t x F t t x x e
x d
x P iL x
t P
n P
P t t t
n
t rel n t
n
n
n ′ ′ ′ ′ ′
+ ′
〉
〈
〉
∂ 〈
∂
− ′
′
∞
′
∫
−∫
′∑
µα ε ϕ βα α α
(13)
where
(
ˆ( ) (, ) ( ; )(1, (, ;)))
)ˆ , ˆ (1,
= ) ,
; , (
t x t x I t t T x P iL
N T
N I
t t x x
N rel n
n P
Pnn
′
′
′
× ′
′
′
′
′
ρ ϕ
α
α
(14) are the generalized transport kernels (the memory functions), which describe dissipative processes in the system. To de- monstrate the structure of the transport equations (13) and the 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 concrete example of diffusion processes of the particle in non-homogeneous media.
III. G
ENERALIZEDD
IFFUSIONE
QUATIONS WITHF
RACTIONALD
ERIVATIVESOne of main parameters of the reduced description to des- cribe the diffusion processes of the particles in non-homo- geneous media with fractal structure is the nonequilibrium density of the particle numbers, 〈Pˆn(x)〉αt : n(r;t)=〈nˆ(r)〉tα, where ˆ( )= ( )
1
= j
N
j r r
r
n
∑
δ − is the microscopic density of the particles. The corresponding generalized diffusion equation for n(r;t) can be obtained on base of Eqs. (8), (11), (13),) ,
;' ) (
,
; , ( )
( ) =
ˆ(
'
* ' ' ) (
' dt
r t t r
t r r D e r r d
t r n
q t t t t
∂ ′
′
⋅∂
∂ ⋅
∂
∂
∂ ′−
∞
−
∫
∫
α ε α αα α
α µ βν
(15) where
t rel
q r r t t v r T t t v r
D (,'; , ′)=〈ˆ() (, ′)ˆ(')〉α,
(16)
76
ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic
is the generalized coefficient diffusion of the particles within the Renyi statistics. Averaging in Eq. (16) is performed with the power-law Renyi distribution,
(
( ) ( ; )ˆ( ))
,1 1 ) (
= 1 ) (
1 1
* −
− −
−
∫
qR
rel H d r r t nr
q q t
t Z β µ ν
ρ α (17)
where
(
( ) *( ; )ˆ( ))
111 1
) , ˆ(1, ) , ˆ (1,
= ) (
−
− − −
×
∫
qR
r n t r r d q H
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 (0<q<1).
Parameter ν(r;t) is the chemical potential of the particles, which is determined from the self-consistency condition,
. ) ˆ(
= ) ˆ(
, t
rel
t nr
r
n α α (19) T
kB 1/
β = (kB is the Boltzmann constant), T is the equilib- rium value of temperature, ˆ( )= ( )
1
= j j
N
j v r r
r
v
∑
δ − is the mic- roscopic flux density of the particles. At q=1, the generali- zed diffusion equation within the Renyi statistics goes into the generalized diffusion equation within the Gibbs statistics with fractional derivatives. If q=1 and α=1, we obtain the generalized diffusion equation within the Gibbs statistics. In the Markov approximation, the generalized coefficient of diffusion in time and space has the form) ' ( ) ( ) ,
; ' ,
(r r t t D t t r r
Dq ′ ≈ qδ − ′δ − . And by excluding the pa- rameter ν*(r';t′) via the self-consistency condition, we ob- tain the diffusion equation with fractional derivatives from Eq. (15)
).
; ' (
= )
ˆ( *
2 2
t r r
D r
t n b q
t ′
∂
∂
∂
∂
∑
αν α
α (20)
The generalized diffusion equation takes into account spa- tial fractality of the system and memory effects in the genera- lized coefficient of diffusion Dq(r,r';t,t′) within the Renyi statistics. Obviously, spatial fractality of system influences on transport processes of the particles that can show up as mul- tifractal time with characteristic relaxation times. It is known that the nonequilibrium correlation functions Dq(r,r';t,t′) can not be exactly calculated, therefore the some approxima- tions 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 par- ticles 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 t t W t t D r r
Dq q
′ ′ ′ ′ (21) where W(t,t′) can be defined as the time memory function.
In view of this, Eq. (15) can be represented as
, )
; ( ) , (
= )
ˆ(r e ( )W t t r t dt t n
t t t
t ′Ψ ′ ′
∂
∂ ′−
∞
−
∫
ε
α (22)
where
).
; ( )
, ( )
(
= )
;
( ' * r' t
r r r r D r d t
r q ′
∂
⋅ ∂
⋅ ′
∂
′ ∂
Ψ ′
∫
µα αα αα βν (23)Further we apply the Fourier transform to Eq. (22), and as a result we get in frequency representation
).
; ( ) (
= )
;
( ω ω ω
ωnr W r
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
W i (25)
where the introduced relaxation time τa characterizes the particles transport processes in the system. Then Eq. (24) can be represented as
).
; ( ) (
= )
; ( )
(1+iωτ iωnrω iω 1−ξΨ rω (26) Further we use the Fourier transform to fractional derivati- ves of functions,
(
0D1ξf(t);iω)
=(iω)1ξL(f(t);iω).L t− − (27)
By using it, the inverse transformation of Eq. (26) to time re- presentation gives the Cattaneo-type generalized diffusion equation with taking into account spatial fractality,
),
; (
= )
; (
= )
; ( )
;
( 1
1 1
2 0 2
t t r
t r D t r tn t r
t n t
Ψ
∂ Ψ ∂
∂ + ∂
∂
∂
−
− −
ξ ξ ξ
τ (28)
which is the new Cattaneo-type generalized equation within the Renyi statistics with multifractal time and spatial frac- tality. At q=1 from Eq. (29), we get the Cattaneo-type gene- ralized equation within the Gibbs statistics with multifractal time and spatial fractality,
),
; ' (
) , ( )
(
= )
; ( )
; (
'
1 2 0
2
t r r
r r r D r d D t r tn t r
t n t
βν
µ τ
α α
α α ξ α
∂
⋅ ∂
⋅ ′
∂
′ ∂
∂ + ∂
∂
∂ −
∫
(29)
Eqs. (28), (29) contain significant spatial non-homogeneity in Dq(r,r′). If we neglect spatial non-homogeneity,
), (
= ) ,
(r r D r r
Dq ′ qδ −′ (30) we get the Cattaneo-type diffusion equation with fractality of space-time and the constant coefficients of the diffusion within the Renyi statistics,
),
; (
= )
; ( )
;
( 2 *
2 1
2 0 2
t r r
D D t r tn t r
t n t q
βν
τ ξ αα
∂
∂
∂ + ∂
∂
∂ − (31)
At q=1, 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
2 0 2
t r r D D t r tn t r
t n t b
ν
τ ξ αα
∂
∂
∂ + ∂
∂
∂ −
∑
(32)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- on equations, which were obtained in Ref. [18],
).
; (
= )
; ( )
;
( 2
2 1 2 0
2
t r r D D t r tn t r
t n t
ν
τ ξ
∂
∂
∂ + ∂
∂
∂ −
(33)
77
ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic
At τ =0, we get an important particular case — the gene- ralized diffusion equation of particles with taking into ac- count fractality of space-time,
),
; ( )
, ( )
(
= )
;
( 0 1 ' * r' t
r r r r D r d D t r
tn t q
ξ µα αα αα βν
∂
⋅ ∂
⋅ ′
∂
′ ∂
∂
∂ −
∫
(34)and by neglecting spatial non-homogeneity of the diffusion coefficients Dq(r,r′), we also get the diffusion equation with the constant coefficients of the diffusion with the fractional derivatives within the Renyi statistics,
),
; (
= )
;
( 2 *
2 1
0 r t
D r D t r
tn t q
ξ αα βν
∂
∂
∂
∂ −
(35) At α=1, τ=0, we get the diffusion equation with the constant coefficients of the diffusion without spatial fractality within the Renyi statistics
),
; (
= )
;
( 2 *
2 1
0 r t
D r D t r
tn t q
ξ βν
∂
∂
∂
∂ −
(36) At α=1, τ=0, q=1, ξ =1, we get the usual diffusion equation for the particles within the Gibbs statistics,
).
; (
= )
;
( 2
2
t r r
D t r tn
βν
∂
∂
∂
∂ (37)
Let us consider another model of the memory function ) ,
( 1
)
= ( )
( 1
1
−
−
+ γ
ξ
ωτ ω ω
i
W i (38) then in frequency representation we get
(
1+(iωτ)γ−1)
iωn(r;ω)=(iω)1−ξΨ(r;ω). (39) By using Eq. (27) and inverse transformation of Eq. (39) to the time t, we get the generalized Cattaneo-type diffusion equation with taking into account multifractal time and spatial fractality.IV. C
ONCLUSIONWe presented the general approach for obtaining the gene- ralized 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 fractio- nal 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 fre- quency dependent memory functions, which lead to the known diffusion equations with the fractality of space-time and their generalizations.
R
EFERENCES[1] K. B. Oldham and J. Spanier, The Fractional Calculus:
Theory and Applications of Differentiation and Integration to Arbitrary Order. Dover Books on Mathematics; Dover Publications, 2006.
[2] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and
Applications. Gordon and Breach Science Publishers, 1st ed., 1993.
[3] I. Podlubny and V. T. E. Kenneth, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, 1st ed., 1998.
[4] V. V. Uchaikin, Fractional Derivatives Method.
Artishock-Press, Uljanovsk, 2008.
[5] P. Kostrobij, B. Markovych, O. Viznovych, and M.
Tokarchuk, "Generalized diffusion equation with fractional derivatives within Renyi statistics." Journal of Mathematical Physics, vol. 57, pp. 093301, 2016.
[6] P. Kostrobij, B. Markovych, O. Viznovych, and M.
Tokarchuk, "Generalized electrodiffusion equation with fractality of space-time." Mathematical Modeling and Computing, vol. 3, n. 2, pp. 163–172, 2016.
[7] R. R. Nigmatullin, "To the Theoretical Explanation of the “Universal Response”." Physica Status Solidi (b), vol. 123, pp. 739–745, 1984.
[8] R. R. Nigmatullin, "On the Theory of Relaxation for Systems with “Remnant” Memory." Physica Status Solidi (b), vol. 124, pp. 389–393 1984.
[9] R. R. Nigmatullin, "The realization of the generalized transfer equation in a medium with fractal geometry."
Physica Status Solidi (b), vol. 133, pp. 425–430, 1986.
[10] R. R. Nigmatullin, "Fractional integral and its physical interpretation." Theoretical and Mathematical Physics, vol. 90, pp. 242–251, 1992.
[11] D. N. Zubarev, V. G. Morozov, and G. Röpke, Statistical mechanics of nonequilibrium processes. Fizmatlit, Vol.
1, 2, 2002.
[12] B. B. Markiv, R. M. Tokarchuk, P. P. Kostrobij, and M.
V. Tokarchuk, "Nonequilibrium statistical operator method in Renyi statistics." Physica A: Statistical Mechanics and its Applications, vol. 390, pp. 785–791, 2011.
[13] V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Field and Media. Springer, New York, 2011.\
[14] V. E. Tarasov, "Transport equations from Liouville equations for fractional systems." International Journal of Modern Physics B, vol. 20, pp. 341–353, 2006.
[15] K. Cottrill-Shepherd and M. Naber, "Fractional differential forms." Journal of Mathematical Physics, vol. 42, pp. 2203–2212, 2001.
[16] F. Mainardi, In Fractals and Fractional Calculus in Continuum Mechanics; A. Carpinteri, F.Mainardi, Eds.;
Springer Vienna: Vienna, pp. 291–348, 1997.
[17] M. Caputo and F. Mainardi, "A new dissipation model based on memory mechanism." Pure and Applied Geophysics, vol. 91, pp. 134–147, 1971.
[18] A. Compte and R. Metzler, "The generalized Cattaneo equation for the description of anomalous transport processes." Journal of Physics A: Mathematical and General, vol. 30, pp. 7277–7289, 1997.
