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 ) + ∑Drα 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
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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
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