Macroscopic parameters of Fokker-Planck flows

∞ ∞ ∞
3
=
2iF i . (2π )3
(22)
4. Fokker - Planck equation and conservation laws Fokker - Planck equation with damping force is: ∂n ∂n ∂n ∂2 n + vk − α vk −3 αn = k ; ∂t ∂xk ∂v k ∂v m ∂v m where α - damping coefficient. Multiplying (23) by exp(−i v k q k ) and integrating over velocities, we obtain
For our simple model of non-interacting particles following relation between density of internal energy and pressure is valid p=− where p - the hydrostatic pressure. The current of energy is Fi = 1 1 1 1 v k v k vi = ρ u k u k ui + w k w k ui + u k w k wi + w k w k wi = 2 2 2 2 = ( K + E ) ui − u k σ ki + 1 w w w. 2 k k i (16) 1 2 σ kk = E ; 3 3 (15)
From macroscopical point of view the tensor of stresses describes the interaction between portions of continuum. It is remarkable, that though interaction between particles is missing, macroscopic cells are interacting nevertheless. This interaction is due to chaotic momentum transfer from one cell to another. For interacting particles additional term should arise in (9). The volume density of energy is
-31 vk vk ; 2
e=
(11)
because the only form of energy for non-interacting particles is their kinetic energy. Decomposition (7) of velocities leads to decomposition of full energy e= 1 1 ρ uk uk + wk wk = K + E . 2 2 (12)
−∞ −∞ −∞
∫ ∫ ∫ nv dv dv dv
k 1 2
∞ ∞ ∞
3
=
−i ρ u k ; (2π )3
(19)
- current of momentum tensor ∂2 M −1 (t , x 1 , x 2 , x 3 , 0, 0, 0) = ∂q i ∂q j (2π )3
−∞ −∞ −∞
k k 1 2
∞ ∞ ∞
3
=
−e ; (2π )3
(21)
- current of energy is ∂3 M i (t , x 1 , x 2 , x 3 , 0, 0, 0) = ∂q i ∂q∫ ∫ nv v v dv dv dv
i k k 1 2
∫ ∫ ∫ nv v dv dv dv
i j 1 2
∞ ∞ ∞
3
=
− J ij ; (2π )3
(20)
- volume density of energy −1 ∂2 M (t , x 1 , x 2 , x 3 , 0, 0, 0) = (2π )3 ∂q k ∂q k
−∞ −∞ −∞
∫ ∫ ∫ nv v dv dv dv
∫ ∫ ∫ f dx dx dx .
1 2 3 D
(2)
-2-
The most interesting functions are functions, which satisfy some conservation laws. That is time derivative of F is a sum of volume and surface integrals dF = dt
−∞ −∞ −∞
∫ ∫ ∫ n exp(−i v q ) dv dv dv .
k k 1 2 3
∞ ∞ ∞
(17)
where q k - velocities momentum variables. Using this definition, we can express macroscopic parameters in terms of Fourier transform of density and its derivatives: - density 1 (2π )3
σ ij = − w i w j ;
(8)
(9)
- recall classic kinetic gas theory formula for pressure. Using this definition, we can write expression for the momentum current tensor in the final form J ij = ρ ui u j − σ ij . (10)
The first term in (12) is kinetic energy density K= 1 ρu u . 2 k k (13)
The second term in (12) is energy of chaotic movement or internal energy volume density E= 1 wk wk . 2 (14)
∫∫∫
D
∂f dx 1 dx 2 dx 3 + ∂t
∫∫
∂D
Fl 1 dx 2 dx 3 + Fl 2 dx 3 dx 1 + Fl 3 dx 1 dx 2 .
(3)
where Fl i is the current of f . Our aim is to construct the set of conservation laws for special case, when distribution function n( x i , v i ) satisfies Fokker - Planck equation. 2. General definitions of macroscopic parameters The density of mass is the average of unity 1 = ρ. The current of mass vector is vi = ρ ui . where ui - average velocity. (5) represents at the same time the vector momentum density. The current of momentum tensor is J ij = v i v j . Decomposition of velocity gives vi = ui + wi ; (7) (6) (5) (4)
3. Expressions for macroscopic parameters through Fourier transform of density Let us denote by M the Fourier transform of density M (t , x i , q j ) = 1 (2π )3
+∞ +∞ +∞
f ( xi ) =
−∞ −∞ −∞
∫ ∫ ∫ n f dv dv dv .
1 2 3
(1)
Defined in such a way average value, depends obviously only on spatial coordinates x i . Macroscopic observer deals with these average values and traits them from the continuum mechanics field theory point of view. He perceives average f as volume density of some additive function F F=
Keywords Fokker-Planck equation, continuum mechanics
1. Introduction The statistical mechanics of systems described by kinetic equations and generalized Fokker - Planck equations is currently subject of active research (see [1]). In this work we study rather special kind of medium without interaction of particles. To describe the state of this medium we use density distribution function n( x i , v i ) in the space of Cartesian coordinates x i and corresponding velocities v i . We define average of any function f of these variables by