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The non-linear Fokker-Planck equation arises in describing the evolution of stochastic system, which is a variant of the Boltzmann equation modeling the evolution of the random system with Brownian motion, where the collision term is replaced by a drift-diffusion operator. This model conserves mass, momentum and energy; the dissipation is much weaker than that in a simplified model we considered before which conserved only mass, thus more difficult to analyze. The macro-micro decomposition of the solution around the local Maxwellian introduced by T.-P. Liu, T. Yang and S.-H. Yu for Boltzmann equation is used, to reformulate the model into a fluid-type system incorporate viscosity and heat diffusion terms, coupled with an equation of the microscopic part. The viscosity and heat diffusion terms can give dissipative mechanism for the analysis of the model.

The non-linear Fokker-Planck equations arise in describing the evolution of stochastic system such as the erratic motions of small particles immersed in fluids, fluctuations of the intensity of laser light, velocity distributions of fluid particles in turbulent flows, or the stochastic behavior of exchange rates. This model has been widely applied in physics, biology, ecology, economy and social science, see [

∂ t f + v ⋅ ∇ x f = ρ α ∇ v ⋅ [ T ∇ v f + ( v − u ) f ] , (1.1)

where the unknown function f = f ( t , x , v ) is the distribution of particles at time t, position x and velocity v for ( t , x , v ) ∈ ℝ + × ℝ 3 × ℝ 3 , α ∈ [ 0 , 1 ] is the friction parameter, ρ , u ¯ , T are the mass density, the mean velocity and the local temperature, which are coupled to f by

ρ ( t , x ) = ∫ ℝ 3 f ( t , x , v ) d v , ρ u ( t , x ) = ∫ ℝ 3 v f ( t , x , v ) d v , 3 ρ T ( t , x ) = ∫ ℝ 3 | v − u | 2 f ( t , x , v ) d v . (1.2)

As a variant of the Boltzmann equation modeling the evolution of the random system with Brownian motion, the collision term is replaced by a drift-diffusion operator. Note that this famous model conserves mass, momentum and energy [

Some related models. When relativistic/quantum effect is included, the relativistic/quantum Fokker-Planck equations are considered in [

Motivation of the present study. Note that this fully non-linear model (1.1) preserves mass, momentum and energy; the dissipation is much weaker than that in the simplified model considered in [

Following [

f ( t , x , v ) = M ( t , x , v ) + G ( t , x , v ) , (2.1)

where the local Maxwellian M = M ( t , x , v ) = M [ ρ , u , T ] ( v ) depending on the macroscopic quantities ( ρ , u , T ) defined in (1.2), is given by

M [ ρ , u , T ] ( v ) = ρ ( 2 π T ) 3 / 2 exp ( − | v − u | 2 2 T ) , (2.2)

and G ( t , x , v ) is the microscopic, non-fluid part. Define the weighted inner product

〈 g , h 〉 M : = ∫ ℝ 3 g ( v ) h ( v ) 1 M d v (2.3)

for two functions g and h of v such that the integral is well defined. Denote

χ 0 M = 1 ρ M , χ i M = v i − u i T ρ M ( i = 1 , 2 , 3 ) , χ 4 M = 1 6 ρ ( | v − u | 2 T − 3 ) M , (2.4)

which are orthonormal with respect to the inner product (2.3), that is,

〈 χ β M , χ γ M 〉 M = δ β γ , β , γ = 0 , 1 , 2 , 3 , 4.

Then the macroscopic projection P 0 M and microscopic projection P 1 M of a function h are defined as [

P 0 M h : = ∑ β = 0 4 〈 h , χ β M 〉 M χ β M , P 1 M h = ( I − P 0 M ) h . (2.5)

The projections will be used to decompose the nonlinear Fokker-Planck Equation (1.1) into a system of fluid equations coupled with a microscopic equation.

Take the right hand side of (1.1) as the “collision term”, it has five collision invariants { ψ β } β = 0 4 , where

ψ 0 = 1 , ψ i = v i ( i = 1 , 2 , 3 ) , ψ 4 = 1 2 | v | 2 , (3.1)

thus, by multiplying Equation (1.1) with collision invariants and integrating, one has five conservation laws

∫ ℝ 3 ψ β ( f t + v ⋅ ∇ x f ) d v = 0 , β = 0 , 1 , 2 , 3 , 4 ,

that is,

ρ t + d i v x ( ρ u ) = 0 , ( ρ u i ) t + ∑ j ∂ x j ( ρ u i u j ) + ∂ x i ( ρ T ) + ∂ x j ∫ ℝ 3 G v i v j d v = 0 , i = 1 , 2 , 3 , [ ρ ( | u | 2 2 + 3 T 2 ) ] t + ∑ j ∂ x j [ u j ( 5 2 ρ T + 1 2 ρ | u | 2 ) ] + ∫ ℝ 3 1 2 | v | 2 v ⋅ ∇ x G d v = 0. (3.2)

This is a fluid-type system, coupled with the microscopic part G, which satisfies the microscopic equation derived by applying the microscopic projection P 1 M to Equation (1.1):

G t + P 1 M ( v ⋅ ∇ x M + v ⋅ ∇ x G ) = ρ α ∇ v ⋅ ( T ∇ v M + ( v − u ) M ) + ρ α ∇ v ⋅ ( T ∇ v G + ( v − u ) G ) = ρ α ∇ v ⋅ ( T ∇ v G + ( v − u ) G ) . (3.3)

Note that

∇ v ( g M ) = ∇ v g M − ( v − u ) g T M , ∇ v ⋅ ( T ∇ v g + ( v − u ) g ) = T ∇ v [ M ⋅ ∇ v ( g M ) ] ,

then (3.3) can be rewritten as

G t + P 1 M ( v ⋅ ∇ x M + v ⋅ ∇ x G ) = : ρ α L M G , (3.4)

where L M is the Fokker-Planck operator given by

L M g : = T ∇ v [ M ⋅ ∇ v ( g M ) ] . (3.5)

In summary, the nonlinear Fokker-Planck Equation (1.1) is decomposed into the fluid-type system (3.2) coupled with the microscopic Equation (3.4). Note that the system (3.2) becomes Euler equations when the microscopic part G is set to be zero, as in the traditional Hilbert expansion. To recover Navier-Stokes type system [

Note that the nonlocal integration terms in the fluid-type system (3.2) are related to the microscopic component G, which are the origin of viscosity and heat diffusion. This is derived by applying Chapman-Enskog expansion as for Boltzmann equation, keeping the leading order term in the microscopic part (see [

Lemma 4.1. The Fokker-Planck operator L M defined in (3.5) satisfies

L M χ 0 M = 0 , L M χ i M = − χ i M ( i = 1 , 2 , 3 ) , L M χ 4 M = − 2 χ 4 M , 〈 L M g , h 〉 M = 〈 g , L M h 〉 M , − 〈 P 1 M h , L M P 1 M h 〉 M ≥ λ 0 ‖ P 1 M h ‖ M , ν 2 for some constant λ 0 > 0 , ‖ L M − 1 P 1 M h ‖ M , ν 2 ≤ 1 λ 0 2 ∬ ℝ 3 × ℝ 3 | P 1 M h | 2 ( 1 + | v | 2 ) − 1 M d v d x , (4.1)

and

L M N 1 = − 2 N 1 , L M N 2 = − 3 N 2 , (4.2)

where

N 1 : = ∑ i j ( v i − u i ) ( v j − u j ) u x j i M T − ∇ x ⋅ u 3 T | v − u | 2 M , N 2 : = ( | v − u | 2 − 5 T ) ( v i − u i ) M .

Proof. The proof of (4.1) is direct. Note that this is also similar to the microscopic version of the H-theorem for Boltzmann equation in [

L M [ ( v i − u i ) ( v j − u j ) M ] = T ∇ v ⋅ [ M ∇ v ( ( v i − u i ) ( v j − u j ) ) ] = ∑ k T ∂ v k [ M δ k j ( v i − u i ) + M δ k j ( v j − u j ) ] = [ − ( v i − u i ) ( v j − u j ) M + ∑ k M T δ k j δ k i ] ∗ 2 ,

that is, for fixed i , j ,

L M [ ( v i − u i ) ( v j − u j ) M ] = − 2 ( v i − u i ) ( v j − u j ) M + 2 δ i j T M . (4.3)

Next,

L M ( | v − u | 2 M ) = T ∇ v ⋅ [ M ∇ v ⋅ ( | v − u | 2 M M ) ] = 2 T ∇ v ⋅ [ ( v − u ) M ] = − 2 | v − u | 2 M + 6 T M . (4.4)

Combining (4.3) and (4.4) to get

L M [ ∑ i j ( v i − u i ) ( v j − u j ) u x j i M T − ∇ x ⋅ u 3 T | v − u | 2 M ] = ∑ i j [ − 2 ( v i − u i ) ( v j − u j ) u x j i M T + 2 δ i j u x j i T M T ] − ∇ x ⋅ u 3 T ( − 2 | v − u | 2 M + 6 T M ) = − 2 [ ∑ i j ( v i − u i ) ( v j − u j ) u x j i M T − ∇ x ⋅ u 3 T | v − u | 2 M ] , (4.5)

thus the first equation in (4.2) is proved. Next,

L M ( ( v i − u i ) M ) = T ∇ v [ M ∇ v ( v i − u i ) ] = ∑ j T ∂ v j [ M ∂ v j ( v i − u i ) ] = T ∂ v i M = − ( v i − u i ) M , (4.6)

then

L M [ 5 T ( v i − u i ) M ] = − 5 ( v i − u i ) T M . (4.7)

Furthermore,

L M ( ( v i − u i ) | v − u | 2 M ) = T ∇ v ⋅ [ M ∇ v ( ( v i − u i ) | v − u | 2 ) ] = T ∂ v j [ M δ i j | v − u | 2 + 2 M ( v i − u i ) ( v j − u j ) ] = T ∂ v j ( M | v − u | 2 ) + 2 T ∂ v j ( M ( v i − u i ) ( v j − u j ) ) = − ( v i − u i ) | v − u | 2 M + 2 ( v i − u i ) T M − 2 | v − u | 2 ( v i − u i ) M + 6 T ( v i − u i ) M + 2 T M ( v i − u i ) = − 3 ( v i − u i ) | v − u | 2 M + 10 ( v i − u i ) T M . (4.8)

Combine (4.7)-(4.8):

L M [ ( | v − u | 2 − 5 T ) ( v i − u i ) M ] = − 3 ( | v − u | 2 − 5 T ) ( v i − u i ) M , (4.9)

this is the second equation in (4.2).

Now, using the properties of the Fokker-Planck operator L M , one can invert L M on (3.4) to get

G = ρ − α L M − 1 ( P 1 M ( v ⋅ ∇ x M ) ) + ρ − α L M − 1 π , π : = G t + P 1 M ( v ⋅ ∇ x G ) . (4.10)

By substituting (4.10) into (3.2), one has the fluid-type equations

ρ t + d i v x ( ρ u ) = 0 , ( ρ u i ) t + ∑ j ∂ x j ( ρ u i u j ) + ∂ x i ( ρ T ) = − ∑ j ∂ x j ∫ ℝ 3 ρ − α L M − 1 π v i v j d v − ∑ j ∂ x j ∫ ℝ 3 ρ − α L M − 1 ( P 1 M ( v ⋅ ∇ x M ) ) v i v j d v , [ ρ ( | u | 2 2 + 3 T 2 ) ] t + ∑ j ∂ x j [ u j ( 5 2 ρ T + 1 2 ρ | u | 2 ) ] = − ∑ i ∂ x j ∫ ℝ 3 1 2 | v | 2 v i ρ − α L M − 1 π d v − ∑ i ∂ x i ∫ ℝ 3 1 2 | v | 2 v i ρ − α L M − 1 ( P 1 M ( v ⋅ ∇ x M ) ) d v . (4.11)

As the same to the Boltzmann equation, the terms containing L M − 1 ( P 1 M ( v ⋅ ∇ x M ) ) in above yield the viscosity and heat conductivity, which can be specified more precisely, by using the following lemma.

Lemma 4.2. The Fokker-Planck operator L M defined in (3.5) satisfies

L M − 1 P 1 M ( v ⋅ ∇ x M ) = − 1 6 T 2 ( | v − u | 2 − 5 T ) ( v − u ) ⋅ ∇ T M − 1 2 T [ ∑ i j ( v i − u i ) ( v j − u j ) u x j i − ∇ ⋅ u 3 | v − u | 2 M ] . (4.12)

Proof. Notice that the x-derivative of the local Maxwellian M in (2.2) is

∂ x i M = [ ρ x i ρ − 3 2 T x i T − ( | v − u | 2 2 T ) x i ] M = [ ρ x i ρ − 3 2 T x i T + T x i 2 T 2 | v − u | 2 + ( v − u ) u x i T ] M ,

then

P 1 M ( v ⋅ ∇ x M ) = P 1 M ( ( v − u ) ⋅ ∇ x M ) + P 1 M ( u ⋅ ∇ x M ) = P 1 M ( ( v i − u i ) T x i 2 T 2 | v − u | 2 M ) + P 1 M ( ( v i − u i ) ( v j − u j ) u x j i M T ) : = I 1 + I 2 . (4.13)

Use the definition of the microscopic projection, one has

I 1 = ( v − u ) ⋅ ∇ T | v − u | 2 2 T 2 M − ∫ ℝ 3 ( v j − u j ) ( v i − u i ) T x i 2 T 2 | v − u | 2 M d v ⋅ v j − u j T ρ M = ( | v − u | 2 2 T 2 − 5 2 T ) ( v − u ) ⋅ ∇ T M . (4.14)

and

I 2 = ( v i − u i ) ( v j − u j ) u x j i M T − ( ∫ ℝ 3 ( v i − u i ) ( v j − u j ) u x j i M T d v ) M ρ − ( ∫ ℝ 3 ( v i − u i ) ( v j − u j ) u x j i M T ⋅ ( | v − u | 2 T − 3 ) d v ) 1 6 ρ ( | v − u | 2 T − 3 ) M : = I 20 + I 21 + I 22 . (4.15)

Note that

∫ ℝ 3 | v i − u i | 2 M d v = 1 3 ∫ ℝ 3 | v − u | 2 M d v = ρ T , ∫ ℝ 3 | v i − u i | 4 M d v = 15 ρ T 2 ,

then

I 21 = ∑ i ∫ ℝ 3 | v i − u i | 2 M d v ⋅ 1 T u x i i ⋅ M ρ = ( ∇ ⋅ u ) M , (4.16)

and

I 22 = ∑ i ( ∫ ℝ 3 | v i − u i | 2 ( | v − u | 2 T − 3 ) M d v ) u x i i 6 ρ T ( | v − u | 2 T − 3 ) M = 1 3 ( ∫ ℝ 3 | v − u | 2 ( | v − u | 2 T − 3 ) M d v ) ∇ ⋅ u 6 ρ T ( | v − u | 2 T − 3 ) M = ∇ ⋅ u 3 ( | v − u | 2 T − 3 ) M . (4.17)

Combining the above to get

I 2 = ∑ i j ( v i − u i ) ( v j − u j ) u x j i M T − ∇ ⋅ u 3 T | v − u | 2 M . (4.18)

Plug (4.14) and (4.18) into (4.13), one gets

P 1 M ( v ⋅ ∇ x M ) = ( | v − u | 2 2 T 2 − 5 2 T ) ( v − u ) ⋅ ∇ T M + ∑ i j ( v i − u i ) ( v j − u j ) u x j i M T − ∇ ⋅ u 3 T | v − u | 2 M . (4.19)

Then (4.12) is proved with the help of Lemma 4.1.

Now we compute the viscosity and heat conductivity, which are given in below.

Theorem 4.3. For each index i , j , we have

∫ ℝ 3 L M − 1 P 1 M ( v ⋅ ∇ x M ) v i v j d v = − ρ T 2 ( u x j i + u x i j − 2 3 ( d i v u ) δ i j ) . (4.20)

∫ ℝ 3 L M − 1 P 1 M ( v ⋅ ∇ x M ) v j | v | 2 d v = − 5 3 ρ T ∂ x j T − ρ T ∑ i u i ( u x j i + u x i j − 2 3 ( ∇ ⋅ u ) δ i j ) . (4.21)

Remark: The right hand sides of (4.20) and (4.21), respectively, give the viscosity and heat conductivity to the fluid-type system.

Proof. Using (4.12), one compute

∫ ℝ 3 L M − 1 P 1 M ( v ⋅ ∇ x M ) v i v j d v = − 1 6 T 2 ∫ ℝ 3 ( | v − u | 2 − 5 T ) ( v − u ) ⋅ ∇ T v i v j M d v − ∫ ℝ 3 1 2 T [ ∑ k l ( v k − u k ) ( v l − u l ) u x k l − ∇ ⋅ u 3 | v − u | 2 v i v j M d v ] : = J 1 + J 2 , (4.22)

where

J 1 = − 1 6 T 2 ∫ ℝ 3 ( | v − u | 2 − 5 T ) ( v − u ) ⋅ ∇ T ( v i − u i ) ( v j − u j ) M d v − 1 6 T 2 ∫ ℝ 3 ( | v − u | 2 − 5 T ) ( v − u ) ⋅ ∇ T ( u i ( v j − u j ) + ( v i − u i ) u j + u i u j ) M d v = − u i 6 T 2 ∫ ℝ 3 ( | v − u | 2 − 5 T ) ( v − u ) ⋅ ∇ T ( v j − u j ) M d v − ⋯ ( i ↔ j ) = − u i ∂ x j T 6 T 2 ∫ ℝ 3 ( | v − u | 2 − 5 T ) | v j − u j | 2 M d v − ⋯ = − u i ∂ x j T 18 T 2 ∫ ℝ 3 ( | v − u | 4 − 5 T | v − u | 2 ) M d v = 0 , (4.23)

and

J 2 = − ∫ ℝ 3 1 2 T [ ∑ k l ( v k − u k ) ( v l − u l ) u x k l − ∇ ⋅ u 3 | v − u | 2 ] ( v i − u i ) ( v j − u j ) M d v − ∫ ℝ 3 1 2 T [ ∑ k l ( v k − u k ) ( v l − u l ) u x k l − ∇ ⋅ u 3 | v − u | 2 ] × ( u i ( v j − u j ) + ( v i − u j ) u i + u i u i ) M d v = − ∫ ℝ 3 1 2 T [ ∑ k l ( v k − u k ) ( v l − u l ) u x k l − ∇ ⋅ u 3 | v − u | 2 ] ( | v i − u i | 2 δ i j + u i u j ) M d v = − ρ T 2 ( u x j i + u x i j − 2 3 ( d i v u ) δ i j ) . (4.24)

Combining (4.22)-(4.24), one has (4.20). Next, for fixed index j,

∫ ℝ 3 L M − 1 P 1 M ( v ⋅ ∇ x M ) v j | v | 2 d v = − ∂ x i T 6 T 2 ∫ ℝ 3 ( | v − u | 2 − 5 T ) ( v i − u i ) v j | v | 2 M d v − ∫ ℝ 3 1 2 T [ ∑ k l ( v k − u k ) ( v l − u l ) u x k l − ∇ ⋅ u 3 | v − u | 2 ] v j | v | 2 M d v : = J 3 + J 4 , (4.25)

in which

J 3 = − ∂ x i T 6 T 2 ∫ ℝ 3 ( | v − u | 2 − 5 T ) ( v i − u i ) ( v j − u j ) | v | 2 M d v − u j ∂ x i T 6 T 2 ∫ ℝ 3 ( | v − u | 2 − 5 T ) ( v i − u i ) | v | 2 M d v = − 5 3 ρ T ∂ x j T , (4.26)

and

J 4 = − 1 2 T ∫ ℝ 3 [ ∑ k l ( v k − u k ) ( v l − u l ) u x k l − ∇ ⋅ u 3 | v − u | 2 ] ( v j − u j ) | v | 2 M d v − u j 2 T ∫ ℝ 3 [ ∑ k l ( v k − u k ) ( v l − u l ) u x k l − ∇ ⋅ u 3 | v − u | 2 ] | v | 2 M d v = − ρ T ∑ i u i ( u x j i + u x i j − 2 3 ( ∇ ⋅ u ) δ i j ) . (4.27)

Put (4.26) and (4.27) into (4.25), one gets (4.21).

Now with the help of Theorem 4.3, the macroscopic Equation (4.11) becomes

ρ t + d i v x ( ρ u ) = 0 , ( ρ u i ) t + ∑ j ∂ x j ( ρ u i u j ) + ∂ x i ( ρ T ) = ∑ j ∂ x j ( ρ 1 − β T 2 ( u x j i + u x i j − 2 3 ( d i v u ) δ i j ) ) − ∑ j ∂ x j ∫ ℝ 3 ρ − α L M − 1 π v i v j d v , [ ρ ( | u | 2 2 + 3 T 2 ) ] t + ∇ ⋅ [ u ( 5 2 ρ T + 1 2 ρ | u | 2 ) ] = − ∑ i ∂ x i ∫ ℝ 3 1 2 | v | 2 v i ρ − α L M − 1 π d v + 5 12 ∇ ⋅ [ ρ 1 − β ∇ ( T 2 ) ] + ∑ i , j ∂ x j ( ρ 1 − β T 2 u i ( u x j i + u x i j − 2 3 ( d i v u ) δ i j ) ) . (4.28)

Note that this is a Navier–Stokes type fluid system, coupled with π defined in (4.10), the remainder term of the microscopic component.

The fully non-linear Fokker-Planck Equation (1.1) preserves mass, momentum and energy; the dissipation is much weaker than that in the simplified model considered in [

This research is partially supported by National Natural Science Foundation of China (Nos. 11871335 and 11971008) and USRP of East China University of Science and Technology (No. X20251).

The authors declare no conflicts of interest regarding the publication of this paper.

Ren, Y.A., Yu, L.J. and Liao, J. (2020) The Hydrodynamic Limit of Nonlinear Fokker-Planck Equation. Journal of Applied Mathematics and Physics, 8, 2488-2499. https://doi.org/10.4236/jamp.2020.811184

the distribution of particles at time t, position x and velocity v: f ( t , x , v ) ,

collision invariants: ψ 0 = 1 , ψ i = v i ( i = 1 , 2 , 3 ) , ψ 4 = 1 2 | v | 2 ,

mass density, the mean velocity and the local temperature: ( ρ , u ˜ , T ) ,

perturbation of mass density, the mean velocity and the local temperature: ( σ , u , T ˜ ) ,

local/global Maxwellian: M ( t , x , v ) , M ¯ ( t , x , v ) ,

weighted inner product with respect to a Maxwellian M:

〈 g , h 〉 M : = ∫ ℝ 3 g ( v ) h ( v ) 1 M d v ,

basis function of the macroscopic quantities:

χ 0 M = 1 ρ M , χ i M = v i − u i T ρ M ( i = 1 , 2 , 3 ) , χ 4 M = 1 6 ρ ( | v − u | 2 T − 3 ) M ,

macroscopic projection P 0 M and microscopic projection P 1 M of a function h:

P 0 M h : = ∑ β = 0 4 〈 h , χ β M 〉 M χ β M , P 1 M h = ( I − P 0 M ) h ,

macro-micro decomposition of f: f ( t , x , v ) = M ( t , x , v ) + G ( t , x , v ) ,

Fokker-Planck operator: L M g : = T ∇ v [ M ⋅ ∇ v ( g M ) ] .