Uniform-in-time propagation of chaos for second order interacting particle systems
Yun Gong Zhenfu Wang Pengzhi Xie








Uniform-in-time propagation of chaos for second order interacting particle systems
    Yun Gong [Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China. 1901110013@pku.edu.cn.] 
	  Zhenfu Wang[Beijing International Center for Mathematical Research, Peking University, Beijing 100871,  China. zwang@bicmr.pku.edu.cn.]   Pengzhi Xie[Department of Finance and Control Sciences, School  of Mathematical Sciences, Fudan University, Shanghai 200433, China. 22110180046@m.fudan.edu.cn.]

    September 9, 2024
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We study the long time behavior of second order particle systems interacting through global Lipschitz kernels. Combining hypocoercivity method in <cit.> and relative entropy method in <cit.>, we are able to overcome the degeneracy of diffusion in position direction by controlling the relative entropy and relative Fisher information together. This implies the uniform-in-time propagation of chaos through the strong convergence of all marginals. Our method works at the level of Liouville equation and relies on the log Sobolev inequality of equilibrium of Vlasov-Fokker-Planck equation. 








§ INTRODUCTION

equationsection



 §.§ Framework


In this article, we consider the stochastic second order particle systems for N indistinguishable point-particles, subject to a confining external force - ∇ V and an interacting kernel K,



    { x_i(t)    = v_i(t)  t, 
     v_i(t)     = - ∇ V(x_i(t))  t + 1/N∑_j ≠ iK(x_i(t) - x_j(t))  t - γ v_i(t)  t + √(2 σ) B_t^i,
    .

where i = 1,2,...,N. We take the position x_i in Ω, which may be the whole space ^d or the periodic torus 𝕋^d, while each velocity  v_i lies in ^d. Note that  { (B_·^i ) }_i=1^N denotes  N independent copies of Wiener processes on ^d. We take the diffusion coefficient before the Brownian motions √(2 σ) as a constant for simplicity. Usually one can take  σ = γβ^-1, where the constant γ > 0 denotes the friction parameter, and β is the inverse temperature. In more general models, those σ  may depend on the number of particles N, the position of particles x_i, etc. 



In many important models, the kernel is given  by K = -  ∇ W, where W is an interacting potential. A well-known example for W is  the Coulomb potential. We recall that  the Hamiltonian energy H: (Ω×^d)^N→ associated to the particle system  (<ref>) reads as 

    H(x_1,v_1,...,x_N,v_N) = 1/2∑_i=1^N v_i^2 + ∑_i=1^N V(x_i) + 1/2N∑_i=1^N ∑_j ≠ i  W(x_i - x_j),

which is the sum of the kinetic energy 1/2∑_i=1^N v_i^2 and the potential energy U(x_1,...,x_N) defined by

    U(x_1,...,x_N):= ∑_i=1^N V(x_i) + 1/2N∑_i=1^N ∑_j ≠ i  W(x_i - x_j).


We recall the Liouville equation as  in <cit.>, which describes the joint distribution f_N of the particle system  (<ref>) on (Ω×^d)^N, 

    ∂_t f_N + 
    ∑_i=1^N v_i ·∇_x_i f_N     + ∑_i=1^N ( - ∇_x_i V(x_i) + 1/N∑_j ≠ i K(x_i - x_j) ) ·∇_v_i f_N 
        = σ∑_i=1^N Δ_v_i f_N + γ∑_i=1^N∇_v_i· (v_i f_N).

We define the associated Liouville operator as

    L_N f_N =    ∑_i=1^N v_i ·∇_x_if_N + ∑_i=1^N ( -∇_x_i V(x_i) + 1/N∑_j ≠ i K(x_i - x_j) ) ·∇_v_if_N 
        - σ∑_i=1^N Δ_v_i f_N - γ∑_i=1^N∇_v_i· (v_i f_N).

As the mean field theory indicates,  when  N →∞, the limiting behavior of any  particle in (<ref>) is described by the McKean-Vlasov SDE 

    {  x(t)    = v(t)   t, 
     v(t)     = - ∇ V(x(t)) + K ∗ρ(x(t))  t - γ v(t)  t + √(2σ) B_t, 
    .

where (x,v) ∈Ω×^d, (B_·) denotes a standard Brownian motion on ^d. We then denote its phase space density by f_t = (x(t), v(t))  and the spatial density by 

    ρ(t,x) = ∫_^d f(t,x,v)   v.

The law  f_t further satisfies the mean-field equation or the nonlinear Fokker-Planck equation

    ∂_t f + v ·∇_x f - ∇ V ·∇_v f + (K ∗ρ) ·∇_vf = σΔ_vf + γ∇_v · (vf).

If σ = γβ^-1, one can check  that its nonlinear equilibrium satisfies the following equation

    f_∞ = 1/Z e^- β (V(x) + 1/2v^2 + W ∗ρ_∞),

where Z is the constant to make f_∞ a probability density. 
In the literature, one may use the “formal equilibrium" of Eq.(<ref>) at time t to refer 

    f̂_t = 1/Ẑ e^- β (V(x) + 1/2v^2 + W ∗ρ_t),

again where Ẑ is the constant to make f̂_t a probability density. 


It is well-known that the large N limit of particle system (<ref>) is mathematically formalized by the notion of propagation  of   chaos originating in <cit.>. See also the surveys for instance  <cit.>.  Recently, many important works have been done in quantifying propagation of chaos for different kinds of first order interacting particle systems with singualr kernels. See for instance  <cit.> for detailed descriptions  of recent development. However, for 2nd order systems or Newton's dynamcis for interacting particles, the natural question is to consider the mean field limit of a purely deterministic problem, that is σ=0 as in Eq. (<ref>) or our setting with the diffusion only acts on the velocity variables.  Thus the limiting equation (<ref>) has the Laplacian term only in its velocity variable. Consequently, we cannot treat more general kernels K and long time propagation of chaos for 2nd order systems by exploiting entropy dissipation as easy as those in the first-order setting for instance as in <cit.>.    

The main purposes of this article are two-fold. Firstly, we study the long time convergence of  the solution to the N-particle Liouville equation (<ref>) toward its unique equilibrium (i.e. the Gibbs measure) uniformly in N. Secondly,  we establish propagation of chaos of  (<ref>) (or (<ref>) )toward (<ref>) (or (<ref>) ) uniformly in time. We combine the entropy method developed in <cit.> with the hypocoercivity method refined in <cit.> to overcome the degeneracy of diffusion in position direction. Here we deal with the cases where the interaction kernel is globally Lipschitz and the strength of diffusion σ is large enough. Uniform-in-time propagation of chaos cannot hold in full generality, one of the most critical issues is that the non-linear equilibrium of (<ref>) might not be unique. Furthermore, for systems  (<ref>) with singular interactions K, for instance the Coulomb interactions, even the mean-field limit/propagation of chaos for general initial data on a fixed finite time horizon, for instance t ∈ [0, T], is still widely open. The recent progress can be found for instance in <cit.>.   Uniform-in-N convergence of (<ref>) toward its equilibrium is seldom investigated for systems even with very mild singularities. Those will be the objects of our further study. 

Let us briefly describe how we combine the relative entropy with hypocoercivity to establish propagation of chaos.  To illustrate the ideas, we pretend  all solutions are classical  and there is no regularity issue when we take derivatives. We first recall the evolution of relative entropy in <cit.> for systems with bounded kernel K, 

    / t H_N(f^t_N|f_t^⊗ N) ≤    - 1/N∫_(Ω×^d)^N f^t_N R_N  Z 
        - σ/N∫_(Ω×^d)^N f^t_N | ∇_V logf^t_N/f_t^⊗ N|^2  Z,

where H_N(f^t_N|f_t^⊗ N) is  the normalized relative entropy defined as 

    H_N(f^t_N|f_t^⊗ N) = 1/N∫_(Ω×^d)^N f^t_N logf^t_N/f_t^⊗ N Z,

and R_N is the error term defined as

    R_N = ∑_i=1^N ∇_v_ilog f_t(x_i,v_i) ·{1/N∑_j,j≠ i K(x_i - x_j) - K ∗ρ_t(x_i) }.

 Hereinafter we write  that Z= (x_1, v_1, x_2, v_2, ⋯, x_N, v_N) ∈ (Ω×^d)^N, X= (x_1, x_2, ⋯, x_N) ∈Ω^N and V= (v_1, v_2, ⋯, v_N) ∈ (ℝ^d)^N.   Then by integration by parts, one can rewrite (<ref>) as

    / t H_N(f^t_N|f_t^⊗ N) =     - 1/N∫_(Ω×^d)^N∇_V logf^t_N/f_t^⊗ N· R^0_N  Z 
        - σ/N∫_(Ω×^d)^N f^t_N | ∇_V logf^t_N/f_t^⊗ N|^2  Z,

where R^0_N is a dN-dimensional vector defined as 

    R^0_N = {1/N∑_j=1,j≠ i^N K(x_i - x_j) - K ∗ρ_t(x_i) }_i=1^N.

Inspired by the strategy used in <cit.> to deal with the first order dynamics on torus, we expect to find the term

    - 1/N∫_(Ω×^d)^N f^t_N | ∇_X logf^t_N/f_t^⊗ N|^2  Z

in some way to use log-Sobolev inequality for the second order system (<ref>). Fortunately, hypocoercivity in entropy sense in <cit.> motivates us to take the derivative  of normalized relative Fisher information I^M_N(f^t_N|f_t^⊗ N), which  is defined as

    I_N^M(f^t_N|f_t^⊗ N) = 1/N∫_(Ω×^d)^N f^t_N | √(M)∇logf^t_N/f_t^⊗ N|^2  Z,

where M is a 2Nd × 2Nd positive defined matrix. If we choose the matrix M as the form

    M = 
    (
    [ E F; F G ]),

where E,F,G are Nd × Nd positive definite matrices to be specified later, then we obtain

    / t I_N^M(f^t_N|f_t^⊗ N) ≤    - c_1/N∫_(Ω×^d)^N f^t_N |√(E)∇_X logf^t_N/f_t^⊗ N|^2  Z 
        +  
    c_2/N∫_(Ω×^d)^N f^t_N | √(G)∇_V logf^t_N/f_t^⊗ N|^2  Z 
        + ,

where c_1, c_2 are two positive constants that depend on M, σ, γ, K and V. The concrete form of error terms will be given in Lemma <ref> and  Lemma <ref>. Roughly speaking, time derivative  of  relative entropy only provides us dissipation in the direction of velocity and time derivative of relative Fisher information provides us further dissipation in the direction of position. The reference measure of (<ref>) and (<ref>) could be more general. For insatnce, if we replace f_t^⊗ N by f_N,∞, we can show the convergence from f^t_N towards f_N,∞ as t →∞. 

Now we combine these two quantities, relative entropy and Fisher information, to form a new “modulated energy" as 

    ℰ^M_N(f^t_N|f̅_̅N̅) = H_N(f^t_N|f̅_̅N̅) + I^M_N(f^t_N|f̅_̅N̅),

where f̅_̅N̅ may take f_t^⊗ N or f_N,∞. The remaining argument is to appropriately control error terms in (<ref>). When f̅_̅N̅ = f_N,∞, linear structure of the Liouville equation (<ref>) makes error terms vanish. When f̅_̅N̅ = f_t^⊗ N, error terms in (<ref>) can be written as

    -2 ∫_(Ω×^d)^N f^t_N ∇R_N, M ∇logf^t_N/f_t^⊗ N Z

for suitable selection of M (See Corollary <ref>), here ∇ = (∇_X, ∇_V). However, the terms ∇_x ∇_v log f_t and ∇_v ∇_v log f_t are too difficult to control  to obtain some uniform-in-time estimates for (<ref>)  (See Remark <ref>). The key observation is that ∇_x ∇_v log f_t and ∇_v ∇_v log f_t are trivial terms if we replace f_t by its non-linear equilibrium, then we can show

    / tℰ^M_N(f^t_N|f_∞^⊗ N) ≤    - c'_1/N∫_(Ω×^d)^N f^t_N | ∇_X logf^t_N/f_∞^⊗ N|^2  Z 
        -  
    c'_2/N∫_(Ω×^d)^N f^t_N | ∇_V logf^t_N/f_∞^⊗ N|^2  Z 
        + C/N,

where c'_1, c'_2 > 0, C > 0 are some constants that only depend on M, σ, γ, K and V (See Theorem <ref>). To control the error between ℰ^M_N(f^t_N|f_t^⊗ N) and ℰ^M_N(f^t_N|f_∞^⊗ N), we use the convergence result from f_t to its equilibrium f_∞. There exist some results studying this type of convergence . In this article,  we adapt the arguments as  in <cit.> for K with small ∇ K _∞ and <cit.> for the case when K = - ∇ W and the  interaction energy is convex to establish the following estimate

    H(f_t|f_∞) ≤ Ce^-c'_3t,

where c'_3 > 0 depends on σ, γ, K and C > 0 depends on initial data f_0.
We give the sketch of proof of (<ref>) in Theorem <ref> and Theorem <ref> in the appendix. In the end, we combine (<ref>) with (<ref>) through 2-Wasserstein distance to conclude.    

Finally, we give some comments about terms ∇_x ∇_v log f_t and ∇_v ∇_v log f_t. Unfortunately we cannot directly obtain  uniform-in-time estimates about these terms, otherwise we would  have 

    / tℰ^M_N(f^t_N|f_t^⊗ N) ≤    - c'_1/N∫_(Ω×^d)^N f^t_N | ∇_X logf^t_N/f_t^⊗ N|^2   Z 
        -  
    c'_2/N∫_(Ω×^d)^N f^t_N | ∇_V logf^t_N/f_t^⊗ N|^2  Z 
        + C/N,

for some c_1', c_2', C > 0 and then finish the proof by Gronwall's  inequality. Uniform-in-time propagation of chaos can only be expected when the limiting PDE (<ref>) has a unique equilibrium. Indeed, we obtain Theorem <ref> under the assumptions in  Theorem <ref> and some assumption on W which prevents the existence of multiple equilibria of Eq.(<ref>). Similar arguments/estimates have also appeared in the  first-order setting. For instance, the authors of  <cit.>  obtained Li-Yau type growth estimates of ∇^2 logρ_t with respect to the position x for any fixed time horizon in particular in the case of 2d Navier-Stokes equation, with a universal constant growing with time t. Those type estimates are very useful when one treats the mean field limit problem set on the whole space <cit.>. 





 §.§ Main results and examples

Let us fix some notations first. We denote z = (x,v) ∈Ω×^d as a phase space configuration of one particle, and X = (x_1,...,x_N), V = (v_1,...,v_N) and Z= (X, V) for configurations in position, velocity  and phase space of N particles, respectively. We also use that z_i = (x_i,v_i) and Z = (X,V) = (z_1,...,z_N). As the same spirit, we write  operator ∇_X = (∇_x_1,..., ∇_x_N), ∇_V = (∇_v_1,..., ∇_v_N) and Δ_V = ∑_i=1^NΔ_v_i. Those are operators on ℝ^dN. 

For two probability measures μ, ν on Ω×^d,  we denote by Π(μ,ν) theset of all  couplings between μ and ν. We define L^2 Wasserstein distance as 

    𝒲_2(μ,ν) =  ( inf_π∈Π(μ,ν)∫_(Ω×^d)^2 |z_1 - z_2|^2 π(z_1,z_2) )^1/2.

For a measure μ and a positive integrable function f such that ∫_Ω×^d f|log f| μ < ∞, we define the  entropy of f with respect to  μ as

    Ent_μ(f) = ∫_Ω×^d f log f μ - ( ∫_Ω×^d f μ) log( ∫_Ω×^d f  μ).

For two probability measures μ and ν, we define the relative entropy between μ and ν as 

    H(μ|ν) = {   ∫_Ω×^d h log h  ν,     μ << ν  h = μ/ν, 
        + ∞,     , 
    .

and the normalized version of relative entropy for probability measures μ' and ν' on (Ω×^d)^N as 

    H_N(μ'|ν') = 1/NH(μ'|ν').

We also define relative Fisher information between μ and ν as

    I^M(μ|ν) = {   ∫_Ω×^d M ∇ h, ∇ h/hν,    μ << ν  h = μ/ν, 
       + ∞     otherwise,
    .

where M(z) is a positive definite matrix-valued function such that M(z) ≥κ I for some κ > 0 independent on z ∈Ω×^d. Similarly, the normalized version of relative Fisher information between two probability measures μ' and ν' on (Ω×^d)^N reads as

    I^M_N(μ'|ν') = 1/N I^M(μ'|ν').

We abuse the notation of probability measures f_N and f as probability densities if they have densities. Finally, we give definitions of some functional inequalities that we will use in the following.

(log-Sobolev inequality) We call that  the probability measure 
μ on Ω×^d satisfies the log-Sobolev inequality  if there exists some constant ρ_ls(μ) > 0 such that  for all smooth function g with ∫ g^2 μ = 1, it holds that 

    Ent_μ(g^2) ≤ρ_ls(μ) ∫_Ω×^d (|∇_x g|^2 + |∇_v g|^2) μ.



(Weighted log-Sobolev inequality) Let H(z) be a smooth function on ^d ×^d. We call that  the probability measure μ on ^d ×^d satisfies the weighted log-Sobolev inequality with weight H if there exists some constant θ > 0, ρ_wls(μ) > 0  such that for all smooth function g with ∫ g^2 μ = 1, it holds that 

    Ent_μ(g^2) ≤ρ_wls(μ) ∫_^d ×^d  (H^-2θ|∇_x g|^2 + |∇_v g|^2)  μ.
 

In the following, we use ρ_ls (ρ_wls) to denote the (weighted) log-Sobolev inequality constant of the equilibrium  measure  f_∞ of the limiting PDE (<ref>) and  ρ_LS to denote the uniform-in-N log-Sobolev inequality constant of the stationary measure  f_N,∞ of the Liouville equation (<ref>). 


Now we detail assumptions about interaction potential W and confining potential V.


Suppose that  V(x) ∈ C^2(Ω) and there exist  λ > 0, M > 0 such that V(x) ≥λ |x|^2 - M.

The first condition means that V goes to infinity at infinity and is bounded below. It can be implied by

    1/2∇ V (x) · x ≥ 6λ (V(x) + x^2/2) - A,   x ∈Ω

for some A > 0. This expression implies that the force -∇ V drags particles back to some compact set. Detailed proof can be found in <cit.>. 


The second assumption implies  that  the potential V grows at most quadratically  on Ω. 


Suppose that V(x) ∈ C^2(Ω) and there exists C_V > 0 such that ∇^2 V _L^∞≤ C_V < ∞.




We also treat more general confining potentials when Ω = ^d.




Suppose that V(x) ∈ C^2(^d) and there exist θ > 0 and C_V^θ such that V^-2θ∇^2 V_L^∞≤ C_V^θ < ∞. Moreover, outside a compact domain on ^d, we assume that V satisfies  

(i) Δ V ≤κ_1 |∇ V|^2 for some κ_1 ∈ (0,1); 

(ii) |∇ V|^2 ≥κ_2 V^2θ + 1 for some positive constant κ_2 > 0.



Those conditions in Assumption 3  have been explored in <cit.> for kinetic Langevin process with confining potentials greater than quadratic growth at infinity. The boundedness of  V^-2θ∇^2 V_L^∞ extend the quadratic growth condition and the other two conditions guarantee the weighted log-Sobolev inequality of f_∞ (See Section <ref>). We also use multiplier method developed in <cit.> to deal with this type of confining potentials.


Some important examples have been provided in <cit.>. The first kind of examples is V(x) = |x|^k, k ≥ 2. Then we have Δ_x V = (dk + k^2 - 2k)|x|^k-2, |∇_x V|^2 = k^2|x|^2k-2 and V^-2θ∇^2 V _L^∞∼ |x|^k-2kθ - 2. Finally,  we take θ = 1/2 - 1/k. Then all conditions above can be satisfied. 



Another kind of examples is V(x) = e^a|x|^k provided in <cit.>, which shows that the limit growth of V must be below the exponential growth. Observing that Δ_x V ∼ a k^2|x|^2(k-1)e^a|x|^k, |∇_x V|^2 = a^2k^2e^2a|x|^k and V^-2θ∇^2 V _L^∞∼ e^a(1-2θ)|x|^k, the conditions above imply k < 1 and θ = 1/2.





Suppose that W(x) ∈ C^2(Ω) and there exists 0 < C_K < 1/2λ such that ∇^2 W_L^∞≤ C_K < ∞. 



The mean field functional or the interaction energy  F: 𝒫_2(^d) → defined as

    F(ρ) = ∫_Ω×Ω W(x-y) ρ(x) ρ(y)

is functional convex, i.e. for every t ∈ [0,1] and every ν_1, ν_2 ∈𝒫_2(^d),

    F((1-t)ν_1 + tν_2) ≤ (1-t)F(ν_1) + t F(ν_2).




The harmonic interaction potential 

    W(x) = L_W/2|x|^2
 
with L_W ≤λ/2 is covered by Assumption <ref>. The mollified Coulomb potential when d=3,

    W(x) = a/(|x|^k + b^k)^1/k   or   W(x) = arctan(|x|/r_0) 1/|x|

with some constant a, b>0 or r_0 > 0. The later form satisfies Assumption <ref> (See Section 3 in <cit.>).



The harmonic interaction potential does not satisfy Assumption <ref>.
Let us take 

    F(ρ) = ∫_Ω×Ω (x-y)^2 ρ(x) ρ(y),

then for t ∈ [0,1],

    F((1-t)ν_1 + tν_2) ≤ (1-t)F(ν_1) + t F(ν_2) 
    ⟺   ∫_Ω×Ω (x-y)^2 (ν_1 - ν_2)^⊗ 2(x,y) ≥ 0 
    ⟺    2 (∫_Ω x^2 ν_1 ) (∫_Ω x^2 ν_2 ) ≥(∫_Ω x^2 ν_1 )^2 + (∫_Ω x^2 ν_2 )^2,

which does not hold in general.


With those specific statement of assumptions as above, we can now state our main results.  

Suppose that V satisfies Assumption <ref> and <ref>, and W satisfies Assumption <ref> with C_K < 1. Then for initial data f_N^0 of the Liouville equation (<ref>) 
such that  ℰ^M_N(f^0_N|f_N,∞) < ∞, we have

    ℰ^M_N(f^t_N|f_N,∞) ≤ e^-ctℰ^M_N(f^0_N|f_N,∞),

where c = c(f_N,∞, M) > 0 is explicit and independent of N.


 We can choose matrix M as the following

    M = 
    (
    [ E F; F G ]),

where

    E = diag{δ a^3,...,δ a^3},    F = diag{δ a^2,...,δ a^2},    G = diag{2δ a,...,2δ a},

and two constants a and δ satisfy 

    a ≤2γ/C_K + C_V,    δ≤σ/2(4 + 8a γ)^2.

Then the constant c can be taken as c = 1/2(1 + ρ_LS)min{3/2δ a^2, σ/2}. By the selection of δ, we observe that c ∼σ, i.e. the larger diffusion strength we have, the faster convergence from f_N^t to f_N,∞. 


Our second contribution is the uniform-in-N exponential convergence from f_N to f_∞^⊗ N.

Suppose that  V and W satisfy one of the following two cases,

(i) V satisfies Assumption <ref> and <ref>, W satisfies Assumption <ref>;

(ii) V satisfies Assumption <ref> and <ref>, W satisfies Assumption <ref> and ∇ W _L^∞ < ∞.

For the first case, we take M_1 as a constant matrix, then for initial data f_N^0 of Eq.(<ref>) such that ℰ_N^M_1(f^0_N|f^⊗ N_∞) < ∞ and σ≥σ^∗_1 > 0, we have

    H_N(f^t_N|f_∞^⊗ N) ≤ℰ_N^M_1(f^t_N|f^⊗ N_∞) ≤ e^-c_1tℰ_N^M_1(f^0_N|f^⊗ N_∞) + C_1/N,

where c_1 = c_1(f_∞, M_1) > 0 and C_1 = C_1(f_∞, σ, γ, C_K, C_V) > 0 are explicit and independent of N. 

For the second case, we take M_2 as a matrix function on ^2d, then for initial data f_N^0 of Eq.(<ref>) such that ℰ_N^M_2(f^0_N|f^⊗ N_∞) < ∞ and σ≥σ^∗_2 > 0, we have

    H_N(f^t_N|f_∞^⊗ N) ≤ℰ_N^M_2(f^t_N|f^⊗ N_∞) ≤ e^-c_2tℰ_N^M_2(f^0_N|f^⊗ N_∞) + C_2/N,

where c_2 = c_2(f_∞, M_2), C_2 = C_2(f_∞, σ, γ, C_K, C^θ_V) > 0 are explicit and independent of N.




For the first case, we also choose M_1 as in  Remark <ref>, but two constants a and δ now should satisfy

    {    a ≤min{2 γ/C_K + C_V, 1/4C_K + 2,  γ/5120 eρ_ls (C_K+1)^2}, 
       δ≤σ/4[8 + a + 28aγ + 32a^2 γ^2],
    .

then the constant c_1 can be taken as c_1 = δ a^2/16(ρ_ls + 1) which implies c_1 ∼σ, and the lower bound of diffusion constant σ^∗_1 satisfies 

    σ_1^∗≥max{160[10 + 28 γ + 32 γ^2]ρ_lse/a^2 γ, 3200ρ_lseγ}C_K^2.

For the second case, we choose M_2 as

    {    E = diag{e(z_1)Id_d × d,...,e(z_N)Id_d × d}, 
        F = diag{f(z_1)Id_d × d,...,f(z_N)Id_d × d}, 
        G = diag{g(z_1)Id_d × d,...,g(z_N)Id_d × d}, 
    .

where E,F,G are Nd × Nd diagonal matrices. We choose e(z), f(z) and g(z) as

    e(z) = δ a^3(H(z))^-3θ,  
    b(z) = δ a^2(H(z))^-2θ,   c(z) = 2 δ a(H(z))^-θ,

where 

    H(z) = v^2/2 + V(x) + H_0,    H_0 > 0,

and two constants a and δ satisfy

    {    a ≤min{1/4C_K + 6θ + 2, γ/C_V^θ + C_K, γ/6400 eρ_wls (C_K+1)^2}, 
       δ≤3 σ/8 + 32C_K + m_2',
    .

where m_2' = [4 + 6γ a + 4aθ(2γ + ∇ W_L^∞)]^2 + a[6γ + θ(2γ + ∇ W_L^∞)], then the constant c_2 can be taken as c_2 = δ a^2/16(ρ_wls + 1), which implies c_2 ∼σ, and the lower bound of diffusion constant σ^∗_2 satisfies 

    σ_2^∗≥max{800(40+m_2”)ρ_wlse/a^2 γ, 3200ρ_wlseγ}·max{C_K^2, C_K^3},

where m_2” = [4 + 6γ + 4θ(2γ + ∇ W_L^∞)]^2 + [6γ + θ(2γ + ∇ W_L^∞)].


Theorem <ref> implies that second order particle system (<ref>) not only exponentially converges to its equilibrium, but also converges to the unique mean field equilibrium as N →∞. If we take t →∞, the results (<ref>) and (<ref>) imply that, 

    H_N(f_N,∞|f^⊗ N_∞) ≤ℰ_N^M_i(f_N,∞|f^⊗ N_∞) ≤C/N,   i =1,2,

which offers us a kind of dynamical approach to prove the concentration of the Gibbs or stationary measure of the second order particle system around the nonlinear equilibrium of the limiting equation (<ref>). 


Combining the exponential convergence from f_t to f_∞, we could replace f_∞ by f_t in last theorem so that avoid the estimates about ∇log f_t and ∇^2 log f_t. Based on this observation, we establish the uniform-in-time propagation of chaos  both in the sense of the Wasserstein distance and  relative entropy.


Suppose that V and W satisfy one of the following two cases,

(i) V satisfies Assumption <ref> and <ref>, W satisfies Assumption <ref>. Moreover, we assume either C_K is small or interaction functional F satisfies Assumption <ref>.

(ii) V satisfies Assumption <ref> and <ref>, W satisfies Assumption <ref> and ∇ W _L^∞ < ∞. Moreover, we assume either C_K is small or interaction functional F satisfies Assumption <ref>.

For the first case, for initial data f_N^0 of Eq.(<ref>) such that ℰ_N^M_1(f^0_N|f^⊗ N_∞) < ∞, f^0 of Eq.(<ref>) such that ℰ^M(f_0|f̂_0) < ∞, and σ≥σ_1^∗, we have

    𝒲_2^2(f_N,k, f^⊗ k) ≤ C_1'k e^-c'_1t + C_1 k/N,

where c'_1 = min{c,c_1} > 0 and C'_1 = C'_1(f_N^0, f_0, f_∞, ρ_LS) > 0 are explicit and independent of N. 

For the second case, for initial data f_N^0 of Eq.(<ref>) such that ℰ_N^M_2(f^0_N|f^⊗ N_∞) < ∞, f^0 of Eq.(<ref>) such that ℰ^M_2'(f_0|f̂_0) < ∞, and σ≥σ_2^∗, we have

    𝒲_2^2(f_N,k, f^⊗ k) ≤ C_2'ke^-c'_2t + C_2k/N,

where c'_2 = min{c_2, c_2”} > 0 and  C'_2 = C'_2(f_N^0, f_0, f_∞) > 0 are explicit and independent of N.


The constants C_1' and C_2' are taken as 

    C_1'     = (1+ρ_LS)[ℰ_N^M_1(f^0_N|f^⊗ N_∞) + ℰ^M(f_0|f̂_0)], 
    
    C_2'     = ℰ_N^M_2(f^0_N|f^⊗ N_∞) + ℰ^M'_2(f_0|f̂_0),

and c_2”, M_2' can be found in Theorem <ref>.  



The main difference of assumptions in Theorem <ref> compared with  those in Theorem <ref> is that C_K is small or the  interaction functional F is convex. Those  two  conditions make sure that f_t exponentially converges to f_∞. The small condition of C_K
comes from Theorem 10 in <cit.>, which obtains the exponential convergence from f_t to f_∞ by uniform log Sobolev inequality of f_N,∞. The convexity condition of F is inspired by Theorem 2.1 in <cit.>, this kind of condition avoid the smallness assumption on  C_K. We extend their result to more general confining potentials in the Appendix.



The uniform-in-time propagation of chaos of second order particle system (<ref>) has been investigated in  <cit.> and <cit.>. Compared with <cit.> that exploits the coupling method, our result applies to more general confining potentials. In terms of Theorem 2.3 in <cit.>, we do not need the uniform-in-m log-Sobolev inequality of measure proportional to e^-δ F/δ m(m,x), which is not very easy to verify. 





 §.§ Related literature


Hypocoercivity.  Hypocoercivity is an important analytical tool to study the long time behavior of Langevin dynamics and the corresponding kinetic Fokker-Planck equation. It was  initiated by Villani <cit.> and and then later advanced by Dolbeault, Mouhot and Schmeiser  in <cit.> and <cit.>. However, those now well-known results are only restricted to the one particle dynamics without any interactions. For the N particle system given by the Liouville equation (<ref>), the natural stationary measure is simply the Gibbs measure given by the following form  

    f_N,∞ = 1/Z_N,β e^-β H(z_1,...,z_N).

A natural problem is that whether or not the convergence rate  from f_N toward f_N,∞ depends on the number of particles N. Many researchers have contributed to this problem. Guillin, etc study the uniform in N functional inequalities in <cit.>. Guillin and Monmarché show uniform-in-N exponential decay rate in <cit.> and <cit.> by “Generalized Γ calculus" developed in <cit.> and uniform log-Sobolev inequality in <cit.>. Guillin, etc also use H^1 type norm to show the uniform-in-N exponential decay rate by hypocoercivity and uniform Poincaré inequality in <cit.>. These result are all restricted to potentials with smallness of ∇^2 U _L^∞. There are also some results that treat systems with singular potentials. Baudoin, Gordina and Herzog showed convergence to equilibrium by Gamma calculus in <cit.> with singular potentials. Lu and Mattingly constructed new Lyapunov function to show egodicity for systems (<ref>) with Coulomb potential in the sense of weighted total variation distance in <cit.>. However, the convergence rates, if they provides one,   all depend on N. 

 


Propagation  of  chaos  for  kinetic Vlasov equation.   The main result presented in this article is a further development of the relative entropy method introduced in <cit.>, where Jabin and and the 2nd author proved a quantitative propagation of chaos for Newton's system with  bounded  interaction kernel in terms of relative entropy.  Lacker  <cit.> then developed an approach based on the  BBGKY hierarchy and the entropy dissipation to optimize  the local convergence rate of k-marginals towards the limiting law.   Bresch, Jabin and Soler  <cit.> exploited the BBGKY hierarchy approach to firstly include the 2d Vlasov-Possion-Fokker-Planck case.  More recently, Bresch, Jabin and Duerinckx <cit.> introduced a duality approach to cover the arbitrary square-integrable interaction forces at possibly vanishing temperature. Up to now, the  mean field limit or the propagation of chaos results are still very limited for second order particle system with singular interaction forces. See also for the results in <cit.> and the review <cit.> for more detailed discussions. 

For long time propagation of chaos, Monmarché showed   uniform-in-time propagation of chaos in the sense of Wasserstein distance of one marginal for systems with convex potentials, i.e.

    𝒲_2(f_N,1,f) ≤C/N^α,

for some contant α > 0, C > 0 independent on N and t. The sharp rate with α = 1/2 for the case W(x) = c|x|^2 has also been established there. Guillin and Monmarché <cit.> later improved the convergence result to all marginals but without optimality in terms of of α, i.e.

    𝒲_2(f_N,k,f^⊗ k) ≤C √(k)/N^α,

for some contant α > 0, C > 0 independent on N and t. Thanks to the reflection coupling method, Guillin, Bris and Monmarché  <cit.> proved the optimal convergence rate of N for all marginals with convex or non-convex interaction potentials, i.e.

    𝒲_2(f_N,k,f^⊗ k) ≤C √(k)/√(N),

for some constant C > 0 independent on N and t, with the smallness assumption of the Lipschitz constant of interaction force K.  Recently, Chen, Lin, Ren and Wang <cit.> showed  uniform-in-time propagation of chaos with functional convexity condition. Even though they do not  need smallness of ∇ K_L^∞ , they require some  uniform-in-time Poincaré inequality for  the solution of the limiting PDE (<ref>) . To the best of our knowledge, there is no result of uniform-in-time propagation of chaos for second order systems with singular interaction forces yet. We leave this topic for our further study.  



Equilibrium of VlasovFokker Planck equation. Uniform-in-time propagation of chaos cannot hold in general. One critical obstacle is that the non-linear Vlasov-Fokker-Planck equation (<ref>) may have multiple equilibria and  hence exhibit phrase transition. The convergence from f_t towards f_∞ prevents the phrase transition or the presence of multiple equilibria of the limiting system. There are some results about this kind of convergence but with very limited conditions about potentials. See for instance <cit.> and the reference therein.  Villani <cit.> proved that  f_t converge to the Maxwellian

    1/(2π)^d e^-βv^2/2

on 𝕋^d with any polynomial order in the sense of L^1 norm, which requires that W ∈ C^∞ and W _L^∞ is small enough. Guillin and Monmarché showed that  f converges to f_∞ in the sense of “mean-field entropy" in <cit.>, which defines as 

    H_W(ν) = E(ν) - inf_μ∈𝒫(Ω×^d) E(μ)

for probability measure ν,  where
E(ν) = H(ν|α) + 1/2 F(ν) and α∝ e^-1/2v^2 -V(x). Baudoin, Feng and Li <cit.> established  that f converges
to f_∞ with exponential decay rate in the sense of free energy combining with “relative Fisher Information" (by our notation)

    ℰ^M(f_t|f̂_t) = ℱ(f_t) - ℱ(f_∞) + I^M(f_t|f̂_t),

where M is a constant matrix and f̂∝ e^-v^2/2 - V(x) - W ∗ρ_t. The free energy they used is  defined as 

    ℱ(f) = 1/2∫_Ω×^d v^2 f  x  v + ∫_Ω×^d f log f  x  v + ∫_Ω×^d Vf  x  v + 1/2 F(f).

They used Γ-calculus to overcome the dissipation degeneracy in x direction with convexity and smallness of ∇^2 V and ∇^2 W.  Chen, Lin, Ren and Wang also exploited the quantity (<ref>) to prove ℰ^M(f_t|f̂_t) exponentially converges to 0 in <cit.> under conditions ∇^2 (W ∗ρ_t + V)_L^∞ < ∞ and the functional convexity of  F (Assumption <ref>). These two groups both used  the so called free energy  to quantify the convergence from f_t to f_∞, i.e.

    H_W(f_t) = ℱ(f_t) - ℱ(f_∞).

Finally, let us recall the convergence result in <cit.> and extend Theorem 2.1 in <cit.> to more general confining potentials. By <cit.>, we have

Suppose that  V satisfies Assumption <ref> and <ref>, W satisfies Assumption <ref>. Suppose f_N,∞ satisfies uniform log Sobolev inequality with constant ρ_LS. Then for the solution f_t of Eq.(<ref>) with initial data f_0 such that H_W(f_0) < ∞ and ∫_Ω×^d z^2 f_0(z)  
z < ∞, we have

    H_W(f_t) ≤ e^-ctH_W(f_0),
 
and 

    𝒲_2^2(f_t, f_∞) ≤ρ_LS H_W(f_t) ≤ρ_LSℰ^M(f_0|f̂_0) e^-ct.

where c>0 is the same as Theorem <ref>.

We extend Theorem 2.1 in <cit.> to more general confining potentials,

Suppose that  V satisfies Assumption <ref> and <ref>, W satisfies Assumption <ref> and Assumption <ref>. Then for solution f_t of Eq.(<ref>) with initial data f_0 such that ℰ^M_2'(f_0|f̂_0) < ∞, we have

    𝒲_2^2(f_t, f_∞) ≤ℰ^M(f_t|f̂_t) ≤ e^-c”_2tℰ^M(f_0|f̂_0),

where c”_2 = δ a^2/16 + 16ρ_wls with some choice of M_2', δ and a (detailed in the Appendix).


We will give the sketch of proofs of these two theorems in the Appendix.




 §.§ Outline of the article

The paper is then organized as follow: In Section <ref>, we develop the basic tools we will use throughout this article. In Section 2.1, we introduce the normalized relative Fisher information and compute its  time evolution  under the kinetic dynamics (<ref>) and (<ref>).  In Section 2.2, we select the nontrivial matrix M for relative Fisher information to deal with the confining potentials greater than a quadratic  function at infinity, where the crucial idea  “entropy multipliers" is inspired by  the one particle case as   in <cit.>. In Section 2.3, we introduce the weighted log-Sobolev inequality, which is essentially obtained  with the entropy multiplier method. In Section 2.4, we prove a new Law of Large Number estimates for systems with Lipschitz interaction force K. In Section 3, we give the complete proof of our main results, Theorem <ref>, <ref> and <ref>. In the  Appendix, we prove the convergence from  f to f_∞ under some conditions on V and W. 






§ PRELIMINARY

Let us define some linear operators in (Ω×^d)^N we will use in this section. We denote A = (0, ∇_V) on (Ω×^d)^N and A_i = (0, ∇_v_i) on Ω×^d. The operator B collects all of one order part of the Liouville operator L_N in (<ref>), i.e.

    B = ∑_i=1^N B_i,   B_i = v_i ·∇_x_i - ∇_x_iU ·∇_v_i - γ v_i ·∇_v_i,

where U is defined  in  (<ref>). We write  the infinitesimal generator of N-particle system (<ref>) as 

    L^∗_N = B + σΔ_V.


We recall the time evolution of the relative entropy as in <cit.>. 

Assume that f_N is a solution of Eq.(<ref>). Assume further that f(t,z) ∈ W^1, ∞(×Ω×^d) solves Eq.(<ref>) with f(t,z) > 0 and ∫_Ω×^d f(t,z)dz = 1. Then

    d/dt H_N(f_N|f̅_̅N̅) =     - σ/N∫_(Ω×^d)^N f_N | ∇_V logf_N/f̅_̅N̅|^2  Z 
        - 1/N∫_(Ω×^d)^N∇_V logf_N/f̅_̅N̅· R^0_N  Z,

where R^0_N is a dN-dimensional vector defined as 

    R^0_N = {1/N∑_j=1,j≠ i^N K(x_i - x_j) - K ∗ρ_t(x_i) }_i=1^N

if we take f̅_̅N̅ = f^⊗ N(t,Z), and the second line of (<ref>) vanishes if we take f̅_̅N̅ = f_N,∞.



The computation is standard. We recommend <cit.> and <cit.> for the detailed computation when f_N = f_N,∞ and f^⊗ N. Using Young inequality when f_N = f^⊗ N, we also have

    d/dt H_N(f_N|f̅_̅N̅) ≤    - 3 σ/41/N∫_(Ω×^d)^N f_N | ∇_V logf_N/f̅_̅N̅|^2  Z 
        + 4/σ1/N∫_(Ω×^d)^N |R^0_N|^2   Z.




In the next, we turn to the argument about Fisher information.



 §.§ Hypocoercivity in entropy sense


In this subsection, we extend hypocoercivity in entropy sense in <cit.> to N particle system with nontrivial interaction force. We also use more general reference measure in (Ω×^d)^N —– invariant measure f_N,∞ or N times tensor product of limiting measure f^⊗ N, corresponding to uniform egodicity problem and uniform-in-time propagation of chaos problem.

In the following, we use notation h = f_N/f̅_̅N̅ and u = log h for convenience, f̅_̅N̅ may take f_N,∞ or f^⊗ N.
Before tedious manipulations, we firstly derive the equation of u.


Assume that f_N is a solution of Eq.(<ref>), and assume that f(t,z) ∈ W^2, ∞(×Ω×^d) solves Eq.(<ref>), then

    ∂_t u = -Bu - σ A^∗Au + σ∇_V log f_N · A u - R_N,

where R_N takes

    R_N = ∑_i=1^N∇_v_ilog f(x_i,v_i) ·{1/N∑_j=1, j≠ i^N K(x_i-x_j) - K⋆ρ(x_i) },

if f̅_̅N̅ = f^⊗ N and R_N = 0 if f̅_̅N̅ = f_N,∞.


The proof is direct computation. In terms of  Eq.(<ref>), we have

    ∂_t log f_N  =  -B log f_N + σΔ_V f_N/f_N + γ Nd,

and for Eq.(<ref>), we have

    ∂_t logf̅_̅N̅ = -B logf̅_̅N̅ + σΔ_V f̅_̅N̅/f̅_̅N̅ + γ Nd + R_N,

we could understand R_N as the difference of drift part between particle system (<ref>) and McKean-Vlasov system (<ref>). Combine Eq.(<ref>) and Eq.(<ref>), we have

    ∂_t u = -Bu   + σ{Δ_V f_N/f_N - Δ_V f̅_̅N̅/f̅_̅N̅} - R_N,

using identity 

    Δ_V F/F = Δ_V log F + |∇_V log F|^2,

therefore, u satisfies the equation

    ∂_t u = -Bu + σΔ_V u + σ∇_V log(f_N f̅_̅N̅) ·∇_V u - R_N,

recall A = (0,∇_V), now we regard f̅_̅N̅ as reference measure and use Proposition 3 of <cit.>, for vector function g: (Ω×^d)^N →^Nd, we have

    A^∗g = - A · g - ∇_V logf̅_̅N̅, g ,

now we rewrite Eq.(<ref>) as following

    ∂_t u = -Bu - σ A^∗Au + σ∇_V log f_N · A u - R_N,

we complete the proof.


For f̅_̅N̅ = f^⊗ N, the conjugation relationship (<ref>) is in a series of Hilbert space L^2(f̅_̅N̅) depends on time t whose associated measure satisfy Eq.(<ref>). If we take reference measure as equilibrium of Eq.(<ref>), then the equation about u is the same as (<ref>) but R_N = 0: 

    ∂_t u = -Bu - σ A^∗Au + σ∇_V log f_N · A u,

this is why we start our computation from Eq.(<ref>).


Now let us compute the time derivation of relative Fisher Information. We omit the integration domain (Ω×^d)^N for convenience.


Assume that f_N is a solution of Eq.(<ref>). Assume that f(t,z) ∈ W^2, ∞(×Ω×^d) solves Eq.(<ref>) with f(t,z) > 0 and ∫_Ω×^d f(t,z)dz = 1. Let B, C, C' be linear differential operators on (Ω×^d)^N, where

    B = ∑_i=1^N B_i,   B_i = v_i ·∇_x_i - ∇_x_iU ·∇_v_i - γ v_i ·∇_v_i,

and C, C' are to be confirmed, then


    d/dt{∫ f_N ⟨ Cu, C'u⟩} =     ∫ f_N ⟨ [B, C]u, C'u ⟩ + ∫ f_N ⟨ Cu, [B, C']u ⟩
        - ∫ f_N ⟨  C R_N, C'u ⟩  - ∫ f_N ⟨  Cu, C'R_N ⟩
        - 2 σ∫ f_N  CAu, C'Au  + σ Q_C,A + σ Q_C',A,

where

    Q_C,A =     -∫ f_N  [C,A^∗]Au, C'u 
    - ∫ f_N  [∇_V logf̅_̅N̅· A, C]u, C'u 
        -∫ f_N  [C,A]u, Au ⊗ C'u  - ∫ f_N  [A,C]u, C'Au ,



    Q_C',A =     -∫ f_N  Cu, [C',A^∗]Au - ∫ f_N  Cu, [∇_V logf̅_̅N̅· A, C']u 
        - ∫ f_N  Au ⊗ Cu, [C',A]u - ∫ f_N  CAu, [A, C']u .



B is a d-tuple differential operator, but C, C' are 2Nd-tuple differential operators. We denote C_i and C_i' as (C_i_j)_1 ≤ i ≤ N, 1 ≤ j ≤ d and (C'_i_j)_1 ≤ i ≤ N, 1 ≤ j ≤ d in the sense of coordinate z_j = (x_j,v_j). We omit the index j for convenience in the following, i.e. C = (C_i)_1 ≤ i ≤ N and C' = (C'_i)_1 ≤ i ≤ N. Each of them can be identified with a vector field D_i, in such a way that C_i f = D_i ·∇ f, so D can be seen as a map valued in (2Nd × 2Nd) matrix. The inner products above should be understood as Cu, C'u = ∑_i=1^N  C_i u, C_i'u _^2d, CAu, C'Au = ∑_i,j=1^N  C_i A_ju, C_i'A_ju _^2d × 2d.



Let us explain the commutators we use. [B,C] is a 2Nd-tuple operator, understood as [B,C]_i_j = [B, C_i_j]. But [C,A] is a operator with 2Nd × 2Nd components, understood as [C,A]_ij = [C_i,A_j], and others follow. If C, C' are commutative with A, the only nontrivial operators are [C,A^∗] and [∇_V logf̅_̅N̅· A, C], we will compute them later.  




Step1. In this step, we claim

    d/dt∫ f_N  Cu, C'u  =    ∫ f_N ⟨ [B, C]u, C'u ⟩ + ∫ f_N ⟨ Cu, [B, C']u ⟩
        -  ∫ f_N ⟨  C R_N, C'u ⟩  - ∫ f_N ⟨  Cu, C'R_N ⟩ + σ Q_σ,

here Q_σ collects all terms with coefficient σ and reads as

    Q_σ =    ∫ f_N Δ_V  Cu, C'u 
        - ∫ f_N  CA^∗A u, C'u + ∫ f_N  C(∇_V log f_N · Au), C'u
        - ∫ f_N  Cu, C'A^∗A u + ∫ f_N  Cu, C'(∇_V log f_N · Au).

There terms comes from diffusion part of Eq.(<ref>) and Eq.(<ref>), we will deal with them in next step.

We directly take derivative and split into three terms:

    d/dt∫ f_N ⟨ Cu, C'u⟩  =    ∫∂_t f_N ⟨ Cu, C'u⟩
        + ∫ f_N ⟨ C(∂_t u), C'u ⟩ + ∫ f_N ⟨ Cu, C'(∂_t u)⟩.

For the first term, we use Eq.(<ref>) and integral by parts,

    ∫∂_t f_N ⟨ Cu, C'u ⟩ =    ∫ -L_N f_N ⟨ Cu, C'u⟩
     =    ∫ f_N (B ⟨ Cu, C'u⟩) + σ∫ f_N (Δ_V ⟨ Cu, C'u⟩) 
     =    ∫ f_N  BCu, C'u + ∫ f_N  Cu, BC'u + σ∫ f_N (Δ_V ⟨ Cu, C'u⟩).

For second term,

    ∫ f_N ⟨ C(∂_t u), C'u ⟩ =     - ∫ f_N ⟨ C(B u), C'u ⟩ - ∫ f_N  CR_N, C'u 
        - σ∫ f_N  CA^∗A u, C'u + σ∫ f_N  C(∇_V log f_N · Au), C'u.

Similarly, for third term,

    ∫ f_N ⟨ Cu, C'(∂_t u) ⟩ =     - ∫ f_N ⟨ Cu, C'(Bu) ⟩ - ∫ f_N  Cu, C'R_N 
        - σ∫ f_N  Cu, C'A^∗A u + σ∫ f_N  Cu, C'(∇_V log f_N · Au).

Combine (<ref>), (<ref>), (<ref>), we complete the claim in this step.



Step2. In this step, we deal with the diffusion part, we claim 


    Q_σ =
    - 2∫ f_N  CAu, C'Au  + Q_C,A + Q_C',A,

where

    Q_C,A =     -∫ f_N  [C,A]u, Au ⊗ C'u  - ∫ f_N  [A,C]u, C'Au 
        -∫ f_N  [C,A^∗]Au, C'u 
    - ∫ f_N  [∇_V logf̅_̅N̅· A, C]u, C'u ,



    Q_C',A =     - ∫ f_N  Au ⊗ Cu, [C',A]u - ∫ f_N  CAu, [A, C']u 
        -∫ f_N  Cu, [C',A^∗]Au - ∫ f_N  Cu, [∇_V logf̅_̅N̅· A, C']u .

The terms Q_C,A and Q_C',A collect all terms including commutators, we will take suitable operators C and C' to simplify these commutators.

For the first term of (<ref>), we use integral by parts with respect to Lebesgue measure,   

    ∫Δ_V f_N  Cu, C'u 
     =     - ∫ f_N A u  · A  Cu, C'u  - ∫ f_N ∇_V logf̅_̅N̅· A  Cu, C'u.

Recall we denote A = (0,∇_V), we continue the first term of (<ref>)

    - ∫ f_N A u  · A  Cu, C'u 
     =     -∫ f_N  ACu, Au ⊗ C'u  - ∫ f_N  Au ⊗ Cu, AC'u
     =     -∫ f_N  CAu, Au ⊗ C'u  - ∫ f_N  Au ⊗ Cu, C'Au
        -∫ f_N  [C,A]u, Au ⊗ C'u  - ∫ f_N  Au ⊗ Cu, [C',A]u.

Let us explain the notation we use: ACu, Au ⊗ C'u should be understood as 

    ∑_i,j=1^N C_iA_ju,  A_ju, C'_i u,
 
and 

    [C,A]u, Au ⊗ C'u  = ∑_i,j=1^N  [C_i,A_j]u,  A_ju, C'_iu .

Moreover, we take the conjugate operator of [C,A] w.r.t measure f̅_̅N̅, we have

    -∫ f_N  [C,A]u, Au ⊗ C'u  =      - ∫ f_N  [C,A]^∗ Au ⊗ C'u 
     =     - ∑_i,j=1^N ∫ f_N  [C_i,A_j]^∗ A_ju, C'_iu .

For the second term of (<ref>),
we rewrite it as 

    - ∫ f_N  CA^∗A u, C'u
     =     - ∫ f_N  A^∗CAu, C'u - ∫ f_N  [C,A^∗]Au, C'u,

and

    - ∫ f_N  A^∗CAu, C'u =     - ∫f̅_̅N̅ A^∗CAu, C'h
     =     - ∫f̅_̅N̅ CAu, AC'h
     =     - ∫f̅_̅N̅ CAu, AhC'u
     =     - ∫ f_N  CAu, AC'u  - ∫ f_N  CAu, Au ⊗ C'u,

then we have

    - ∫ f_N  CA^∗A u, C'u =     - ∫ f_N  CAu, C'Au  - ∫ f_N  CAu, [A, C']u 
        -∫ f_N  [C,A^∗]Au, C'u - ∫ f_N  CAu, Au ⊗ C'u.

Similarly, for the fourth term of (<ref>), up to the exchange of C and C', we have

    - ∫ f_N  Cu, C'A^∗Au =     - ∫ f_N  CAu, C'Au  - ∫ f_N  [A,C]u, C'Au 
        -∫ f_N  Cu, [C',A^∗]Au - ∫ f_N  Au ⊗ Cu, C'Au.

Finally, all of left terms are third and fifth term in (<ref>) and underlined term in (<ref>), we collect them as below, 

    - ∫ f_N ∇_V logf̅_̅N̅· A  Cu, C'u
        + ∫ f_N  C(∇_V log f_N · Au), C'u
        + ∫ f_N  Cu, C'(∇_V log f_N · Au).

We deal with the first term of (<ref>) as following,

    - ∫ f_N ∇_V logf̅_̅N̅· A  Cu, C'u
     =     - ∫ f_N ∇_V logf̅_̅N̅· A (Cu), C'u - ∫ f_N  Cu, ∇_V logf̅_̅N̅· A (C'u)
     =     - ∫ f_N  C(∇_V logf̅_̅N̅· Au), C'u - ∫ f_N  [∇_V logf̅_̅N̅· A, C]u, C'u 
        - ∫ f_N  Cu, C'(∇_V logf̅_̅N̅· A u) - ∫ f_N  Cu, [∇_V logf̅_̅N̅· A, C']u ,

combine with the last two term of (<ref>), we have

    - ∫ f_N ∇_V logf̅_̅N̅· A  Cu, C'u
        + ∫ f_N  C(∇_V log f_N · Au), C'u
        + ∫ f_N  Cu, C'( ∇_V log f_N · Au)
     =    ∫ f_N  C|Au|^2, C'u + ∫ f_N  Cu, C'|Au|^2
        - ∫ f_N  [∇_V logf̅_̅N̅· A, C]u, C'u  - ∫ f_N  Cu, [∇_V logf̅_̅N̅· A, C']u 
     =     2∫ f_N  CAu, Au ⊗ C'u + 2∫ f_N  Au ⊗ Cu, C'Au
        - ∫ f_N  [∇_V logf̅_̅N̅· A, C]u, C'u  - ∫ f_N  Cu, [∇_V logf̅_̅N̅· A, C']u .

After gathering (<ref>), (<ref>), (<ref>) and (<ref>), we find

    - 2∫ f_N  CAu, C'Au 
       -∫ f_N  [C,A]u, Au ⊗ C'u  - ∫ f_N  Au ⊗ Cu, [C',A]u
        - ∫ f_N  [A,C]u, C'Au  - ∫ f_N  CAu, [A, C']u 
        -∫ f_N  [C,A^∗]Au, C'u -∫ f_N  Cu, [C',A^∗]Au
        - ∫ f_N  [∇_V logf̅_̅N̅· A, C]u, C'u  - ∫ f_N  Cu, [∇_V logf̅_̅N̅· A, C']u .

Based on the conjugate rule (<ref>) for all similar terms, we complete the proof.



Let f̅_N = f^⊗ N, if A is commutative with C and C', we have

    Q_C,A = 2 ∑_i=1^N ∫ f_N (C_i ∇_v_ilog f) A_iu, C_iu_^2d,

and 

    Q_C',A = 2 ∑_i=1^N ∫ f_N (C'_i ∇_v_ilog f) A_iu, C'_iu_^2d.




We recall that

    Q_C,A = -∫ f_N  [C,A^∗]Au, C'u 
    - ∫ f_N  [∇_V logf̅_̅N̅· A, C]u, C'u .

For the second term of (<ref>), we direct compute the commutator [∇_V logf̅_̅N̅· A, C], i.e.

    - ∫ f_N  [∇_V logf̅_̅N̅· A, C]u, C'u 
     =     - ∑_i,j=1^N ∫ f_N  [∇_v_jlogf̅_̅N̅· A_j, C_i]u, C'_i u 
     =    ∑_i,j=1^N ∫ f_N  C_i ∇_v_jlogf̅_̅N̅, A_j u ⊗ C_i'u
     =    ∑_i=1^N ∫ f_N (C_i ∇_v_ilog f) A_iu, C_iu_^2d,

by C_i ∇_v_jlogf̅_̅N̅ = 0 if i ≠ j. For the first term of (<ref>), we understand the commutator [C,A^∗] as the row of operators 

    [C,A^∗] = ([C_1,A^∗],...,[C_N,A^∗] ),

then we have

    -∫ f_N  [C,A^∗]Au, C'u
     =     - ∑_i=1^N ∫ f_N  [C_i,A^∗]Au, C_i'u 
     =     - ∑_i=1^N ∫ f_N  C_i(A^∗Au)- A^∗(C_iAu), C'_iu ,

here we understand C_iAu as
a N-tuple vector reads as (C_iA_1u, ... ,C_iA_Nu), which can be operated by A^∗. Recall 

    A^∗g = - A · g - ∇_V logf̅_̅N̅, g ,

then we have

    -∫ f_N  [C,A^∗]Au, C'u
     =    ∑_i,j=1^N ∑_k,l=1^d ∫ f_N { C_i_lA_j_kA_j_ku + C_i_l{(∇_v_j_klogf̅_̅N̅) (A_j_ku)}, C'_i_lu}
        - ∑_i,j=1^N ∑_k,l=1^d ∫ f_N { A_j_kC_i_lA_j_ku + {(∇_v_j_klogf̅_̅N̅) (C_i_lA_j_ku)}, C'_i_lu }
     =    ∑_i=1^N ∫ f_N (C_i ∇_v_ilog f) A_iu, C'_iu_^2d.

Up to change the position of C and C', we complete the proof.





 §.§ Entropy multipliers


In order to deal with more general potentials V and W, we develop the method of entropy multipliers for N-particle relative Fisher information. We recommend Part 1, section 8 in <cit.> and <cit.> for “one particle version” without interaction potentials, and they only consider the invariant measure of single particle Fokker Planck equation as reference measure. Now let us consider a 2Nd × 2Nd weight matrix M(t,Z) to distort the relative Fisher Information, i.e.

    1/N∫ f_N Cu, M_t C'u dZ,

we evolve this quantity along time t in the following lemma, many ideas of manipulation are similar with Lemma <ref>.



Assume that f_N is a solution of Eq.(<ref>). Assume that f(t,z) ∈ W^1, ∞(×Ω×^d) solves Eq.(<ref>) with f(t,z) > 0 and ∫_Ω×^d f(t,z)dz = 1. Let B, C, C' be differential operators on (Ω×^d)^N, where

    B = ∑_i=1^N B_i,   B_i = v_i ·∇_x_i - ∇_x_iU ·∇_v_i - γ v_i ·∇_v_i,

and C, C' are to be confirmed. Let M_t(Z): × (Ω×^d)^N →^2Nd × 2Nd be a matrix valued function smooth enough for all variables, then

    d/dt{∫ f_N ⟨ Cu, M_tC'u⟩} =     ∫ f_N ⟨ M_t[B, C]u, C'u ⟩ + ∫ f_N ⟨ Cu, M_t[B, C']u ⟩
        - ∫ f_N ⟨  CR_N, M_tC'u ⟩  - ∫ f_N ⟨  M_tCu, C'R_N ⟩
        - 2 σ∫ f_N  CAu, M_tC'Au  + σ Q_C,A + σ Q_C',A
        + ∫ f_N  Cu, (∂_t M_t + BM_t + σΔ_V M_t)C'u ,

where

    Q_C,A =      -∫ f_N  [C,A^∗]Au, M_tC'u 
    - ∫ f_N  [∇_V logf̅_̅N̅· A, C]u, M_tC'u 
        -∫ f_N  [C,A]u, Au ⊗ M_tC'u  - ∫ f_N  M_t[A,C]u, C'Au 
        + ∫ f_N  [A,C]u, (AM_t)C'u ,



    Q_C',A =     -∫ f_N  M_tCu, [C',A^∗]Au - ∫ f_N  M_tCu, [∇_V logf̅_̅N̅· A, C']u 
        - ∫ f_N  Au ⊗ M_tCu, [C',A]u - ∫ f_N  CAu, M_t[A, C']u 
        + ∫ f_N  (AM_t)Cu, [A,C']u .



The notations of inner product and commutator operators appeared above are the same as in Remark <ref> and <ref>. Let us explain the notations associated with M_t we used above. We denote Cu,[B,C]u, CR_N are Nd-tuple vectors, then it is reasonable to multiply M_t with them. CAu, M_tC'Au should be understood as

    ∑_i,j,l=1^N  C_i A_l u, M_i jC'_j A_lu_^2d

in the sense of coordinate z_j ∈Ω×^d. The derivative of M_t (i.e. ∂_t M_t, AM_t, BM_t, Δ_V M_t) is taken componentwise. 




We use the similar argument with Lemma <ref>. We directly take derivative and obtain,  

    d/dt∫ f_N ⟨ Cu, M_tC'u⟩  =    ∫∂_t f_N ⟨ Cu, M_tC'u⟩
        + ∫ f_N ⟨ C(∂_t u), M_tC'u ⟩ + ∫ f_N ⟨ Cu, M_tC'(∂_t u)⟩
        + ∫ f_N  Cu, (∂_t M_t)C'u.

Observe that the last term

    ∫ f_N  Cu, (∂_t M_t)C'u

is new. Next we use the similar step in Lemma <ref> to go on. 


Step1. In this step, we claim the following

    d/dt∫ f_N  Cu, M_tC'u  =    ∫ f_N ⟨ [B, C]u, M_tC'u ⟩ + ∫ f_N ⟨ Cu, M_t[B, C']u ⟩
        -  ∫ f_N ⟨  CR_N, M_tC'u ⟩  - ∫ f_N ⟨  Cu, M_tC'R_N ⟩ + σ Q_σ
        + ∫ f_N
     Cu, (∂_t M_t + BM_t)C'u,

where Q_σ collects all terms with coefficient σ, which reads as

    Q_σ =    ∫ f_N Δ_V  Cu, M_tC'u 
        - ∫ f_N  CA^∗A u, M_tC'u + ∫ f_N  C(∇_V log f_N · Au), M_tC'u
        - ∫ f_N  Cu, M_tC'A^∗A u + ∫ f_N  Cu, M_tC'(∇_V log f_N · Au).



Let us explain some notations we use. Since [B,C']u, C'(A^∗Au) and C'(∇_V log f_N · Au) are all 2Nd-tuple vectors, it is reasonable to multiple them with M_t. After that, M_t[B,C']u, M_tC'(A^∗Au) and M_t C'(∇_V log f_N · Au) are 2Nd-tuple vectors.    


For the first term of (<ref>), recall Liouville operator (<ref>) we have 

    ∫∂_t f_N ⟨ Cu, M_tC'u ⟩ =    ∫ -L_N f_N ⟨ Cu, M_tC'u⟩
     =    ∫ f_N (B ⟨ Cu, M_tC'u⟩) + σ∫ f_N (Δ_V ⟨ Cu, M_tC'u⟩) 
     =    ∫ f_N  BCu, M_tC'u + ∫ f_N  Cu, M_tBC'u + σ∫ f_N (Δ_V ⟨ Cu, M_tC'u⟩) 
        + ∫ f_N  Cu,(BM_t)C'u.

For the second term and third term of (<ref>), using Eq.(<ref>), we have

    ∫ f_N ⟨ C(∂_t u), M_tC'u ⟩ =     - ∫ f_N ⟨ C(B u), M_tC'u ⟩ - ∫ f_N  CR_N, M_tC'u 
        - σ∫ f_N  CA^∗A u, M_tC'u + σ∫ f_N  C(∇_V log f_N · Au), M_tC'u.




    ∫ f_N ⟨ Cu, M_tC'(∂_t u) ⟩ =     - ∫ f_N ⟨ Cu, M_tC'(Bu) ⟩ - ∫ f_N  Cu, M_tC'R_N 
        - σ∫ f_N  Cu, M_tC'A^∗A u + σ∫ f_N  Cu, M_tC'(∇_V log f_N · Au),

Combine terms (<ref>),(<ref>) and (<ref>), we obtain the (<ref>). 
 


Step2. In this step, we focus on the term Q_σ as before, whose terms all come from diffusion. Now we claim,

    Q_σ =     
    - 2∫ f_N  CAu, M_tC'Au  + Q_C,A + Q_C',A
        + ∫ f_N  Cu, (Δ_V M_t)C'u ,

where

    Q_C,A =      -∫ f_N  [C,A^∗]Au, M_tC'u 
    - ∫ f_N  [∇_V logf̅_̅N̅· A, C]u, M_tC'u 
        -∫ f_N  [C,A]u, Au ⊗ M_tC'u  - ∫ f_N  M_t[A,C]u, C'Au 
        + ∫ f_N  [A,C]u, (AM_t)C'u ,



    Q_C',A =     -∫ f_N  M_tCu, [C',A^∗]Au - ∫ f_N  M_tCu, [∇_V logf̅_̅N̅· A, C']u 
        - ∫ f_N  Au ⊗ M_tCu, [C',A]u - ∫ f_N  CAu, M_t[A, C']u 
        + ∫ f_N  (AM_t)Cu, [A,C']u ,

The terms Q_C,A and Q_C',A collect all terms including commutators, we will take suitable operators C and C' to simplify these commutators.

For the first term of (<ref>), using integral by parts with respect to Lebesgue measure,   

    ∫Δ_V f_N  Cu, M_tC'u 
     =     - ∫ f_N A u  · A  Cu, M_tC'u  - ∫ f_N ∇_V logf̅_̅N̅· A  Cu, M_tC'u.

Recall we denote A = (0, ∇_V), we continue the first term of (<ref>) 

    - ∫ f_N A u  · A  Cu, M_tC'u 
     =     -∫ f_N  ACu, Au ⊗ M_tC'u  - ∫ f_N  Au ⊗ M_tCu, AC'u
        - ∫ f_N Au · Cu, (AM_t)C'u
     =     -∫ f_N  CAu, Au ⊗ M_tC'u  - ∫ f_N  Au ⊗ M_tCu, C'Au
        + ∫ f_N  [C,A]u, Au ⊗ M_tC'u  + ∫ f_N  Au ⊗ M_tCu, [C',A]u
        - ∫ f_N Au · Cu, (AM_t)C'u.

For the second terms of (<ref>), we  rewrite it as

    - ∫ f_N  CA^∗A u, M_tC'u
     =     - ∫ f_N  A^∗CAu,M_tC'u - ∫ f_N  [C,A^∗]Au, M_tC'u,

and we continue the first term of (<ref>), 

    - ∫ f_N  A^∗CAu, M_tC'u
     =     - ∫f̅_̅N̅ A^∗CAu, M_tC'h
     =     - ∫f̅_̅N̅ CAu, M_t(AC'h) - ∫f̅_̅N̅ CAu, (AM_t)C'h 
     =     - ∫ f_N  CAu, M_t(AC'u)  - ∫f̅_̅N̅ CAu, Ah ⊗ M_tC'u - ∫f̅_̅N̅ CAu, (AM_t)C'h 
     =     - ∫ f_N  CAu, M_t(AC'u)  - ∫ f_N  CAu, Au ⊗ M_tC'u - ∫ f_N  CAu, (AM_t)C'u ,

now we have

    - ∫ f_N  CA^∗A u, M_tC'u =     - ∫ f_N  CAu, M_t(C'Au)  - ∫ f_N  CAu, M_t[A, C']u 
        -∫ f_N  [C,A^∗]Au, M_tC'u - ∫ f_N  CAu, Au ⊗ M_tC'u
        - ∫ f_N  CAu, (AM_t)C'u .

Similarly, for the fourth term of (<ref>), since M_t is symmetric, using its conjugation in ^dN, 

    - ∫ f_N  Cu, M_tC'A^∗Au = - ∫ f_N  M_tCu, C'A^∗Au,

up to the exchange of C and C', we have

    - ∫ f_N  Cu, M_tC'A^∗Au =     - ∫ f_N  M_t(CAu), C'Au  - ∫ f_N  M_t[A,C]u, C'Au 
        -∫ f_N  M_tCu, [C',A^∗]Au - ∫ f_N  Au ⊗ M_tCu, C'Au
        - ∫ f_N  (AM_t)Cu, C'Au .

Finally, all of left terms are third and fifth term in (<ref>), and underlined term in (<ref>), we collect them as below,

    - ∫ f_N ∇_V logf̅_̅N̅· A  Cu, M_tC'u
        + ∫ f_N  C(∇_V log f_N · Au), M_tC'u
        + ∫ f_N  M_tCu, C'(∇_V log f_N · Au).

We deal with the first term of (<ref>) as following,

    - ∫ f_N ∇_V logf̅_̅N̅· A  Cu, M_tC'u
     =     - ∫ f_N ∇_V logf̅_̅N̅· A (Cu), M_tC'u - ∫ f_N  M_tCu, ∇_V logf̅_̅N̅· A (C'u)
        - ∫ f_N  Cu, (∇_V log f_N · AM_t)C'u 
     =     - ∫ f_N  C(∇_V logf̅_̅N̅· Au), M_tC'u - ∫ f_N  [∇_V logf̅_̅N̅· A, C]u, M_tC'u 
        - ∫ f_N  M_tCu, C'(∇_V logf̅_̅N̅· A u) - ∫ f_N  M_tCu, [∇_V logf̅_̅N̅· A, C']u 
        - ∫ f_N  Cu, (∇_V log f_N · AM_t)C'u ,

by similar argument with (<ref>) and (<ref>) in Lemma <ref>, we have 

    2∫ f_N  CAu, Au ⊗ C'u + 2∫ f_N  Au ⊗ Cu, C'Au
        - ∫ f_N  [∇_V logf̅_̅N̅· A, C]u, C'u  - ∫ f_N  Cu, [∇_V logf̅_̅N̅· A, C']u 
        - ∫ f_N  Cu, (∇_V log f_N · AM_t)C'u .

Until now, let us collect all terms don't appear in the Lemma <ref>:

    - ∫ f_N Au · Cu, (AM_t)C'u
        - ∫ f_N  CAu, (AM_t)C'u  - ∫ f_N  (AM_t)Cu, C'Au 
        -∫ f_N  Cu, (∇_V logf̅_̅N̅· A M_t + ∂_t M_t + BM_t)C'u.

We compute the first term more precisely,

    -∫ f_N Au · Cu, (AM_t)C'u
     = - ∫f̅_̅N̅ Ah · Cu, (AM_t)C'u,

using the conjugate relationship w.r.t. measure f̅_N, we have

    - ∫f̅_̅N̅ Ah · Cu, (AM_t)C'u = - ∫ f_N A^∗ Cu, (AM_t)C'u,

recall A^∗ = - A · g - ∇_V logf̅_̅N̅, g, then we have

    - ∫ f_N A^∗ Cu, (AM_t)C'u
     =    ∑_i=1^N ∫ f_N A_i · Cu, (A_iM_t)C'u  + ∫ f_N ∇_V logf̅_̅N̅· Cu, (AM_t)C'u
     =    ∑_i=1^N∫ f_N  A_iCu, (A_iM_t)C'u  + ∑_i=1^N ∫ f_N  (A_iM_t)Cu, A_iC'u 
        + ∫ f_N  Cu, (Δ_V M) C'u + ∫ f_N  Cu, (∇_V logf̅_̅N̅· A M_t)C'u
     =    ∫ f_N  ACu, (AM_t)C'u  + ∫ f_N  (AM_t)Cu, AC'u 
        + ∫ f_N  Cu, (Δ_V M) C'u + ∫ f_N  Cu, (∇_V logf̅_̅N̅· A M_t)C'u.

Combine with last three terms of (<ref>), all of remaining terms compared with (<ref>) in the proof of Lemma <ref> are  

    ∫ f_N  Cu, (Δ_V M_t)C'u 
        + ∫ f_N  [A,C]u, (AM_t)C'u  + ∫ f_N  (AM_t)Cu, [A,C']u .

Together with (<ref>) in Step1, we finish the proof.



If we take M as a block positive defined matrix, i.e.

    M = 
    (
    [ E F; F G ]),

where E,F,G: × (Ω×^d)^N →^Nd × Nd are matrix-valued functions smooth enough, and take C = C' = (∇_X, ∇_V), then we have

    d/dt{∫ f_N  Cu, M_t Cu} =     - ∫ Cu, S_tCu  - 2 ∫ f_N  CR_N, M_t Cu
        - 2 σ∫ f_N  CAu, M_tC'Au ,

where S_t = S(t,Z) is a matrix reads as

    S_t = 
    (
    [                      2F - ∂_t E - L^∗_NE  D_1 - ∂_t F - L^∗_NF - 2E ∇^2 U + 2 γ F;                      2G - ∂_t F - L^∗_NF D_2 - ∂_t G - L^∗_N G - 2F ∇^2 U + 2 γ G ])

and 

    D_1 = -4 σ (F ∇_V ∇_V logf̅_̅N̅ + E ∇_X ∇_V logf̅_̅N̅),


    D_2 = -4 σ (F ∇_X ∇_V logf̅_̅N̅ + G ∇_V
    ∇_V logf̅_̅N̅).



A easy computation shows that  

    ∇^2 U =
    [ ∇^2 V(x_1) + 1/N∑_j ≠ 1^N ∇^2 W(x_1 - x_j)                                          ⋯                       -1/N∇^2 W(x_1 - x_N);                                          ⋮                                          ⋱                                          ⋮;                       -1/N∇^2 W(x_N - x_1)                                          ⋯ ∇^2 V(x_N) + 1/N∑_j ≠ N^N ∇^2 W(x_N - x_j) ],

which is a Nd × Nd matrix.


The main idea is to analyse every term in (<ref>). Let us compute what commutator [B,C] is firstly. For each component of operator C = (∇_X,∇_V), we have

    [∇_X, B]_i = ∑_k=1^N [∇_x_i, B_k] = - ∑_k=1^N [∇_x_i,  ∇_x_k U ·∇_v_k] = - ∑_k=1^N ∇^2_x_i,x_k U ·∇_v_k,

and

    [∇_V, B]_i = ∑_k=1^N [∇_v_i, B_k]     = ∑_k=1^N [∇_v_i, v_k ·∇_x_k - γ v_k ·∇_v_k] 
        = ∑_k=1^Nδ_ik (∇_x_k - γ∇_v_k) = ∇_x_i - γ∇_v_i.

In other word, we have

    [B,C]^⊤ = - (-∇^2 U ·∇_V, ∇_X - γ∇_V)^⊤ =
    (
    [     0 ∇^2 U;   -Id  γ Id ])
    (
    [ ∇_X; ∇_V ]).

Then we regard M_t[B, C]u, Cu  +  Cu, M_t[B, C]u as a quadratic form with matrix

    (
    [               -2F E ∇^2 U - γ F - G; E ∇^2 U - γ F - G  2F ∇^2 U - 2 γ G ]).

Since M_t is a positive defined matrix, - CAu, M_tCAu must be negative, so we just keep it. Moreover, C, C' are commutative with A, only nontrivial terms in (<ref>) and (<ref>) are first line two terms. We recall [C,A^∗] and [∇_V logf̅_̅N̅· A, C] in Corollary <ref>, and we regard Q_C,A + Q_C',A as a quadratic form with following matrix,

    (
    [ E F; F G ])
    (
    [                    0 4 σ∇_X ∇_V logf̅_̅N̅;                    0 4 σ∇_V ∇_V logf̅_̅N̅ ])
    =
    (
    [                                               0 4 σ (E ∇_X ∇_V logf̅_̅N̅ + F ∇_V ∇_V logf̅_̅N̅);                                               0 4 σ (F ∇_X ∇_V logf̅_̅N̅ + G ∇_V ∇_V logf̅_̅N̅) ]).

For the last line in (<ref>), we recall duality of Liouville operator (<ref>), then we have

    ∂_t M_t + BM_t + σ M_t = 
    (
    [ ∂_t E + L_N^∗E ∂_t F + L_N^∗F; ∂_t F + L_N^∗F ∂_t G + L_N^∗G ]).

Combining (<ref>),(<ref>) and (<ref>), we finish the proof.





 §.§ Weighted log Sobolev inequality


In this section, we establish the weighted Log-Sobolev inequality for nonlinear equilibrium f_∞ defined by Eq.(<ref>), then we extend to the weighted N-particle version by verifying tensorized invariance of weighted Log-Sobolev inequality. Before that, let us talk about the situation of first order particle system. Guillin et al. consider the uniform-in-time propagation of chaos with the following limiting equation on 𝕋^d in <cit.>,

    ∂_t ρ + div_x(ρ K ∗ρ ) = σΔ_x ρ,    x ∈𝕋^d,

where ρ(t, x): ×Ω→ is a probability density. An very useful observation in <cit.> is that, if there exists some constant λ > 1 such that

    1/λ≤ρ_0 ≤λ,

for initial data ρ_0 of Eq.(<ref>) on 𝕋^d, then they propagate this property to all time t uniformly, i.e.

    1/λ≤ρ(t) ≤λ.

After controlling the upper bound of  ∇^n ρ_L^∞ by standard energy estimates, they obtain the upper bound of ∇logρ_L^∞ and ∇^2 logρ_L^∞, which is essential for the proof of Theorem 1 in <cit.>. Another important observation in <cit.> is that ρ_t(x) satisfies Log-Sobolev inequality uniformly in t as a result of perturbation of uniform distribution by (<ref>) (See Proposition 5.1.6, <cit.>). These two facts help them obtain uniform-in-time propagation of chaos even for Biot-Savart kernel. But in the case of Vlasov-Fokker-Planck equation (<ref>), the situation becomes totally different. The best we can expect to initial data of Eq.(<ref>) is 

    f_0(x,t) ≥ Ce^-av^2/2,
       (x,v) ∈Ω×^d

for some constant C > 0, a > 0. The lack of positive lower bound of (<ref>) makes the uniform-in-time upper bound of ∇^2 log f _L^∞ fail by the same strategy in <cit.>, that is the reason why we replace the reference measure f by f_∞. 

Inspired by the argument of Gibbs measure for one particle in <cit.>,

    dμ = 1/Ze^- β H(x,v) x  v,    (x,v) ∈Ω×^d,

where H(x,v) = v^2/2 + V(x) is the Hamiltonian defined on Ω×^d and Z is partition function. We omit the temperature constant β in the following and recall some related results in <cit.>.



μ satisfies the following weighted Log-Sobolev inequality in ^d ×^d if there exists some constant ρ_wls(μ) > 0 s.t. for all smooth enough g with ∫ g^2 dμ = 1:

    Ent_μ(g^2) ≤ρ_wls(μ) ∫ (H^-2η|∇_x g|^2 + |∇_v g|^2) μ.


The weighted Log-Sobolev inequality (<ref>) associates with a new second order operator L_η on ^d ×^d, 

    L_η := H^-2ηΔ_x + Δ_v - H^-2η( 2 η∇_x H/H + ∇_x H ) ·∇_x - ∇_v H ·∇_v,

which is symmetric in L^2(μ) and satisfies

    ∫ f (L_ηg)dμ = - ∫ (H^-2η∇_x f ·∇_x g + ∇_v f ·∇_v g) μ.

The following theorem tells us how to verifying the weighted Log-Sobolev inequality for suitable condition of function H.

Assume that H goes to infinity at infinity and that there exists a > 0 such that e^aH∈ L^1(μ). 

(1) If μ satisfies the weighted Log-Sobolev inequality (<ref>), then, there exists a Lyapunov function, i.e. a smooth function W such that W(x,y) ≥ w > 0 for all (x,v), two positive constant λ and b such that

    L_ηW ≤ - λ H W + b.


(2) Conversely, assume that there exists a Lyapunov function satisfying (<ref>) and that |∇ H| ≥ c > 0 for |(x,y)| large enough. Define

    θ(r) = sup_z ∈∂ A_rmax_i,j=1,...,2d|∂^2 H/∂ z_i ∂ z_j|

and assume that θ(r) ≤ ce^C_0r with some positive constants C_0 and c for r sufficiently large. Then μ satisfies the weighted Log-Sobolev inequality (<ref>).


A natural choice of Lyapunov function is W(x,v) = e^α(v^2/2 + V(x)) with α∈ (0,1), we refer <cit.> and <cit.> to readers for more details. A simple perturbation argument in <cit.> could extend the weighted Log-Sobolev inequality to f_∞.

Assume that μ satisfies the weighted Log-Sobolev inequality (<ref>), let μ_1 be a probability measure with density h with respect to μ such that 1/λ≤ h ≤λ for some constant λ > 0, then μ_1 satisfies

    Ent_μ_1(g^2) ≤ρ_wls(μ) λ^2 ∫ (H^-2η|∇_x g|^2 + |∇_v g|^2) μ_1.




We use the following lemma in <cit.> (Page240, Lemma 5.1.7),

Let ϕ: I → on some open interval I ⊂ be convex of class C^2. For every bounded or suitably integrable measurable function f: E → with values in I,

    ∫_E ϕ(f)dμ - ϕ( ∫_Ef dμ) = inf_r ∈ I∫_E [ϕ(f) - ϕ(r) - ϕ'(r)(f - r)] μ.


For the function ϕ(r) = rlogr on I = (0, ∞), we can easily establish (<ref>) by

    Ent_μ_1(g^2) = inf_r ∈ I∫_E [ϕ(g^2) - ϕ(r) - ϕ'(r)(g^2 - r)] dμ_1 ≤λ Ent_μ(g^2),

and 

    λ Ent_μ(g^2)    ≤ρ_wls(μ) λ∫ (H^-2η|∇_x g|^2 + |∇_v g|^2) μ
       ≤ρ_wls(μ) λ^2∫ (H^-2η|∇_x g|^2 + |∇_v g|^2) μ_1.



Now we verify the invariance of weighted Log-Sobolev constant, by which we extend weighted Log-Sobolev inequality (<ref>) to the measure of tensorized form μ^⊗ N.   


Assume that μ_1, μ_2 satisfy the weighted Log-Sobolev inequality (<ref>) with constant ρ_wls(μ_1) and ρ_wls(μ_2), then μ_1 ⊗μ_2 satisfies 

    Ent_μ_1 ⊗μ_2(f^2) ≤ρ_wls∑_i=1^2 ∫_E_1 × E_2 (H^-2η(z_i)|∇_x_i f|^2 + |∇_v_i f|^2) μ_1 ⊗μ_2

with ρ_wls = max{ρ_wls(μ_1), ρ_wls(μ_1)}.


We denote

    g^2(z_1) = ∫_E_2 f^2(z_1,z_2) μ_2(z_2),

and of course g ≥ 0, then

    Ent_μ_1 ⊗μ_2(f^2) =      Ent_μ_1(g^2) 
     +    ∫_E_1( ∫_E_2 f^2(z_1,z_2) log f^2(z_1,z_2) μ_2(z_2) . 
       . - ∫_E_2 f^2(z_1,z_2) μ_2(z_2) log∫_E_2f^2(z_1,z_2) μ_2(z_2) ) μ_1(z_1).

Observe that

    Eut_μ_2(f^2) =    ∫_E_2 f^2(z_1,z_2) log f^2(z_1,z_2) μ_2(z_2) 
        - ∫_E_2 f^2(z_1,z_2) μ_2(z_2) log∫_E_2f^2(z_1,z_2) μ_2(z_2) 
    ≤   ρ_wls(μ_2) ∫_E_2 (H^-2η(z_2
    )|∇_x_2f|^2 + |∇_v_2f|^2) μ_2(z_2).

Next we estimate Ent_μ_1(g^2), 

    Ent_μ_1(g^2) ≤ρ_wls(μ_1) ∫_E_2 (H^-2η(z_1)|∇_x_1g|^2 + |∇_v_1g|^2) μ_1(z_1),

for terms on the right hand side, observe the weight H(z_1) only concerns the first variable,  

    |H^-η(z_1)∇_x_1g|^2      = | H^-η(z_1) ∇_x_1√(∫_E_2 f^2(z_1,z_2) μ(z_2))|^2 
       ≤(∫_E_2 f (H^-η(z_1) |∇_x_1f|) μ(z_2))^2/∫_E_2 f^2(z_1,z_2) μ(z_2),

and use Cauchy-Schwarz inequality, we have

    |H^-η(z_1)∇_x_1g|^2 ≤∫_E_2 H^-2η(z_1)|∇_x_1f|^2 μ_2(z_2).

Similarly,

    |∇_v_1g|^2 ≤∫_E_2 |∇_v_1f|^2 μ_2(z_2),

then we finish the proof.



Assume that V satisfies (i) of Assumption <ref>, and W satisfies Assumption <ref>, then f_∞ satisfies weighted Log-Sobolev inequality (<ref>). Moreover, f_∞^⊗ N satisfies weighted Log-Sobolev inequality (<ref>) with the same constant. 


According to Assumption <ref>, |∇ W(x)| ≤λ/2|x|, then we have

    |W(x)| ≤ |W(0)| + λ/2|x|^2,

hence

    |W ∗ρ_∞(x)|    ≤ |W(0)| + ∫ |y|^2 f_∞(y,v)  y  
    v + λ/2|x|^2 ≤ C + λ/2|x|^2,

for some constant C > 0 with 𝔼_ρ_∞|x|^2 < ∞. By (i) of Assumption <ref> for V, we have

    a μ≤ f_∞≤ b μ,

for some a, b > 0 and μ = 1/Z e^-V(x) - 1/2v^2. Using Proposition <ref>, we obtain the weighted Log-Sobolev inequality for f_∞, then Proposition <ref> makes sure the same fact about f_∞^⊗ N. 





 §.§ Large deviation estimates

In this subsection, we deal with the error terms we mentioned before. For relative entropy, we recall Lemma <ref> and the error term reads as,

    4/σ∑_i=1^N ∫_(Ω×^d)^N f^t_N | 1/N∑_j,j≠ iK(x_i-x_j) - K⋆ρ_∞(x_i) |^2  Z.

For relative fisher information, we recall Corollary <ref> and we estimate the following term,

    ∫_(Ω×^d)^N f^t_N  CR_N, M_t Cu Z.

Our main tool comes from Jabin-Wang's large derivation type estimates for propagation of chaos for singular kernels in series of paper <cit.>, <cit.>. We refer Theorem 3 in <cit.> or Theorem 3 and Theorem 4 in <cit.> to readers for more details to this kind of technique, now we apply it to our cases — bounded or lipschtiz kernels.

The following lemma gives more precise analysis to error term (<ref>). Here we take C = (∇_X, ∇_V).


Let M: Ω×^d →^2Nd × 2Nd be a positive defined matrix function which takes as (<ref>). Assume that E, F, G are block diagonal positive defined matrices, i.e.
E = diag{E^1,...,E^N}, F = diag{F^1,...,F^N} and G = diag{G^1,...,G^N}, where E^i, F^i and G^i are d × d positive defined matrices. Then we have

    ∫ f_N ∇R_N, M ∇ u≤   ∑_i=1^N 4/σ∫ f_N |√(E^i) R_N,i^1|^2 + 4/σ∫ f_N |(F^i)^3/4R_N,i^1|^2 
       ∑_i=1^N γ^2/2σ^2∫ f_N |√(E^i)R^3_N,i|^2 + γ^2/2σ∫ f_N |√(F^i) R^3_N,i|^2 
       ∑_i=1^N γ^2/2σ^2∫ f_N |√(F^i) R^2_N,i|^2 + 4 γ^2/σ^3∫ f_N |G^i R^2_N,i|^2
    
        + C_K ∫ f_N |√(E)∇_V u|^2 + (C_K + 1/2) ∫ f_N |√(F)∇_V u|^2 + σ/4∫ f_N |∇_V u|^2 
        + (C_K + 1/2) ∫ f_N |√(E)∇_X u|^2 + 1/4∫ f_N |√(F)∇_X u|^2 
        + ∑_i=1^N σ/16∫ f_N | √(E^i)∇_v_i∇_x_i u|_^2d^2 + σ/16∫ f_N |(F^i)^1/4∇_v_i∇_v_i u|_^2d^2 
        - ∑_i=1^N ∫ f_N ⟨ R^1_N,i, (∇_v_i E^i) ∇_x_i u ⟩_^2d - ∫ f_N ⟨ R^1_N,i, (∇_v_i F^i) ∇_v_i u ⟩_^2d.

where σ is diffusion constant, C_K = ∇ K_L^∞ and 

    R^1_N,i = 1/N∑_j≠ i∇ K(x_i-x_j) - ∇ K⋆ρ_∞(x_i),



    R^3_N,i = 1/N∑_j, j≠ i∇ K(x_j - x_i) · [v ∗ρ^v_∞(v_i) - (v_i - v_j)],   ρ^v_∞(v) = ∫ f_∞ x,



    R^2_N,i = - 1/N∑_j≠ iK(x_i-x_j) - K⋆ρ_∞(x_i).



Here we understand ∇ K or R^1_NX,i as a d × d matrix, ∇ K · v and K are d dimensional vectors, we omit the summation about these components for convenience. The meaning of this lemma is to divide the error term of relative Fisher Information into L^2 norm of R_N^i, i=1,2,3 and small terms we can absorb by negative part in (<ref>).


We directly compute ∇R_N and use the Young inequality. For each component i in position direction, we have

    ∇_x_iR_N     = ∇_v_ilog f(x_i, v_i) ·{1/N∑_j=1,j≠ i^N ∇ K(x_i - x_j) - ∇ K⋆ρ(x_i) }
          + ∇_x_i∇_v_ilog f(x_i,v_i) ·{1/N∑_j=1,j≠ i^N K(x_i-x_j) - K⋆ρ(x_i) }
          - 1/N∑_j=1,j≠ i^N ∇ K(x_j - x_i) ·∇_v_jlog f(x_j, v_j),

we denote these three lines above by

    ∇_x_iR_N = ∇_v_ilog f(x_i,v_i) · R^1_N,i + ∇_x_i∇_v_ilog f(x_i,v_i) · R^2_N,i + γ/σ R^3_N,i,

where

    R^1_N,i = {1/N∑_j=1,j≠ i^N ∇ K(x_i-x_j) - ∇ K⋆ρ_∞(x_i) }

is a d × d matrix,

    R_N,i^2 = {1/N∑_j=1,j≠ i^N K(x_i-x_j) - K⋆ρ_∞(x_i) }

is a d dimensional vector and 

    R^3_N,i    =  - σ/γ1/N∑_j=1,j≠ i^N ∇ K(x_j - x_i) ·∇_v_jlog f(x_j, v_j)

is a d dimensional vector. Recall that we take M as a block matrix 

    M = 
    (
    [ E F; F G ]),

and E, F, G are d × d diagonal matrices. Hence, we use integral by parts for each component of the first term of (<ref>), we have

    ∑_i=1^N ∫ f_N ⟨∇_v_ilog f · R_N,i^1, E^i ∇_x_i u ⟩
     =     - ∑_i=1^N ∫ f_N ⟨∇_v_i u · R_N,i^1, E^i ∇_x_i u ⟩ + ∑_i=1^N ∫ f_N ⟨∇_v_ilog f_N · R_N,i^1, E^i ∇_x_i u ⟩
     =     - ∑_i=1^N ∫ f_N ⟨∇_v_i u · R_N,i^1, E^i ∇_x_i u ⟩ + ∑_i=1^N ∫⟨∇_v_i f_N · R^1_N,i, E^i ∇_x_i u ⟩
     =     - ∑_i=1^N ∫ f_N ⟨∇_v_i u · R_N,i^1, E^i ∇_x_i u ⟩ - ∑_i=1^N ∫ f_N ⟨ R^1_N,i, E^i ∇_v_i∇_x_i u ⟩_^2d
        - ∑_i=1^N ∫ f_N ⟨ R^1_N,i, (∇_v_i E^i) ∇_x_i u ⟩_^2d.

Using Young inequality, we have, 

    ∑_i=1^N ∫ f_N ⟨∇_v_ilog f · R_N,i^1, E^i ∇_x_i u ⟩
    ≤    C_K ∫ f_N |√(E)∇_V u|^2 + C_K ∫ f_N |√(E)∇_X u|^2 
        + ∑_i=1^N σ/16∫ f_N | √(E^i)∇_v_i∇_x_i u|_^2d^2 + ∑_i=1^N 4/σ∫ f_N |√(E^i) R_N,i^1|_^2d^2 
        - ∑_i=1^N ∫ f_N ⟨ R^1_N,i, (∇_v_i E^i) ∇_x_i u ⟩_^2d,

similarly,

    ∑_i=1^N ∫ f_N ⟨∇_v_ilog f · R_N,i^1, F^i ∇_v_i u ⟩
    ≤    C_K ∫ f_N |√(F)∇_V u|^2 
        + ∑_i=1^N σ/16∫ f_N | (F^i)^1/4∇_v_i∇_v_i u|_^2d^2 + ∑_i=1^N 4/σ∫ f_N |(F^i)^3/4 R_N,i^1|_^2d^2 
        - ∑_i=1^N ∫ f_N ⟨ R^1_N,i, (∇_v_i F^i) ∇_v_i u ⟩_^2d.

Actually the computation above is the same if we take f = f_∞, then we finish the estimates. For the third term of (<ref>), by Young inequality, we have

    ∑_i=1^N ∫ f_N ⟨ R^3_N,i, E^i ∇_x_i u ⟩≤∑_i=1^N γ^2/2σ^2∫ f_N |√(E^i)R^3_N,i|^2 + 1/2∫ f_N |√(E)∇_X u|^2,


    ∑_i=1^N ∫ f_N ⟨ R^3_N,i, F^i ∇_v_i u ⟩≤∑_i=1^N γ^2/2σ^2∫ f_N |√(F^i) R^3_N,i|^2 + 1/2∫ f_N |√(F)∇_V u|^2,

moreover, we rewrite R^3_N,i as 

    R^3_N,i = 1/N∑_j=1, j ≠ i^N ∇ K(x_j - x_i) · [v ∗ρ_∞^v - (v_i - v_j)],    ρ_∞^v(v) = ∫ f_∞dx,

by ∇_v_ilog f_∞ = -γ/σ v_i. In the end, the second term  ∇_x_i∇_v_ilog f · R^2_N,i disappears by ∇_x_i∇_v_ilog f_∞ = 0 if we take f = f_∞. Now we finish the estimates of all error terms in position direction. 
 

For each component i in velocity direction, we simply have only one term

    ∇_v_iR_N = ∇_v_i∇_v_ilog f ·{1/N∑_j≠ iK(x_i-x_j) - K⋆ρ(x_i) } = ∇_v_i∇_v_ilog f · R^2_N,i.

We take ∇_v_i∇_v_ilog f(x_i,v_i) = ∇_v_i∇_v_ilog f_∞(x_i,v_i) = - γ/σ Id_d× d and use Young inequality, then we obtain

    ∑_i=1^N ∫ f_N ⟨∇_v_i∇_v_ilog f_∞·  R^2_N,i, F^i ∇_x_i u ⟩
    ≤   ∑_i=1^N ∫ f_N |∇_v_i∇_v_ilog f_∞·√(F^i) R^2_N,i|^2 + 1/4∫ f_N |√(F)∇_X u|^2,


    ∑_i=1^N ∫ f_N ⟨∇_v_i∇_v_ilog f_∞· R^2_N,i, G^i ∇_v_i u ⟩
    ≤   ∑_i=1^N 1/σ∫ f_N |∇_v_i∇_v_ilog f_∞· G^i R^2_N,i|^2 + σ/4∫ f_N |∇_V u|^2,

now we finish the estimates of error terms on velocity direction.

Finally, we collect all terms of (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and(<ref>), we finish the proof.



For error terms of relative Fisher Information, the most difficult part is to show the uniform-in-time estimates for the following terms,

    ∑_i,k=1^N ∫ f_N ⟨∇_v_i∇_v_ilog f · R^2_N,i, M_t^ik C_k u ⟩,


    ∑_i,k=1^N ∫ f_N ⟨∇_x_i∇_v_ilog f · R^2_N,i, M_t^ik C_k u ⟩.

It is not easy to deal with the terms ∇_v_i∇_v_ilog f and ∇_x_i∇_v_ilog f by uniform-in-time way, that is one of the most important reason we select the reference measure as f_∞. It is natural to conjecture that |∇^2 log f| ≤ C(1 + 1/2v^2 + V(x))^k for some positive order k>0. We might explore how to control these terms better in further study.


Now let us focus on three error terms in Lemma <ref>. We rewrite them as following as Theorem 3 in <cit.> has talked about,

    ∑_i=1^N ∫ f_N |R^1_N,i|^2 = ∫ f_N |1/√(N)∑_j,j≠ 1∇ K(x_1-x_j) - ∇ K⋆ρ_∞(x_1) |^2,



    ∑_i=1^N ∫ f_N |R^3_N,i|^2 = ∫ f_N | 1/√(N)∑_j, j≠ 1∇ K(x_j - x_1) · [v ∗ρ^v_∞(v_1) - (v_1 - v_j)]|^2,



    ∑_i=1^N ∫ f_N |R^2_N,i|^2 = ∫ f_N | 1/√(N)∑_j,j≠ 1K(x_1-x_j) - K⋆ρ_∞(x_1) |^2,

for terms (<ref>), (<ref>) and (<ref>), we define 

    ψ_1(x_1,x_j) = ∇ K(x_1-x_j) - ∇ K⋆ρ_∞(x_1),



    ψ_2(x_1,x_j) = K(x_1-x_j) - K⋆ρ_∞(x_1),



    ψ_3(v_1,v_j) = ∇ K(x_j - x_1) · [v ∗ρ^v_∞(v_1) - (v_1 - v_j)].

We also define the 2p moment of f_∞ for any p ≥ 1,

    M_p = ( ∫ f_∞ (1 + 1/2v^2 + 1/2x^2)^p )^1/p.

By similar argument in Section 1.4 in <cit.>, we have

    sup_p M_p/p := Λ < 1/e λ,

where λ satisfies

    ∫ f_∞ e^λ(1 + 
    1/2v^2 + 1/2x^2) z < ∞,

We easily take λ∈ (0,β) in the following.


By Lemma 1 in <cit.>, i.e. Gibbs inequality, we have 

    1/N∫ f_N |R^k_N|^2 dZ ≤1/νH_N(f_N| f̅_̅N̅) + 1/N∫f̅_̅N̅exp(ν |R^k_N|^2) Z

for some ν > 0 and k = 1,2,3, it is sufficient to show that

    ∫f̅_̅N̅exp(
    ν/N∑_j_1,j_2=1^N ψ_i(x_1, x_j_1) ψ_i(x_1,x_j_2))  Z < ∞,    i = 1,2,

and

    ∫f̅_̅N̅exp(
    ν/N∑_j_1,j_2=1^N ψ_3(v_1, v_j_1) ψ_3(v_1,v_j_2))  Z < ∞

for some ν > 0. We observe that ψ_1(x_1,x_j) is bounded by Assumption <ref> and ∫ψ_1(z,x) dx = 0, which implies that we can directly use  Theorem 3 in <cit.> for this type of ψ_1(x_1 - x_j). We recall  the following proposition in <cit.>,

Assume that ψ∈ L^∞ with ψ_L^∞ < 1/2e, and for any fixed y,

    ∫ψ(y,x) f_∞ z = 0,

then 

    ∫f̅_̅N̅exp(
    1/N∑_j_1,j_2=1^N ψ(x_1, x_j_1) ψ(x_1,x_j_2))  Z < C,

where f̅_̅N̅ = f_∞^⊗ N and C > 0 is independent of N.


Theorem 3 of <cit.>.

Now let ψ = νψ_1 for some ν > 0, we obtain the desired result. For the cases ψ_2 and ψ_3, we prove the following proposition:

Assume that |ψ(m,n)| ≤ C(1 + |n|) with C ≤1/8e^2 Λ, and for any fixed m, ∫ψ(m,n) f_∞dz = 0. Here (m,n) takes position pair (y,x) ∈Ω×Ω or velocity pair (u,v) ∈^d ×^d, then we have

    ∫f̅_̅N̅exp (
    1/N∑_j_1,j_2=1^N ψ(m_1, n_j_1) ψ(m_1,n_j_2))  Z < C,

where f̅_̅N̅ = f_∞^⊗ N and C > 0 is independent of N.



We use (m,n) to denote position pair (y,x) ∈Ω×Ω or velocity pair (u,v), and we show the proof of these two cases together. Since

    exp(A) ≤exp(A) + exp(-A) = 2 ∑_k=0^∞1/(2k)! A^2k,

it suffices only to bound the series with even terms,

    2 ∑_k=0^∞1/(2k)!∫f̅_̅N̅(
    1/N∑_j_1,j_2=1^N ψ(m_1, n_j_1) ψ(m_1,n_j_2) )^2k Z,

where in general the k-th term can be expanded as

    1/(2k)!∫f̅_̅N̅(
    1/N∑_j_1,j_2=1^N ψ(m_1, n_j_1) ψ(m_1,n_j_2))^2k Z 
     =    1/(2k)!1/N^2k∑_j_1,...,j_4k=1^N
    ∫f̅_̅N̅ψ(m_1,n_j_1)... ψ(m_1,n_j_4k)  Z.

Now we divide the proof in two different cases: where k is small compared to N and in the simpler case where k is comparable to or larger than N. 


Case: 3 ≤ 3k ≤ N. In this case, for each term of (<ref>), we have

    ∫f̅_̅N̅ψ(m_1,n_j_1)... ψ(m_1,n_j_4k)dZ ≤ C^4k∫f̅_̅N̅(1 + |n_j_1|)...(1 + |n_j_4k|)  Z ,

since we have cancellation condition 
∫ψ(m,n) f_∞dz = 0, the whole estimates relies on how many choices of multi-indices (j_1,...,j_4k) leading to a non-vanishing term. By notations of Theorem 3 and Theorem 4 in <cit.>, we could denote the set of multi-indices (j_1,...,j_4k) s.t.

    ∫f̅_̅N̅ψ(m_1,n_j_1)... ψ(m_1,n_j_4k)  Z ≠ 0,

by ℰ_N-1,4k since j_i ≠ 1 for all 1 ≤ i ≤ 4k, we also denote (a_1,...,a_N) the multiplicity for (j_1,...,j_4k), i.e.

    a_l = |{ν∈{1,...,4k}, j_ν = l}|.

Actually, if there exists l ≠ 1 s.t. a_l = 1, then the variable x_l enters exactly once in the integration, assume for simplicity that j_1 = l then

    ∫f̅_̅N̅ψ(m_1,n_j_1)... ψ(m_1,n_j_4k)  Z 
     =    ∫ψ(m_1,n_j_1)... ψ(m_1,n_j_4k)Π_i≠ l f_∞(z_i) ∫ψ(m_1,n_l)f_∞(z_l) = 0,

by the assumption of vanishing condition for ψ, provided l = j_1 ≠ 1. Using these notations, we have 

    1/(2k)!1/N^2k∑_j_1,...,j_4k = 1^N ∫f̅_̅N̅ψ(m_1,n_j_1)... ψ(m_1,n_j_4k)  Z 
     =    1/(2k)!1/N^2k∑_(j_1,...,j_4k) ∈ℰ_N-1,4k∫f̅_̅N̅ψ(m_1,n_j_1)... ψ(m_1,n_j_4k)  Z 
    ≤   1/(2k)!1/N^2k∑_(j_1,...,j_4k) ∈ℰ_N-1,4k( ∑_(j_2k+1,...,j_4k) ∈ℰ_N-1,4k 1 ) ∫f̅_̅N̅ |ψ(m_1,n_j_1)|^2... |ψ(m_1,n_j_2k)|^2  Z 
        +  1/(2k)!1/N^2k∑_(j_1,...,j_4k) ∈ℰ_N-1,4k( ∑_(j_1,...,j_2k) ∈ℰ_N-1,4k 1 ) ∫f̅_̅N̅ |ψ(m_1,n_j_2k+1)|^2... |ψ(m_1,n_j_4k)|^2  Z,

where ( ∑_(j_2k+1,...,j_4k) ∈ℰ_N-1,4k 1 ) for the first line, we mean that for each fixed former 2k indices in (j_1,...,j_4k) ∈ℰ_N-1,4k, we add all possible cases about later 2k indices in ℰ_N-1,4k, moreover, we have

    ∫f̅_̅N̅ |ψ(m_1, n_j_1)|^2...|ψ(m_1, n_j_2k)|^2dZ =     C^4k∫f̅_̅N̅ [(1 + |n_2|)^2]^a_2...[(1 + |n_N|)^2]^a_N Z 
     =      C^4k M_a_1^a_1M_a_2^a_2...M_a_N^a_N≤ (CΛ)^2k a_1^a_1...a_N^a_N,

with the convention that 0^0 = 1, then we have

    1/(2k)!1/N^2k∑_(j_1,...,j_4k) ∈ℰ_N-1,4k( ∑_(j_2k+1,...,j_4k) ∈ℰ_N-1,4k 1 ) ∫f̅_̅N̅ |ψ(x_1, x_j_1)|^2...|ψ(x_1, x_j_2k)|^2  Z 
    ≤   1/(2k)!1/N^2k(2C^2Λ)^2k∑_l=1^2k∑_J_4k∈ℰ_N-1,4k, |S(J_2k)| = l( ∑_(j_2k+1,...,j_4k) ∈ℰ_N-1,4k 1 ) a_1^a_1...a_N^a_N,

and 

    ( ∑_(j_2k+1,...,j_4k) ∈ℰ_N-1,4k 1 ) ≤∑_a'_1 + ... + a'_l = 2k, a'_1 ≥ 1,...,a'_l ≥ 1(2k)!/a'_1!...a'_l!≤ l^2k,

we denote

    ∑_l=1^2k∑_J_4k∈ℰ_N-1,4k, |S(J_2k)| = l l^2k a_1^a_1...a_N^a_N = ∑_l=1^2k l^2k C_N^l U^l_N,2k,

where

    U_N,2k^l := ∑_a_1 +... + a_l = 2k, a_1 ≥ 1,...,a_l ≥ 1(2k)!/a_1!...a_l! a_1^a_1...a_l^a_l,

using (2.13) and Lemma 6 of <cit.>, we have

    U_N,2k^l ≤ e^2k (2k)! ( ∑_a_1 +... + a_l = 2k, a_1 ≥ 1,...,a_l ≥ 1 1 ) ≤1/√(k) (2e)^2k(2k)!,

Insert (<ref>) and (<ref>) into (<ref>), we obtain that

    1/(2k)!1/N^2k∑_j_1,...,j_4k = 1^N ∫f̅_̅N̅ψ(m_1,n_j_1)... ψ(m_1,n_j_4k)  Z 
    ≤   (4eC^2Λ)^2k/(2k)!2/N^2k  (2k)!
    1/√(k)∑_l=1^2k C_N^l l^2k
     =    2/√(k) (8e^2C^2Λ)^2k∑_l=1^2k C_N^l l^2k N^-2k,

furthermore, by Stirling's formula

    C_N^l l^2k/N^2k = l^2k-l/N^2k-ll^l/N^lN!/(N-l)! l!≤1/√(π l)√(N/N-l)(N/N-l)^N-l,

and by assumption l < 3k ≤ N gives that N/N-l≤3/2, thus

    1/(2k)!1/N^2k∑_j_1,...,j_4k = 1^N ∫f̅_̅N̅ψ(m_1,n_j_1)... ψ(m_1,n_j_4k)  Z ≤ 4k (8e^2C^2Λ)^2k,

we finish the proof of this case. 

 
Case: 3k > N. By Cauchy inequality, we have

    1/N∑_j_1,j_2 = 1^N ψ(m_1, n_j_1) ψ(m_1, n_j_2) ≤C^2/N∑_j_1, j_2^N (1 + |x_j_1|)(1 + |x_j_2|) ≤ C^2 ∑_j=1^N (1 + |x_j|)^2,

then we have

    1/(2k)!1/N^2k∑_j_1,...,j_4k = 1^N ∫f̅_̅N̅ψ(m_1,n_j_1)... ψ(m_1,n_j_4k)  Z 
    ≤   C^4k/(2k)!∫f̅_̅N̅( ∑_j=1^N (1 + |x_j|)^2 )^2k Z ≤ (5e^2C^2 Λ)^2k,

which follows the same argument of Proposition 4 of <cit.>.

Finally, we combine (<ref>) and (<ref>), then we complete the proof.


Now let us take ψ = νψ_2 or νψ_3, we obtain the desired  bound O(1) of (<ref>) and (<ref>) by above proposition. 




§ PROOF OF THE MAIN RESULTS

In this section, we prove three main theorems we claimed in Section <ref>. We roughly divide all three proofs into two steps. 

∙ Step one is show the uniform-in-N estimates for the time evolution of our new quantity ℰ_N(f_N|f̅_N). The key point is to select suitable weight matrix function M_t to estimate the lower bound of new matrix S_t on the right hand side of (<ref>), and the lower bound of matrix S_t should be independent on particle number N. In the case of uniform-in-time propagation of chaos, we also need to deal with the second term on the right hand side of (<ref>),

    -2 ∫_(Ω×^d)^N f_N ∇R_N, M ∇logf_N/f^⊗ N Z,

this is exactly what we do in Lemma <ref>.

∙ Step two is to establish the Gronwall type inequality for time evolution of our new quantity ℰ_N(f_N|f̅_N) by some functional inequalities developed in Section <ref> and large deviation estimates proved in Section <ref>. In the case of uniform-in-time propagation of chaos, we require Log-Sobolev inequality and weighted Log-Sobolev inequality 
for non-linear equilibrium f_∞. However, we require that the N particle invariant measure f_N,∞ satisfies uniform-in-N Log-Sobolev inequality for the long time convergence of Eq.(<ref>), which is more harder than the case of uniform in time propagation of chaos which takes the reference measure as the tensor form f_∞^⊗ N. We refer <cit.> for the study of uniform-in-N functional inequalities.



 §.§ Proof of Theorem <ref>



Step1. Let us take f̅_N = f_N,∞ and R_N = 0. We observe that ∇_X ∇_V log f_N,∞ = ∇_V ∇_X log f_N,∞ = 0 and ∇_V ∇_V log f_N,∞ = - 1/2σ^-1γ Id_Nd × Id, then the matrix S in (<ref>) of Corollary <ref> reads as

    S = 
    (
    [               2F - ∂_t E - L^∗_NE  4γ F - ∂_t F - L^∗_NF + 2E ∇^2 U;               2G - ∂_t F - L^∗_NF 4γ G - ∂_t G - L^∗_N G + 2F ∇^2 U ]).

Now we select the weight matrix M as a constant matrix, 

    E = diag{δ a^3,...,δ a^3},    F = diag{δ a^2,...,δ a^2},    G = diag{2δ a,...,2δ a},

where E,F,G are Nd × Nd constant diagonal matrices
and a > 0, δ > 0 are to be confirmed, then the matrix S_t reduces to

    S = 
    (
    [               2F  4γ F + 2E ∇^2 U;               2G 4γ 
    G + 2F ∇^2 U ]).

Observe that the top left corner matrix F implies the entropy dissipation in position direction, and right down corner concerns the entropy dissipation in velocity direction. If ∇^2 U has a positive lower bound, i.e.∇^2 U ≥κ Id_dN × dN for some κ > 0, then we have 4 γ G + 2F ∇^2 U ≥ (8 δ a γ + 2δ a^2 κ) Id_dN × dN. However, our assumptions also treat ∇^2 U doesn't have positive lower bound. Since we assume that ∇^2 V _L^∞ < ∞ and ∇^2 W _L^∞ < ∞, we have
∇^2 U _L^∞≤ C_K + C_V < ∞. Now for right down corner, we choose a such that a∇^2 U_L^∞ < 2 γ, then we have

    a ≤2γ/C_K + C_V,

and

    ⟨∇_V u, (4γ G + 2F ∇^2 U) ∇_Vu ⟩≥ 4 δ a γ|∇_V u|^2.

For top right corner and left down corner, we deal with cross terms.
It is easy to obtain that

    ⟨∇_Xu, [4γ F + 2E ∇^2 U + 2G] ∇_Vu ⟩
    ≤   δ a[4 + 4aγ + 2 a^2(C_K + C_V)]|∇_V u||∇_X u| 
    ≤   δ a(4 + 8aγ) |∇_V u||∇_X u| 
    ≤   1/2δ a^2 |∇_X u|^2 + 1/2δ(4 + 8aγ)^2 |∇_V u|^2

by (<ref>) in the third line and Cauchy inequality in the fourth line. Combining (<ref>) and (<ref>), we have

    - ∫ f_N ∇ u, S∇ u ≤    - 3/2δ a^2 ∫ f_N |∇_X u|^2 - 4δ aγ∫ f_N |∇_V u|^2 
        + m_1 ∫ f_N |∇_V u|^2,

provided m_1 = 1/2δ(4 + 8a γ)^2 > 0. Now we choose δ such that m_1 ≤σ/4, then we have

    δ≤σ/2(4 + 8aγ)^2,

and

    d/dtℰ^M_N(t) ≤ - c_1/N∫ f_N |∇_X u|^2 - c_2/N∫ f_N |∇_V u|^2,

provided c_1 = 3/2δ a^2 and c_2 = 4δ a γ + σ/2. Finally we obtain the entropy dissipation in X and V direction with constants which are independent of N.

Step2. We directly use the results in <cit.> in this step, under Assumption <ref> and <ref> with C_K < 1, we have uniform-in-N Log Sobolev inequality of N particle invariant measure f_N, ∞,

    Ent_μ_N (f^2) ≤ρ_LS∫_^dN
    |∇ f|^2 dμ_N,   f ∈ C^1_b(^dN),

by our notation, let μ_N = f_N,∞, we have

    d/dtℰ^M_N(t) ≤ -c ℰ^M_N(t),

where c = 1/2(1+ρ_LS)min{3/2δ a^2, σ/2} is a constant depending on ρ_LS, U, σ, γ, then we finish the proof.





 §.§ Proof of Theorem <ref>

In this subsection, we prove the particle system (<ref>) approximates the unique equilibrium (<ref>) of Vlasov-Fokker-Planck equation (<ref>) when diffusion strength σ is large enough. We deal with two kinds of confining potentials by selecting different weight matrix M. When confining potentials V satisfy Assumption <ref>, we choose

    E = diag{δ a^3,...,δ a^3},    F = diag{δ a^2,...,δ a^2},    G = diag{2δ a,...,2δ a},

where E,F,G are Nd × Nd constant diagonal matrices
and δ, a > 0 are to be confirmed. When confining potentials V satisfy Assumption <ref>, we choose

    {    E = diag{e(z_1)Id_d × d,...,e(z_N)Id_d × d}, 
        F = diag{f(z_1)Id_d × d,...,f(z_N)Id_d × d}, 
        G = diag{g(z_1)Id_d × d,...,g(z_N)Id_d × d}, 
    .


where E,F,G are also Nd × Nd diagonal matrices, and e(z), f(z), g(z) are d × d diagonal matrices which should be understood as e(z)Id_d× d, f(z)Id_d× d and g(z)Id_d× d, we omit the symbol “Id_d × d" for convenience. We choose e(z), f(z) and g(z) as

    e(z) = δ a^3(H(z))^-3θ,  
    b(z) = δ a^2(H(z))^-2θ,   c(z) = 2 δ a(H(z))^-θ,

where 

    H(z) = v^2/2 + V(x) + H_0,    H_0 > 0,

and δ, a, θ, H_0 > 0 are to be confirmed. In the next, we establish Gronwall type inequality for the quantity ℰ_N^M(f^t_N|f^⊗ N_∞). 



Case 1. The confining potentials V satisfy Assumption <ref> and we select M as constant diagonal matrix defined by (<ref>).

Step 1. The matrix M is exactly the same as we have used in the last subsection, we immediately have 

    - ∫ f_N ∇ u, S ∇ u ≤    - 3/2δ a^2 ∫ f_N |∇_X u|^2 - 4δ aγ∫ f_N |∇_V u|^2 
        + m_1 ∫ f_N |∇_V u|^2,

provided m_1 = 1/2δ(4 + 8a γ)^2 and a ≤2γ/C_K + C_V. By Lemma <ref>, we have

    ∫ f_N ∇R_N, M ∇ u≤    n_1 ∫ f_N |∇_X u|^2 + n_2 ∫ f_N |∇_V u|^2 
        + 8δ a^3/σ∫ f_N |R_N^1|^2 + δ a^2 γ^2(1+a)/2σ^2∫ f_N |R_N^3|^2 
        + ( 2 γ^2 δ a^3/σ^2 + γ^2 δ^2 a^2/σ^3) ∫ f_N |R_N^2|^2 
        + ∑_i=1^N σ/16∫ f_N | √(E^ii)∇_v_i∇_x_i u|_^2d^2 + σ/16∫ f_N | √(F^ii)∇_v_i∇_v_i u|_^2d^2,

provided n_1 = 1/4δ a^2 + (C_K + 1/2)δ a^3 and n_2 = σ/4 + δ a^2(C_K a + C_K + 1/2). By Lemma <ref> and Young inequality, we have

    d/dt H_N(f^t_N|f^⊗ N_∞) ≤ - 3 σ/41/N∫ f_N|∇_V u|^2 + 1/σ1/N∫ f_N|R^0_N|^2.

Combining (<ref>), (<ref>) and (<ref>), we obtain 

    d/dtℰ_N^M(f_N|f_∞^⊗ N) ≤    - c_1/N∫ f_N |∇_X u|^2 - c_2/N∫ f_N |∇_V u|^2 
        + 8δ a^3/σ1/N∫ f_N |R_N^1|^2 + δ a^2 γ^2(1+a)/2σ^21/N∫ f_N |R_N^3|^2 
        + ( 2 γ^2 δ a^3/σ^2 + γ^2 δ^2 a^2/σ^3) 1/N∫ f_N |R_N^2|^2 + 1/σ1/N∫ f_N|R_N|^2,

where c_1 = -1/2δ a^2 + (C_K + 1/2)δ a^3 and c_2 = -σ/2 - 4 δ a γ + δ a^2(C_K a + C_K + 1/2) + m_1. Recall we choose a such that a ≤2 γ/C_K + C_V, now we update a such that

    a ≤min{2 γ/C_K + C_V, 1/4C_K + 2},

then we have c_1 ≤ -1/4δ a^2 and δ a^2(C_K a + C_K + 1/2) ≤δ a. In the next we choose δ such that 

    δ≤σ/4[8 + a + 28aγ + 32a^2 γ^2],

we have c_2 ≤ - σ/4. Finally, we update (<ref>) as

    d/dtℰ_N^M(f_N|f_∞^⊗ N) ≤    - δ a^2/41/N∫ f_N |∇_X u|^2 - σ/41/N∫ f_N |∇_V u|^2 
        + 8δ a^3/σ1/N∫ f_N |R_N^1|^2 + δ a^2 γ^2(1+a)/2σ^21/N∫ f_N |R_N^3|^2 
        + ( 2 γ^2 δ a^3/σ^2 + γ^2 δ^2 a^2/σ^3) 1/N∫ f_N |R_N^2|^2 + 1/σ1/N∫ f_N|R_N|^2.


Step2. We start from (<ref>). By Proposition <ref>, we have

    1/N∫ f_N|R^1_N|^2 ≤1/νH_N(f^t_N|f^⊗ N_∞) + C/N,

and by Proposition <ref>, we have

    1/N∫ f_N|R^i_N|^2 ≤1/νH_N(f^t_N|f^⊗ N_∞) + C/N,

where i = 0,2,3, the constant ν > 0 satisfies (16C^2_Ke^2 Λ) ν < 1 and the constant C > 0 depends on ν, σ, γ, K and V. Using (<ref>) and (<ref>), we have

    d/dtℰ_N^M(f_N|f_∞^⊗ N) ≤    - δ a^2/41/N∫ f_N |∇_X u|^2 - σ/41/N∫ f_N |∇_V u|^2 
        + n_3/ν H_N(f_N|f_∞^⊗ N) + C/N,

where

    n_3 = [8δ a^3/σ + δ a^2 γ^2(1+a)/2σ^2 + 2 γ^2 δ a^3/σ^2 + γ^2 δ^2 a^2/σ^3 + 1/σ].

Now let us determine the value of a > 0, δ > 0 and ν > 0 such that

    n_3 ρ_ls/ν≤min{δ a^2/8, σ/8},

where ρ_ls is the Log-Sobolev inequality constant of f_∞ (See Corollary <ref>).
Recall the constant a > 0 satisfies

    a ≤min{2 γ/C_K + C_V, 1/4C_K + 2} < 1,

now we choose

    δ = σ/4[10 + 28 γ + 32 γ^2]≤σ/4[8 + a + 28aγ + 32a^2 γ^2]

by (<ref>). We also choose 1/ν = 20C_K^2 e σ/γ with Λ≤1/e λ satisfies λ∈ (0, β) in Proposition <ref> and <ref>. Observing that δ a^2 < σ by a < 1 and δ < σ, we only need

    n_3 ρ_ls/ν≤δ a^2/8.

Now we choose a and σ to make every term of n_3 less than δ a^2 ν/40 ρ_ls,

    {   8δ a^3/σ≤δ a^2 ν/40 ρ_ls⟹ a ≤γ/6400 eρ_ls C_K^2, 
       δ a^2 γ^2/σ^2≤δ a^2 ν/40 ρ_ls⟹σ≥ 3200ρ_lse γ C_K^2, 
       2 γ^2 δ a^2/σ^2≤δ a^2 ν/40 ρ_ls⟹σ≥ 1600ρ_lse γ C_K^2, 
       γ^2 δ a^2/σ^2≤δ a^2 ν/40 ρ_ls⟹σ≥ 3200ρ_lse γ C_K^2, 
       1/σ≤δ a^2 ν/40 ρ_ls⟹σ≥160[10 + 28 γ + 32 γ^2]ρ_lseC_K^2/a^2 γ.
    .

We choose a such that

    a ≤min{2 γ/C_K + C_V, 1/4C_K + 2,  γ/6400 eρ_ls C_K^2},

by the first line above and (<ref>), for example, saying  

    a = min{γ,1}/6400 eρ_ls (C_K+1)^2 + 4C_V + 4C_K + 2,

then we have the range of diffusion constant in our result,

    σ≥max{160[10 + 28 γ + 32 γ^2]ρ_lse/a^2 γ, 3200ρ_lseγ}C_K^2.

Finally, we update (<ref>) as the following

    d/dtℰ_N^M(f^t_N|f_∞^⊗ N) ≤    - δ a^2/161/N∫ f_N |∇_X u|^2 - σ/161/N∫ f_N |∇_V u|^2 
        - δ a^2/16 ρ_lsH_N(f^t_N|f_∞^⊗ N) + C/N
    ≤    -δ a^2/16(ρ_ls + 1)ℰ_N^M(f^t_N|f_∞^⊗ N) + C/N,

we finish the proof of this case. 


Case 2. The confining potentials V satisfy Assumption <ref> and we select M as diagonal mateix function defined by (<ref>).  

Step1. We take E,F,G as the following tensorized matrices,

    E = diag{e(z_1),...,e(z_N)},   F = diag{f(z_1),...,f(z_N)},   G = diag{g(z_1),...,g(z_N)},

we take e,f,g as scalar functions read as

    e(z) = δ a^3 (H(z))^-3θ,  
    f(z) = δ a^2 (H(z))^-2θ,  
    g(z) = 2 δ a (H(z))^-θ,

where 

    H(z) = v^2/2 + V(x) + H_0,    H_0 > 0,

and δ, a, θ, H_0 > 0 are to be confirmed. We recommend <cit.> to readers for similar argument of one particle version. Now let us estimate the lower bound of matrix S in Corollary <ref>. We recall 

    S = 
    (
    [                      2F - ∂_t E - L^∗_NE  D_1 - ∂_t F - L^∗_NF - 2E ∇^2 U + 2 γ F;                      2G - ∂_t F - L^∗_NF D_2 - ∂_t G - L^∗_N G - 2F ∇^2 U + 2 γ G ]),

after we take f̅_̅N̅ = f^⊗ N_∞, we have 

    S = 
    (
    [                2F - L^∗_NE 4 γ F  - L^∗_NF + 2E ∇^2 U;               2G -  L^∗_NF 4 γ G - L^∗_N G + 2F ∇^2 U ]).

For the left top corner element, the diagonal matrix 
F still offers the dissipation in position direction. Observing that 2F - L^∗_N E is a Nd × Nd diagonal matrix, we could estimate them like scalar. Before that, we firstly estimate the upper bound of L^∗_N E, L^∗_N F and L^∗_N G. For some s > 0, we have

    L^∗_N (H^-sθ(z_i))
    
     =    ∑_j=1^N (v_j ·∇_x_j - ∇_x_jU ·∇_v_j - γ v_j ·∇_v_j + σΔ_x_j)H^-sθ(z_i)
    
     =     -(sθ) {σ d/H + σ (-sθ -1)v_i^2/H^2 - γ v_i^2/H - (1/N∑_j:j ≠ i v_i ·∇ W(x_i - x_j))H^-1}H^-s
    θ(z_i).

Recall we assume that ∇ W_L^∞ < ∞ in this case, we have

    |L^∗_N (H^-sθ(z_i))| ≤ sθ(σ d + σ(sθ + 1)/H_0 + γ + ∇ W_L^∞) H^-s θ(z_i),

now we choose H_0 large enough, which satisfies σ(d + 3θ + 1)/H_0≤γ, we have

    |L^∗_N (H(z_i)^-sθ)| ≤ sθ(2 γ + ∇ W_L^∞)H^-s θ.

For i-th component of 2F - L_N^∗E, we have  

    2f(z_i)  - L_N^∗(e(z_i)) =     2 δ a^2 H^-2θ(z_i) - δ a^3 L^∗_N (H^-3θ(z_i)) 
    ≥    2 δ a^2 H^-2θ - 3θδ a^3(2γ + ∇ W_L^∞) H^-3θ,

again we choose H_0 such that 3θ a (2γ + ∇ W_L^∞)/H_0^θ≤ 1, we have

    ⟨∇_Xu, (2F - L_N^∗E)∇_X u ⟩≥δ a^2 ∑_i=1^N H^-2θ(z_i
    )|∇_x_iu|^2,

provided H_0 satisfies

    H_0 ≥max{σ(d + 3 θ + 1)/γ, [3θ a (2γ + ∇ W_L^∞)]^1/θ}.

For the right down corner of S, we recall the formulation of dN × dN matrix ∇^2 U in (<ref>), and we have V^-2θ∇^2 V_L^∞≤ C_V^θ by Assumption <ref>, then we obtain

    H^-2θ∇^2 U_L^∞≤V^-2θ∇^2 V_L^∞ + C_K/H_0≤ C_V^θ + C_K < ∞,

provided H_0 > 1 for convenience. Combining (<ref>) and (<ref>), we immediately have

    - ∇_V u, (4 γ G - L^∗_N G + 2F ∇^2 U) ∇_V u 
    ≤   δ a[4γ + θ(2γ + ∇ W_L^∞) + 2a(C_V^θ + C_K)]|∇_V u|^2,

we choose a such that a(C^θ_V + C_K) ≤γ, we have

    - ∇_V u, (4 γ G - L^∗_N G + 2F ∇^2 U)∇_V u 
    ≤   δ a[6γ + θ(2γ + ∇ W_L^∞)
    ]|∇_V u|^2.

For cross terms, by (<ref>) and similar argument above, we have

    ∇_X u, (4 γ F + 2G - 2 L^∗_NF + 2E ∇^2 U) ∇_V u 
    ≤    [4 γ F + 2G + 2δ a^2|L^∗_N(H^-2θ(z_i))| + 2 δ a^3(C_V^θ + C_K)H^-θ]|∇_Vu||∇_Xu| 
    ≤   δ a H^-θ[4 + 4γ a + 4a θ(2γ + ∇ W_L^∞)H^-θ + 2a^2(C_V^θ + C_K)]|∇_Vu||∇_Xu| 
    ≤   δ a H^-θ[4 + 6γ a + 4aθ(2γ + ∇ W_L^∞)]|∇_Vu||∇_Xu| 
    ≤   1/4δ a^2 H^-2θ|∇_Xu|^2 + δ [4 + 6γ a + 4aθ(2γ + ∇ W_L^∞)]^2|∇_Vu|^2.

Now we collect (<ref>), (<ref>) and (<ref>), we have

    - ∫ f_N∇ u, S ∇ u≤ - 3/4δ a^2 ∑_i=1^N ∫ f_N H^-2θ(z_i)|∇_x_iu|^2 + m_2 ∫ f_N |∇_Vu|^2,

where m_2 = δ [4 + 6γ a + 4aθ(2γ + ∇ W_L^∞)]^2 + δ a[6γ + θ(2γ + ∇ W_L^∞)]. Recall we choose H_0 > 1 for convenience, and again by Lemma <ref>,

    ∫ f_N ∇R_N, M ∇ u≤    n'_1 ∑_i=1^N ∫ f_N H^-2θ(z_i)|∇_x_i u|^2 + n'_2 ∫ f_N |∇_V u|^2 
        + 8δ a^3/σ∫ f_N |R_N^1|^2 + δ a^2 γ^2(1+a)/2σ^2∫ f_N |R_N^3|^2 
        + ( 2 γ^2 δ a^3/σ^2 + γ^2 δ^2 a^2/σ^3) ∫ f_N |R_N^2|^2 
        + ∑_i=1^N σ/16∫ f_N | √(E^ii)∇_v_i∇_x_i u|_^2d^2 + σ/16∫ f_N | (F^ii)^1/4∇_v_i∇_v_i u|_^2d^2 
        + ∑_i=1^N ∫ f_N ⟨ R^1_N,i, (∇_v_i E^ii) ∇_x_i u ⟩_^2d + ∫ f_N ⟨ R^1_N,i, (∇_v_i F^ii) ∇_v_i u ⟩_^2d.

provided n'_1 = 1/4δ a^2 + (C_K + 1/2)δ a^3 and n'_2 = σ/4 + δ a^2(C_K a + C_K + 1/2). 
We only need to estimate the last two term. By the computation (<ref>), we have

    ∫ f_N ⟨ R^1_N,i, (∇_v_i E^ii) ∇_x_i u ⟩_^2d
    ≤   3 θδ a^3/2H^3θ_0∫ f_N |R_N,i^1|^2 + 3 θδ a^3/2∫ f_N H^-2θ(z_i)|∇_x_i u|^2,

and 

    ∫ f_N ⟨ R^1_N,i, (∇_v_i F^ii) ∇_v_i u ⟩_^2d
    ≤   8θδ^2 a^4/H_0^4θσ∫ f_N |R_N,i^1|^2 + σ/16∫ f_N |∇_v_iu|^2.

Now we insert (<ref>) and (<ref>) into (<ref>), and combine with (<ref>), we obtain

    ∫ f_N ∇ u, M ∇ u≤    n_1 ∑_i=1^N ∫ f_N H^-2θ(z_i)|∇_x_i u|^2 + n_2 ∫ f_N |∇_V u|^2 
        + (8δ a^3/σ + 8θδ^2 a^4/H_0^4θσ + 3 θδ a^3/2H^3θ_0) ∫ f_N |R_N^1|^2 
        + ( 2 γ^2 δ a^3/σ^2 + γ^2 δ^2 a^2/σ^3) ∫ f_N |R_N^2|^2 
        + δ a^2 γ^2(1+a)/2σ^2∫ f_N |R_N^3|^2

provided n_1 = - 1/2δ a^2 + (C_K + 1/2 + 3θ/2) δ a^3 and n_2 = 5σ/16 + δ a^2(C_K a + C_K + 1/2) + m_2. By Lemma <ref> and Young inequality, we have

    d/dt H_N(f^t_N|f^⊗ N_∞) ≤ - 3 σ/41/N∫ f_N|∇_V u|^2 + 1/σ1/N∫ f_N|R^0_N|^2.

Finally, we combine (<ref>) and (<ref>), and update the constants a and δ as

    a ≤min{1/4C_K + 6θ + 2, γ/C_V^θ + C_K}

and 

    δ≤3 σ/8 + 32C_K + m_2',

where m_2' = [4 + 6γ a + 4aθ(2γ + ∇ W_L^∞)]^2 + a[6γ + θ(2γ + ∇ W_L^∞)] comes from (<ref>), we obtain,

    d/dtℰ_N^M(f_N^t|f_∞^⊗ N) ≤    - δ a^2/41/N∑_i=1^N ∫ f_N H^-2θ(z_i)|∇_x_i u|^2 - σ/41/N∫ f_N |∇_V u|^2 
        + (8δ a^3/σ + 8θδ^2 a^4/H_0^4θσ + 3 θδ a^3/2H^3θ_0) 1/N∫ f_N |R_N^1|^2 
        + ( 2 γ^2 δ a^3/σ^2 + γ^2 δ^2 a^2/σ^3) 1/N∫ f_N |R_N^2|^2 
        + δ a^2 γ^2(1+a)/2σ^21/N∫ f_N |R_N^3|^2 + 1/σ1/N∫ f_N |R_N^0|^2.



Step2. We start from (<ref>). By similar argument of the last case and using Proposition <ref> and <ref>,

    1/N∫ f_N|R^i_N|^2 ≤1/νH_N(f^t_N|f^⊗ N_∞) + C/N,

where i = 0,1,2,3, we have

    d/dtℰ_N^M(f_N|f_∞^⊗ N) ≤    - δ a^2/41/N∫ f_N |∇_X u|^2 - σ/41/N∫ f_N |∇_V u|^2 
        + n_4/ν H_N(f_N|f_∞^⊗ N) + C/N,

the constant ν > 0 satisfies (16C^2_Ke^2 Λ) ν < 1, the constant C > 0 depends on ν, σ, γ, K and V, and the constant ν collects all coefficients of error terms which reads as

    n_4 = [(8δ a^3/σ + 8θδ^2 a^4/H_0^4θσ + 3 θδ a^3/2H^3θ_0) + δ a^2 γ^2(1+a)/2σ^2 + 2 γ^2 δ a^3/σ^2 + γ^2 δ^2 a^2/σ^3 + 1/σ].

Now let us determine the value of δ, a, H_0, ν > 0 such that

    n_4 ρ_wls/ν≤min{δ a^2/8, σ/8},

where ρ_wls is the constant of weighted Log-Sobolev inequality constant of f_∞ (See Corollary <ref>). Recall the constant a > 0 satisfies

    a ≤min{1/4C_K + 6θ + 2, γ/C_K + C^θ_V} < 1

by (<ref>). Now we choose

    δ = σ/8 + 32C_K + m_2”≤3 σ/8 + 32C_K + m_2'

by (<ref>), where m_2” = [4 + 6γ + 4θ(2γ + ∇ W_L^∞)]^2 + [6γ + θ(2γ + ∇ W_L^∞)]. We also choose 1/ν = 20C_K^2 e σ/γ with Λ≤1/e λ satisfies λ∈ (0, β) in Proposition <ref> and <ref>. Observing that δ a^2 < σ by a < 1 and δ < σ, we only need

    n_4 ρ_ls/ν≤δ a^2/8.

Recall the lower bound of H_0 in Step1,

    H_0 ≥max{σ(d + 3 θ + 1)/γ, [3θ a (2γ + ∇ W_L^∞)]^1/θ, 1},

by (<ref>) and (<ref>), now we choose H_0 such that

    H_0 ≥max{(8θσ)^1/4θ, (2 θσ)^1/3θ},

we have

    (8δ a^3/σ + 8θδ^2 a^4/H_0^4θσ + 3 θδ a^3/2H^3θ_0) ≤10 δ a^3/σ.

In the next, we make every term of n_4 less than δ a^2 ν/40ρ_wls. For the first term of n_4,    

    10 δ a^3/σ≤δ a^2 ν/40 ρ_wls⟹ a ≤γ/8000 eρ_wls C_K^2,

we obtain the range of constant a

    a ≤min{1/4C_K + 6θ + 2, γ/C_V^θ + C_K, γ/8000 eρ_wls C_K^2},

for example, we can take a as

    a = min{γ,1}/8000 eρ_wls C_K^2 + 5C_K + 6θ + C_V^θ + 2.

For other terms, we obtain the range of diffusion constant σ by similar argument of (<ref>) in the last case,

    σ≥max{800(40+m_2”)ρ_wlse/a^2 γ, 3200ρ_wlseγ}·max{C_K^2, C_K^3},

Finally, we update (<ref>) as the following

    d/dtℰ_N^M(f^t_N|f_∞^⊗ N) ≤    - δ a^2/161/N∫ f_N |∇_X u|^2 - σ/161/N∫ f_N |∇_V u|^2 
        - δ a^2/16 ρ_wlsH_N(f^t_N|f_∞^⊗ N) + C/N
    ≤    -δ a^2/16(ρ_wls + 1)ℰ_N^M(f^t_N|f_∞^⊗ N) + C/N,

we finish the proof of this case.




 §.§ Proof of Theorem <ref>

In the last subsection, we have shown that f_N converges to f_∞^⊗ N in the sense of quantity ℰ_N^M(f_N^t|f^⊗ N_∞) as t →∞ and N →∞. In this subsection, we eliminate the effect of replacement of f by f_∞ by proving exponential decay from f_t to f_∞. By triangle inequality of Wasserstein distance, the error between 𝒲_2^2(f^t_N, f_t^⊗ N) and 𝒲_2^2(f^t_N, f^⊗ N_∞) is only a time exponential decay factor. Combining with short time propagation of chaos result, we obtain uniform-in-time propagation of chaos. 



Case1. In this case, we take weight matrix M_1 as (<ref>). We have shown that 

    𝒲_2^2(f^t_N, f^⊗ N_∞) ≤ H(f^t_N|f_∞^⊗ N) ≤  N e^-c_1tℰ_N^M_1(f^0_N|f^⊗ N_∞) + C

in Theorem <ref>, 
where c_1 = δ a^2/16(1 + ρ_ls) and C > 0 depends on σ, M, C_K and C_V.  Moreover, Theorem <ref> implies that

    𝒲_2^2(f_t^⊗ N, f^⊗ N_∞) ≤ H(f_t^⊗ N|  f^⊗ N_∞) ≤ρ_LS N e^-ctℰ^M(f_0|f̂_0)

with some c > 0 under condition C_K < 1. Theorem <ref> also implies the same conclusion under condition that the interaction function F is functional convex (See Assumption <ref>). Hence, 
by triangle inequality of 2-Wasserstein distance, we have

    𝒲^2_2(f^t_N,f_t^⊗ N) ≤ N (1+ρ_LS)[ℰ_N^M_1(f^0_N|f^⊗ N_∞) + ℰ^M(f_0|f̂_0)]e^-c_1't + C,

where constant c_1' = min{c_1, c}. If we take chaotic initial data for Liouville equation (<ref>), i.e. f^0_N = f_0^⊗ N, we have

    𝒲^2_2(f^t_N,f_t^⊗ N) ≤ 2N (1+ρ_LS)ℰ^M(f_0|f̂_0)e^-c_1't + C.

For k-marginal distribution, we have 

    𝒲^2_2(f^t_N,k, f_t^⊗ k) ≤  2k(1+ρ_LS)e^-c_1'tℰ^M(f_0|f̂_0) + Ck/N.


Case2. In this case, we take weight matrix M_2 as (<ref>). The main argument of this case is the same with Case1. By Theorem <ref>, we have

    𝒲_2^2(f^t_N, f^⊗ N_∞) ≤ H(f^t_N|f_∞^⊗ N) ≤  N e^-c_2tℰ_N^M_2(f^0_N|f^⊗ N_∞) + C.

provided c_2 = δ a^2/16(1 + ρ_wls) and C = C(σ, M_2, C_K, C_V^θ). By Theorem <ref>, we have 

    𝒲_2^2(f_t^⊗ N, f^⊗ N_∞) ≤ H(f_t^⊗ N|  f^⊗ N_∞) ≤ N e^-c_2”tℰ^M_2'(f_0|f̂_0)

with constant c_2” > 0 under condition that the interaction function F is functional convex. We can also use the similar result under condition that the constant C_K is small, which have been proved by Guillin, Le Bris and Momarché in <cit.> (See Theorem 1.1). All in all, we have

    𝒲_2^2(f_t^⊗ N, f^⊗ N_∞) ≤ N  e^-c”_2tℰ^M(f_0|f̂_0),

provided some constant c_2” > 0. Now we take c_2' = min{c_2,c_2”}, we have

    𝒲^2_2(f^t_N,f_t^⊗ N) ≤ N  [ℰ_N^M_2(f^0_N|f^⊗ N_∞) + ℰ^M'_2(f_0|f̂_0)]e^-c'_2t + C,

If we take chaotic initial data for Liouville equation (<ref>), i.e. f^0_N = f_0^⊗ N, we have

    𝒲^2_2(f^t_N,f_t^⊗ N) ≤ 2N  ℰ^M'_2(f_0|f̂_0)e^-c'_2t + C.

For k-marginal distribution, we have 

    𝒲^2_2(f^t_N,k, f_t^⊗ k) ≤  2ke^-c_2'tℰ^M'_2(f_0|f̂_0) + C k/N.






§ APPENDIX

In appendix, we give the sketch of proof of Theorem <ref> and Theorem <ref>. The main idea of proof has already appeared in other articles, like <cit.> and <cit.>. For the completeness of the article, we reprove them under the current setting in the following.



 §.§ Proof of Theorem <ref>

The main idea originates from Theorem 10 in <cit.> and Theorem 3 in <cit.>. By the finite time propagation of chaos result for lipschitz force ∇ W and ∇ V, we could conclude that f_N,1(t,z) converges to f(t,z) weakly for each t > 0, then by lower semi-continuity of relative entropy, we have

    H(f|α) ≤lim inf_N →∞ H(f_N,1|α),

where α∝ e^- v^2/2 - V(x), then we get

    lim inf_N →∞ H_N(f_N|f_N,∞) = lim inf_N →∞{ H_N(f_N|α^⊗ N) + 1/2N^2∫ W(x_1 - x_2) f_N,2 + 1/Nlog Z_N,β}.

For the first term, we use Lemma 18 in <cit.>, we have

    H_N(f_N| α^⊗ N) ≥ H(f_N,1|α).

For the second term, we use |W(x)| ≤ C(1 + |x|^2) and f_N,2 weakly converges to f^⊗ 2 for each t > 0, hence

    lim inf_N →∞1/2N^2∫ W(x_1 - x_2) f_N,2 = 1/2∫ W ∗ρρ,   ρ = ∫_^d f(x,v)dv.

For third term, we conclude by <cit.> that

    lim inf_N →∞1/Nlog Z_N, β = - inf_f ∈ℳ_1(Ω) E(f).

Hence we have

    H_W(f) ≤lim inf_N →∞ H_N(f_N|f_N,∞).
 
Now by Theorem <ref> with constant c > 0, we obtain that

    H_W(f) ≤ e^-ctℰ_N^M(f_0^⊗ N|f_N,∞),

where the constant c = c(f_N,∞, M) is the same as Theorem <ref>.
Furthermore, we use Lemma 17 and Lemma 23 in <cit.> with α∝ e^-v^2/2 - V(x), let N →∞, we have

    H_W(f) ≤ e^-ctℰ^M(f_0|f̂_0).

Again, by Theorem 3 in <cit.>, we have

    𝒲_2^2(f_t, f_∞) ≤ρ_LS H_W(f_t) ≤ρ_LSℰ^M(f_0) e^-ct.





 §.§ Proof of Theorem <ref>

In this subsection, we show the convergence from f to f_∞ for limiting equation (<ref>) with more general confinement potential V. We use the quantity ℰ^M(f_t|f̂_t) to show the convergence, which combines free energy and relative Fisher Information of f_t and f̂_t, i.e.

    ℰ^M(f_t|f̂_t) = ℱ(f_t) - ℱ(f_∞) + I^M(f_t|f̂_t),

where f̂_t ∝ e^-1/2v^2 - V(x) - W ∗ρ_t, the later term is

    I^M(f_t| f̂_t) = ∫ f_t ∇ u, M ∇ u  dz,

where ∇ = (∇_x, ∇_v), u = logf_t/f̂_t and M is the weighted matrix which is to be decided. We follow the idea of Theorem 2.1 in <cit.> to deal with the nonlinear term ∇ W ∗ρ_t of Eq.(<ref>). Now let us compute the time evolution of ℰ^M(f_t|f̂_t). We omit the footnote “t" for convenience. For free energy part ℱ(f), we directly use the result in Theorem 2.1 in <cit.>,

    d/dtℱ(f) = - σ∫ f|∇_v u|^2 dz.

For distorted Fisher Information part, we formally write the equation of logf̂ as following

    ∂_t logf̂ + v ·∇_x logf̂ - [∇ V + (K ∗ρ)] ·∇_v logf̂ =    σΔ_v f̂/f̂ + γ (d + v ·∇_v logf̂) 
        - ∂_t (log Z_t + W ∗ρ),

by similar computation of Lemma <ref>, we have the equation of u = logf/f̂,



    ∂_t u = - B_t u - σ A^∗ A + σ∇_v log f · Au + R,

where B_t = v ·∇_x - (∇ V + ∇ W ∗ρ_t) ·∇_v - γ v ·∇_v, A = ∇_v, A^∗ = - ∇_v + v which is conjugate operator of A in the sense of L^2(f̂), R = ∂_t (log Z_t + W ∗ρ). Then by Lemma <ref> and Corollary <ref>, if we take M = 
(
[ e f; f g ]), 
where e,f,g: Ω×^d →^d × d are smooth matrix-valued functions, then we have

    d/dt{∫ f ∇ u, M ∇ u }≤    - ∫ f ∇ u, S ∇ u  + 2 ∫ f ∇ R, M ∇ u,

where S(Z) is a matrix reads as

    S = 
    (
    [               2e - L^∗ e 4 γ f - L^∗ f - 2e ∇^2 U;               2c - L^∗ f 4 γ c - L^∗ c - 2f ∇^2 U ]),

and ∇^2 U = ∇^2 V + ∇^2 W ∗ρ, and L^∗ = B + σΔ_v. Now we follow the similar argument of the Case 2 of Theorem <ref>, we select e(z) = δ a^3 (H(z))^-3θ, 
f(z) = δ a^2 (H(z))^-2θ, 
g(z) = 2δ a(H(z))^-θ where δ > 0, a > 0 are to be confirmed. We firstly estimate the upper bound of L^∗e, L^∗f and L^∗g. For some S>0, we have 

    L^∗(H^-sθ(z)) = -sθ{σ d/H + σ (-3θ -1)v^2/H^2 - γ v^2 + v ·∇ W ∗ρ(x)/H}H^-sθ(z),

then we have

    |L^∗(H^-sθ(z))| ≤ sθ(σ d + σ(sθ + 1)/H_0 + γ + 1 ) H^-s θ(z).

Now we can replace the norm ∇ W_L^∞ by 1 in the Case 2 in Theorem <ref> and use completely the same argument of (<ref>)-(<ref>), 
we have

    - ∫ f∇ u, S ∇ u≤ - 3/4δ a^2 ∫ f H^-2θ(z)|∇_xu|^2 + m_2 ∫ f |∇_vu|^2,

where m_2 = δ [4 + 6γ a + 4aθ(2γ + 1)]^2 + δ a[6γ + θ(2γ + 1)] and a, H_0 satisfy

    H_0 ≥max{σ(d + 3 θ + 1)/γ, [3θ a (2γ + 1)]^1/θ,1 },   a ≤γ/C^θ_V + C_K.

For second term of (<ref>), we have ∇ R = ∇ W ∗ (∂_t ρ), but 

    ∂_t ρ + ∇_x · (v_t^x ρ_t) = 0

where

    v_t^x(x) = ∫ vf(x,v)dv/∫ f(x,v)dv = ∫ (∇_v u) f(x,v)dv/∫ f(x,v)dv.

By (4.14) of <cit.>, we then have

    |∇ W ∗ (∂_t ρ)| = |∇^2 W ∗ (v_t^x ρ_t)| = | ∫∇^2 W(x-y) · v_t^yρ(y)dy | ≤ C_K ∇_v u _L^2(f),

therefore

    ∫ f ∇ R, M ∇ u≤    C_K ∇_v u _L^2(f) M ∇ u _L^2(f)
    ≤   σ/4∫ f |∇_v u|^2 + C^2_K/σ∫ f ∇ u, M^2 ∇ u 
    ≤   σ/4∫ f |∇_v u|^2 + δ^2 a^2 C^2_K/σ[(a^2 + 4)+2(a^2 + 2)^2]∫ f |∇_v u|^2 
        + δ^2 a^4 C^2_K/σ(a^2 + 3)∫ f H^-2θ(z)|∇_x u|^2.

Combining (<ref>), (<ref>) and (<ref>), we have

    d/dtℰ^M(f|f̂) ≤    (-3σ/4 + m_2)∫ f |∇_v u|^2 + δ^2 a^2 C^2_K/σ[(a^2 + 4)+2(a^2 + 2)^2]∫ f |∇_v u|^2 
        + [-3/4δ a^2 + δ^2 a^4 C^2_K/σ(a^2 + 3)]∫ f H^-2θ(z)|∇_x u|^2.

Now we choose δ such that

    {   δ^2 a^2 C^2_K/σ[(a^2 + 4)+2(a^2 + 2)^2] ≤σ/4, 
       δ^2 a^4 C^2_K/σ(a^2 + 3) ≤1/4δ a^2, 
        m_2 ≤σ/4,
    .

i.e.

    δ≤min{σ/2aC_K √([(a^2 + 4)+2(a^2 + 2)^2]), σ/4a^2C^2_K(a^2 + 3), σ/4m_2'},

where m_2' = [4 + 6γ a + 4aθ(2γ + 1)]^2 + a[6γ + θ(2γ + 1)],
then we have

    d/dtℰ^M(f|f̂) ≤ -σ/4∫ f |∇_v u|^2 - 1/4δ a^2 ∫ f H^-2θ(z)|∇_x u|^2.

By weighted Log Sobolev inequality of f̂ and δ a^2 < σ, we have 

    d/dtℰ^M(f_t|f̂_t) ≤ - δ a^2/8 I^M(f|f̂) - δ a^2/8 ρ_wlsH(f|f̂),

furthermore, by convexity of functional F, we have entropy sandwich inequality ℱ(f_t) - ℱ(f_∞) ≤ H(f_t|f̂_t), we have

    d/dtℰ^M(f_t|f̂_t) ≤  - δ a^2/16 + 16 ρ_wlsℰ^M(f_t|f̂_t),

then we finish the proof.




 §.§ Acknowledgements

Zhenfu Wang was partially supported by  the National Key R&D Program of China, Project Number 2021YFA1002800, NSFC grant No.12171009 and  Young
Elite Scientist Sponsorship Program by China Association for Science and Technology (CAST) No. YESS20200028.




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