Uniform-in-time propagation of chaos for second order interacting particle systems

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

$$\left\{\begin{aligned} \mathrm{d}x_{i}(t)&=v_{i}(t)\mathrm{d}t,\\ \mathrm{d}v_{i}(t)&=-\nabla V(x_{i}(t))\mathrm{d}t+\frac{1}{N}\sum_{j\neq i}K(x_{i}(t)-x_{j}(t))\mathrm{d}t-\gamma v_{i}(t)\mathrm{d}t+\sqrt{2\sigma}\mathrm{d}B_{t}^{i},\end{aligned}\right.$$

(1.1)

where $i=1,2,...,N$. We take the position $x_{i}$ in $\Omega$, which may be the whole space $\mathbb{R}^{d}$ or the periodic torus $\mathbb{T}^{d}$, while each velocity $v_{i}$ lies in $\mathbb{R}^{d}$. Note that $\{(B_{\cdot}^{i})\}_{i=1}^{N}$ denotes $N$ independent copies of Wiener processes on $\mathbb{R}^{d}$. We take the diffusion coefficient before the Brownian motions $\sqrt{2\sigma}$ as a constant for simplicity. Usually one can take $\sigma=\gamma\beta^{-1}$, where the constant $\gamma>0$ denotes the friction parameter, and $\beta$ is the inverse temperature. In more general models, those $\sigma$ may depend on the number of particles $N$, the position of particles $x_{i}$, etc.

In many important models, the kernel is given by $K=-\nabla W$, where $W$ is an interacting potential. A well-known example for $W$ is the Coulomb potential. We recall that the Hamiltonian energy $H:(\Omega\times\mathbb{R}^{d})^{N}\rightarrow\mathbb{R}$ associated to the particle system (1.1) reads as

$$H(x_{1},v_{1},...,x_{N},v_{N})=\frac{1}{2}\sum_{i=1}^{N}v_{i}^{2}+\sum_{i=1}^{N}V(x_{i})+\frac{1}{2N}\sum_{i=1}^{N}\sum_{j\neq i}W(x_{i}-x_{j}),$$

(1.2)

which is the sum of the kinetic energy $\frac{1}{2}\sum_{i=1}^{N}v_{i}^{2}$ and the potential energy $U(x_{1},...,x_{N})$ defined by

$$U(x_{1},...,x_{N}):=\sum_{i=1}^{N}V(x_{i})+\frac{1}{2N}\sum_{i=1}^{N}\sum_{j\neq i}W(x_{i}-x_{j}).$$

(1.3)

We recall the Liouville equation as in [26], which describes the joint distribution $f_{N}$ of the particle system (1.1) on $(\Omega\times\mathbb{R}^{d})^{N}$,

$$\begin{split}\partial_{t}f_{N}+\sum_{i=1}^{N}v_{i}\cdot\nabla_{x_{i}}f_{N}&+\sum_{i=1}^{N}\Big{(}-\nabla_{x_{i}}V(x_{i})+\frac{1}{N}\sum_{j\neq i}K(x_{i}-x_{j})\Big{)}\cdot\nabla_{v_{i}}f_{N}\\ &=\sigma\sum_{i=1}^{N}\Delta_{v_{i}}f_{N}+\gamma\sum_{i=1}^{N}\nabla_{v_{i}}\cdot(v_{i}f_{N}).\end{split}$$

(1.4)

We define the associated Liouville operator as

$$\begin{split}L_{N}f_{N}=&\sum_{i=1}^{N}v_{i}\cdot\nabla_{x_{i}}f_{N}+\sum_{i=1}^{N}\Big{(}-\nabla_{x_{i}}V(x_{i})+\frac{1}{N}\sum_{j\neq i}K(x_{i}-x_{j})\Big{)}\cdot\nabla_{v_{i}}f_{N}\\ &-\sigma\sum_{i=1}^{N}\Delta_{v_{i}}f_{N}-\gamma\sum_{i=1}^{N}\nabla_{v_{i}}\cdot(v_{i}f_{N}).\end{split}$$

(1.5)

As the mean field theory indicates, when $N\rightarrow\infty$, the limiting behavior of any particle in (1.1) is described by the McKean-Vlasov SDE

$$\left\{\begin{aligned} \mathrm{d}x(t)&=v(t)\mathrm{d}t,\\ \mathrm{d}v(t)&=-\nabla V(x(t))+K\ast\rho(x(t))\mathrm{d}t-\gamma v(t)\mathrm{d}t+\sqrt{2\sigma}\mathrm{d}B_{t},\end{aligned}\right.$$

(1.6)

where $(x,v)\in\Omega\times\mathbb{R}^{d}$, $(B_{\cdot})$ denotes a standard Brownian motion on $\mathbb{R}^{d}$. We then denote its phase space density by $f_{t}=\mbox{Law}(x(t),v(t))$ and the spatial density by

$$\rho(t,x)=\int_{\mathbb{R}^{d}}f(t,x,v)\mathrm{d}v.$$

The law $f_{t}$ further satisfies the mean-field equation or the nonlinear Fokker-Planck equation

$$\partial_{t}f+v\cdot\nabla_{x}f-\nabla V\cdot\nabla_{v}f+(K\ast\rho)\cdot\nabla_{v}f=\sigma\Delta_{v}f+\gamma\nabla_{v}\cdot(vf).$$

(1.7)

If $\sigma=\gamma\beta^{-1}$, one can check that its nonlinear equilibrium satisfies the following equation

$$f_{\infty}=\frac{1}{Z}e^{-\beta(V(x)+\frac{1}{2}v^{2}+W\ast\rho_{\infty})},$$

(1.8)

where $Z$ is the constant to make $f_{\infty}$ a probability density. In the literature, one may use the “formal equilibrium” of Eq.(1.7) at time $t$ to refer

$$\hat{f}_{t}=\frac{1}{\hat{Z}}e^{-\beta(V(x)+\frac{1}{2}v^{2}+W\ast\rho_{t})},$$

(1.9)

again where $\hat{Z}$ is the constant to make $\hat{f}_{t}$ a probability density.

It is well-known that the large $N$ limit of particle system (1.1) is mathematically formalized by the notion of $propagation\ of\ chaos$ originating in [29]. See also the surveys for instance [36, 18, 25, 27]. 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 [27, 11] 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 $\sigma=0$ as in Eq. (1.1) or our setting with the diffusion only acts on the velocity variables. Thus the limiting equation (1.7) 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 [28].

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 (1.4) toward its unique equilibrium (i.e. the Gibbs measure) uniformly in $N$. Secondly, we establish propagation of chaos of (1.1) (or (1.4) )toward (1.6) (or (1.7) ) uniformly in time. We combine the entropy method developed in [26] with the hypocoercivity method refined in [37] 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 $\sigma$ 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 (1.7) might not be unique. Furthermore, for systems (1.4) 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 $t\in[0,T]$, is still widely open. The recent progress can be found for instance in [23, 24, 26, 31, 7, 6, 5]. Uniform-in-$N$ convergence of (1.4) 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 [26] for systems with bounded kernel $K$,

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}H_{N}(f^{t}_{N}|f_{t}^{\otimes N})\leq$	$\displaystyle-\frac{1}{N}\int_{(\Omega\times\mathbb{R}^{d})^{N}}f^{t}_{N}\overline{R}_{N}\mathrm{d}Z$		(1.10)
		$\displaystyle-\frac{\sigma}{N}\int_{(\Omega\times\mathbb{R}^{d})^{N}}f^{t}_{N}\bigg{|}\nabla_{V}\log\frac{f^{t}_{N}}{f_{t}^{\otimes N}}\bigg{|}^{2}\mathrm{d}Z,$

where $H_{N}(f^{t}_{N}|f_{t}^{\otimes N})$ is the normalized relative entropy defined as

$$H_{N}(f^{t}_{N}|f_{t}^{\otimes N})=\frac{1}{N}\int_{(\Omega\times\mathbb{R}^{d})^{N}}f^{t}_{N}\log\frac{f^{t}_{N}}{f_{t}^{\otimes N}}\mathrm{d}Z,$$

and $\overline{R}_{N}$ is the error term defined as

$$\overline{R}_{N}=\sum_{i=1}^{N}\nabla_{v_{i}}\log f_{t}(x_{i},v_{i})\cdot\bigg{\{}\frac{1}{N}\sum_{j,j\neq i}K(x_{i}-x_{j})-K\ast\rho_{t}(x_{i})\bigg{\}}.$$

Hereinafter we write that $Z=(x_{1},v_{1},x_{2},v_{2},\cdots,x_{N},v_{N})\in(\Omega\times\mathbb{R}^{d})^{N}$, $X=(x_{1},x_{2},\cdots,x_{N})\in\Omega^{N}$ and $V=(v_{1},v_{2},\cdots,v_{N})\in(\mathbb{R}^{d})^{N}$. Then by integration by parts, one can rewrite (1.10) as

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}H_{N}(f^{t}_{N}|f_{t}^{\otimes N})=$	$\displaystyle-\frac{1}{N}\int_{(\Omega\times\mathbb{R}^{d})^{N}}\nabla_{V}\log\frac{f^{t}_{N}}{f_{t}^{\otimes N}}\cdot R^{0}_{N}\mathrm{d}Z$		(1.11)
		$\displaystyle-\frac{\sigma}{N}\int_{(\Omega\times\mathbb{R}^{d})^{N}}f^{t}_{N}\left|\nabla_{V}\log\frac{f^{t}_{N}}{f_{t}^{\otimes N}}\right|^{2}\mathrm{d}Z,$

where $R^{0}_{N}$ is a $dN$-dimensional vector defined as

$$R^{0}_{N}=\bigg{\{}\frac{1}{N}\sum_{j=1,j\neq i}^{N}K(x_{i}-x_{j})-K\ast\rho_{t}(x_{i})\bigg{\}}_{i=1}^{N}.$$

Inspired by the strategy used in [19] to deal with the first order dynamics on torus, we expect to find the term

$$-\frac{1}{N}\int_{(\Omega\times\mathbb{R}^{d})^{N}}f^{t}_{N}\left|\nabla_{X}\log\frac{f^{t}_{N}}{f_{t}^{\otimes N}}\right|^{2}\mathrm{d}Z$$

in some way to use log-Sobolev inequality for the second order system (1.1). Fortunately, hypocoercivity in entropy sense in [37] motivates us to take the derivative of normalized relative Fisher information $I^{M}_{N}(f^{t}_{N}|f_{t}^{\otimes N})$, which is defined as

$$I_{N}^{M}(f^{t}_{N}|f_{t}^{\otimes N})=\frac{1}{N}\int_{(\Omega\times\mathbb{R}^{d})^{N}}f^{t}_{N}\left|\sqrt{M}\nabla\log\frac{f^{t}_{N}}{f_{t}^{\otimes N}}\right|^{2}\mathrm{d}Z,$$

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

$$M=\left(\begin{array}[]{cc}E&F\\ F&G\end{array}\right),$$

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

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}I_{N}^{M}(f^{t}_{N}|f_{t}^{\otimes N})\leq$	$\displaystyle-\frac{c_{1}}{N}\int_{(\Omega\times\mathbb{R}^{d})^{N}}f^{t}_{N}\left|\sqrt{E}\nabla_{X}\log\frac{f^{t}_{N}}{f_{t}^{\otimes N}}\right|^{2}\mathrm{d}Z$		(1.12)
		$\displaystyle+\frac{c_{2}}{N}\int_{(\Omega\times\mathbb{R}^{d})^{N}}f^{t}_{N}\left|\sqrt{G}\nabla_{V}\log\frac{f^{t}_{N}}{f_{t}^{\otimes N}}\right|^{2}\mathrm{d}Z$
		$\displaystyle+\mbox{some error terms},$

where $c_{1},c_{2}$ are two positive constants that depend on $M,\sigma,\gamma$, $K$ and $V$. The concrete form of error terms will be given in Lemma 4 and Lemma 6. 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 (1.10) and (1.12) could be more general. For insatnce, if we replace $f_{t}^{\otimes N}$ by $f_{N,\infty}$, we can show the convergence from $f^{t}_{N}$ towards $f_{N,\infty}$ as $t\rightarrow\infty$.

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

$$\mathcal{E}^{M}_{N}(f^{t}_{N}|\bar{f_{N}})=H_{N}(f^{t}_{N}|\bar{f_{N}})+I^{M}_{N}(f^{t}_{N}|\bar{f_{N}}),$$

where $\bar{f_{N}}$ may take $f_{t}^{\otimes N}$ or $f_{N,\infty}$. The remaining argument is to appropriately control error terms in (1.12). When $\bar{f_{N}}=f_{N,\infty}$, linear structure of the Liouville equation (1.4) makes error terms vanish. When $\bar{f_{N}}=f_{t}^{\otimes N}$, error terms in (1.12) can be written as

$$-2\int_{(\Omega\times\mathbb{R}^{d})^{N}}f^{t}_{N}\langle\nabla\overline{R}_{N},M\nabla\log\frac{f^{t}_{N}}{f_{t}^{\otimes N}}\rangle\mathrm{d}Z$$

(1.13)

for suitable selection of $M$ (See Corollary 2), here $\nabla=(\nabla_{X},\nabla_{V})$. However, the terms $\nabla_{x}\nabla_{v}\log f_{t}$ and $\nabla_{v}\nabla_{v}\log f_{t}$ are too difficult to control to obtain some uniform-in-time estimates for (1.13) (See Remark 15). The key observation is that $\nabla_{x}\nabla_{v}\log f_{t}$ and $\nabla_{v}\nabla_{v}\log f_{t}$ are trivial terms if we replace $f_{t}$ by its non-linear equilibrium, then we can show

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{E}^{M}_{N}(f^{t}_{N}|f_{\infty}^{\otimes N})\leq$	$\displaystyle-\frac{c^{\prime}_{1}}{N}\int_{(\Omega\times\mathbb{R}^{d})^{N}}f^{t}_{N}\left|\nabla_{X}\log\frac{f^{t}_{N}}{f_{\infty}^{\otimes N}}\right|^{2}\mathrm{d}Z$		(1.14)
		$\displaystyle-\frac{c^{\prime}_{2}}{N}\int_{(\Omega\times\mathbb{R}^{d})^{N}}f^{t}_{N}\left|\nabla_{V}\log\frac{f^{t}_{N}}{f_{\infty}^{\otimes N}}\right|^{2}\mathrm{d}Z$
		$\displaystyle+\frac{C}{N},$

where $c^{\prime}_{1},c^{\prime}_{2}>0,C>0$ are some constants that only depend on $M,\sigma,\gamma,K$ and $V$ (See Theorem 1.2). To control the error between $\mathcal{E}^{M}_{N}(f^{t}_{N}|f_{t}^{\otimes N})$ and $\mathcal{E}^{M}_{N}(f^{t}_{N}|f_{\infty}^{\otimes N})$, we use the convergence result from $f_{t}$ to its equilibrium $f_{\infty}$. There exist some results studying this type of convergence . In this article, we adapt the arguments as in [21] for $K$ with small $\|\nabla K\|_{\infty}$ and [13] for the case when $K=-\nabla W$ and the interaction energy is convex to establish the following estimate

$$H(f_{t}|f_{\infty})\leq Ce^{-c^{\prime}_{3}t},$$

(1.15)

where $c^{\prime}_{3}>0$ depends on $\sigma,\gamma,K$ and $C>0$ depends on initial data $f_{0}$. We give the sketch of proof of (1.15) in Theorem 1.4 and Theorem 1.5 in the appendix. In the end, we combine (1.14) with (1.15) through 2-Wasserstein distance to conclude.

Finally, we give some comments about terms $\nabla_{x}\nabla_{v}\log f_{t}$ and $\nabla_{v}\nabla_{v}\log f_{t}$. Unfortunately we cannot directly obtain uniform-in-time estimates about these terms, otherwise we would have

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{E}^{M}_{N}(f^{t}_{N}|f_{t}^{\otimes N})\leq$	$\displaystyle-\frac{c^{\prime}_{1}}{N}\int_{(\Omega\times\mathbb{R}^{d})^{N}}f^{t}_{N}\left|\nabla_{X}\log\frac{f^{t}_{N}}{f_{t}^{\otimes N}}\right|^{2}\mathrm{d}Z$
		$\displaystyle-\frac{c^{\prime}_{2}}{N}\int_{(\Omega\times\mathbb{R}^{d})^{N}}f^{t}_{N}\left|\nabla_{V}\log\frac{f^{t}_{N}}{f_{t}^{\otimes N}}\right|^{2}\mathrm{d}Z$
		$\displaystyle+\frac{C}{N},$

for some $c_{1}^{\prime},c_{2}^{\prime},C>0$ and then finish the proof by Gronwall’s inequality. Uniform-in-time propagation of chaos can only be expected when the limiting PDE (1.7) has a unique equilibrium. Indeed, we obtain Theorem 1.3 under the assumptions in Theorem 1.2 and some assumption on $W$ which prevents the existence of multiple equilibria of Eq.(1.7). Similar arguments/estimates have also appeared in the first-order setting. For instance, the authors of [17] obtained Li-Yau type growth estimates of $\nabla^{2}\log\rho_{t}$ 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 [35, 8].

Let us fix some notations first. We denote $z=(x,v)\in\Omega\times\mathbb{R}^{d}$ 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 $\nabla_{X}=(\nabla_{x_{1}},...,\nabla_{x_{N}})$, $\nabla_{V}=(\nabla_{v_{1}},...,\nabla_{v_{N}})$ and $\Delta_{V}=\sum_{i=1}^{N}\Delta_{v_{i}}$. Those are operators on $\mathbb{R}^{dN}$.

For two probability measures $\mu,\nu$ on $\Omega\times\mathbb{R}^{d}$, we denote by $\Pi(\mu,\nu)$ theset of all couplings between $\mu$ and $\nu$. We define $L^{2}$ Wasserstein distance as

$$\mathcal{W}_{2}(\mu,\nu)=\Big{(}\inf_{\pi\in\Pi(\mu,\nu)}\int_{(\Omega\times\mathbb{R}^{d})^{2}}|z_{1}-z_{2}|^{2}\mathrm{d}\pi(z_{1},z_{2})\Big{)}^{1/2}.$$

For a measure $\mu$ and a positive integrable function $f$ such that $\int_{\Omega\times\mathbb{R}^{d}}f|\log f|\mathrm{d}\mu<\infty$, we define the entropy of $f$ with respect to $\mu$ as

$$Ent_{\mu}(f)=\int_{\Omega\times\mathbb{R}^{d}}f\log f\mathrm{d}\mu-\bigg{(}\int_{\Omega\times\mathbb{R}^{d}}f\mathrm{d}\mu\bigg{)}\log\left(\int_{\Omega\times\mathbb{R}^{d}}f\mathrm{d}\mu\right).$$

For two probability measures $\mu$ and $\nu$, we define the relative entropy between $\mu$ and $\nu$ as

$$H(\mu|\nu)=\left\{\begin{aligned} &\int_{\Omega\times\mathbb{R}^{d}}h\log h\mathrm{d}\nu,&\mbox{ if \, }\mu<<\nu\mbox{ and }h=\frac{\mathrm{d}\mu}{\mathrm{d}\nu},\\ &+\infty,&\mbox{otherwise},\end{aligned}\right.$$

and the normalized version of relative entropy for probability measures $\mu^{\prime}$ and $\nu^{\prime}$ on $(\Omega\times\mathbb{R}^{d})^{N}$ as

$$H_{N}(\mu^{\prime}|\nu^{\prime})=\frac{1}{N}H(\mu^{\prime}|\nu^{\prime}).$$

(1.16)

We also define relative Fisher information between $\mu$ and $\nu$ as

$$I^{M}(\mu|\nu)=\left\{\begin{aligned} &\int_{\Omega\times\mathbb{R}^{d}}\frac{\langle M\nabla h,\nabla h\rangle}{h}\mathrm{d}\nu,&\mbox{ if \, }\mu<<\nu\mbox{ and }h=\frac{\mathrm{d}\mu}{\mathrm{d}\nu},\\ &+\infty&\ \ otherwise,\end{aligned}\right.$$

where $M(z)$ is a positive definite matrix-valued function such that $M(z)\geq\kappa I$ for some $\kappa>0$ independent on $z\in\Omega\times\mathbb{R}^{d}$. Similarly, the normalized version of relative Fisher information between two probability measures $\mu^{\prime}$ and $\nu^{\prime}$ on $(\Omega\times\mathbb{R}^{d})^{N}$ reads as

$$I^{M}_{N}(\mu^{\prime}|\nu^{\prime})=\frac{1}{N}I^{M}(\mu^{\prime}|\nu^{\prime}).$$

(1.17)

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.

In the following, we use $\rho_{ls}$ ($\rho_{wls}$) to denote the (weighted) log-Sobolev inequality constant of the equilibrium measure $f_{\infty}$ of the limiting PDE (1.7) and $\rho_{LS}$ to denote the uniform-in-$N$ log-Sobolev inequality constant of the stationary measure $f_{N,\infty}$ of the Liouville equation (1.4).

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

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

$$\frac{1}{2}\nabla V(x)\cdot x\geq 6\lambda(V(x)+\frac{x^{2}}{2})-A,\ \ x\in\Omega$$

for some $A>0$. This expression implies that the force $-\nabla V$ drags particles back to some compact set. Detailed proof can be found in [20].

The second assumption implies that the potential $V$ grows at most quadratically on $\Omega$.

We also treat more general confining potentials when $\Omega=\mathbb{R}^{d}$.

Those conditions in Assumption 3 have been explored in [10] for kinetic Langevin process with confining potentials greater than quadratic growth at infinity. The boundedness of $\|V^{-2\theta}\nabla^{2}V\|_{L^{\infty}}$ extend the quadratic growth condition and the other two conditions guarantee the weighted log-Sobolev inequality of $f_{\infty}$ (See Section 2.3). We also use multiplier method developed in [10] to deal with this type of confining potentials.

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

Our second contribution is the uniform-in-$N$ exponential convergence from $f_{N}$ to $f_{\infty}^{\otimes N}$.