Ornstein-Uhlenbeck Limit for the Velocity Process of an N-Particle System Interacting Stochastically

Abstract

An N-particle system with stochastic interactions is considered. Interactions are driven by a Brownian noise term and total energy conservation is imposed. The evolution of the system, in velocity space, is a diffusion on a (3N−1)-dimensional sphere with radius fixed by the total energy. In the N→∞ limit, a finite number of velocity components are shown to evolve independently and according to an Ornstein-Uhlenbeck process.

Appendices

Appendix A: Two Representations of the Diffusion on the Sphere, and a Binary Interaction Interpretation

Our representation of the diffusion on the sphere, with the Itô differential equation (8), describes the evolution of the velocity vector in terms of D independent Brownian driving noises B _j , 1≤j≤D, each directly associated with one component of V. Kiessling and Lancellotti [8] recall the representation of the Laplace-Beltrami operator on the unit sphere $\mathbb {S}^{D-1} \subset \mathbb {R}^{D}$ in the form

$$ \triangle_{\mathbb {S}^{D-1}} = \sum_{1 \leq k < l \leq D} (v_k \partial_{v_l} - v_l \partial_{v_k} )^2 $$

(45)

and interpret the Fokker-Planck equation in terms of a model for their original N=1+D/3 particles with two-body and one-body interactions preserving momentum and energy. Here we translate this interpretation to a simple stochastic process, limiting ourselves to the D/3 “effective particles” with interactions preserving only energy.

Diffusion on the sphere may be viewed as a succession of independent infinitesimal rotations. So, consider a family of D(D−1)/2 independent processes Ω _kl (.),1≤k<l≤D, taken to be martingales with initial value 0 and cross-variation 〈Ω _ij ,Ω _kl 〉(t)=δ _ik δ _jl t/D. Complement Ω to an antisymmetric matrix, with Ω _ij =−Ω _ji , so that the differentials dΩ _ij act as infinitesimal rotation generators, and consider the process V′ defined by an initial data V′(0) with $|\mathbf{V}'(0)| = \sqrt{D}$ and the Stratonovich differential equation

$$ \mathrm{d}V'_i = \sum_j V'_j \circ \,\mathrm {d}\varOmega_{ij}. $$

(46)

Its Fokker-Planck operator, acting on measures f on the sphere $\mathbb {S}^{D-1}_{\sqrt{D}}$ with radius $\sqrt{D}$, is the sum $L = - \frac{1}{2} \sum_{1 \leq i < j \leq D} R_{ij}^{*} R_{ij}$, where R _ij is the vector field associated with Ω _ij , and $R_{ij}^{*}$ is its adjoint operator on $\mathbb {R}^{D}$. By (46), the vector field is $R_{ij} = D^{-1/2} (v'_{j} \partial_{v'_{i}} - v'_{i} \partial_{v'_{j}})$. Then $R_{ij}^{*} = - D^{-1/2} ( \partial_{v'_{i}} (v'_{j} \cdot) - \partial _{v'_{j}} (v'_{i} \cdot) ) = - R_{ij}$, so that $L = \frac{1}{2D} \triangle_{\mathbb {S}^{D-1}}$. As the generator determines the law of a diffusion, this proves that the process defined by (46) is the same as the one defined in Sect. 3.

Now, comparing directly the differential equations is also interesting. So, from the Stratonovich form (46), we deduce the Itô equation for V′,

$$\begin{aligned} \mathrm{d}V'_i = & \sum_j V'_j \,\mathrm{d} \varOmega_{ij} + \frac{1}{2} \sum_j \,\mathrm{d}\bigl\langle V'_j, \varOmega_{ij} \bigr \rangle \end{aligned}$$

(47)

$$\begin{aligned} = & \sum_j V'_j \,\mathrm{d}\varOmega_{ij} + \frac{1}{2} \sum _j \sum_{k \neq j} V'_k \,\mathrm{d}\langle\varOmega_{jk}, \varOmega_{ij} \rangle \\ = & \sum_j V'_j \,\mathrm{d}\varOmega_{ij} + \frac{1}{2} \sum _j \sum_{k \neq j} V'_k \bigl(-D^{-1} \delta_{jj} \delta_{ik} \bigr) \,\mathrm{d}t \\ = & \sum_j V'_j \,\mathrm{d}\varOmega_{ij} - \frac{1}{2} V'_i \, \bigl(1 - D^{-1} \bigr) \,\mathrm{d}t. \end{aligned}$$

(48)

For the first equality, we used the relation between Itô and Stratonovich integrals (see e.g. Sect. V.5 in [12]); then we substitute the first term of the r.h.s. of (47) into the cross-variation term, and finally we use the explicit cross-variation of Ω.

In the final expression (48), the drift is exactly the one in (8). The first term appears as coupling the i-th component of vector V′ with all other components, and it is linear with respect to V′ while (8) involves a projection matrix σ orthogonal to V. As this first term is the differential of a martingale, say $\mathrm{d}M'_{i} = \sum_{j} V'_{j} \,\mathrm{d}\varOmega_{ij}$, we now show that it is equivalent (in law) to the first term in (8). Indeed, this martingale M′ is further characterized by its cross-variation process, which has the differential

$$\begin{aligned} \mathrm{d}\bigl\langle M'_i, M'_j \bigr\rangle = & \sum_k \sum _l V'_k V'_l \,\mathrm{d}\langle\varOmega_{ik}, \varOmega_{jl} \rangle \\ = & \sum_k \sum_l V'_k V'_l D^{-1} (\delta_{ij} \delta_{kl} - \delta_{il} \delta_{jk}) \,\mathrm{d}t \\ = & D^{-1} \biggl(\delta_{ij} \sum _k {V'_k}^2 - V'_i V'_j \biggr) \,\mathrm{d}t, \end{aligned}$$

(49)

where one readily recognizes the projector entry σ _ij (V′).

For comparison, the first term in (8) defines a martingale M with dM _i =∑ _j σ _ij (V) dB _j , so that its cross-variation satisfies

$$ \mathrm{d}\langle M_i, M_j\rangle= \sum _k \sum_l \sigma_{ik} \sigma_{jl} \,\mathrm{d}\langle B_k, B_l\rangle= \sigma_{ij}(\mathbf {V})\,\mathrm{d}t $$

(50)

as follows from the cross-variation of B and the fact that σ ²=σ.

As a result, M′ and M have the same cross-variation when generated from the same trajectory (assuming M(0)=M′(0)=0 for definiteness), as befits processes V′ and V having the same law (incidentally, these calculations reformulate the fact that the Fokker-Planck generators are equal). However, the Brownian drivers underlying both processes differ, as those for V are defined from translations along the D components directions while those for V′ are associated with the rotation matrices acting on $\mathbb {R}^{D}$. Interestingly, the non-Abelian nature of rotation compositions is irrelevant to our discussion.

Kiessling and Lancellotti [8] interpret their representation (A.5) of the Laplace-Beltrami operator on the sphere in terms of two types of contributions. Here, one associates with three components, 3k−2≤i≤3k, the Cartesian components of the velocity $\bar{V}'_{k}$ of particle k in $\mathbb {R}^{3}$, so that in the Ω _ij ’s one distinguishes the terms associated with different particles (⌈i/3⌉≠⌈j/3⌉ for the ceiling function ⌈⋅⌉), and terms coupling two velocity components of the same particle (⌈i/3⌉=⌈j/3⌉) so that the particle velocity vector just rotates in $\mathbb {R}^{3}$ as under a gyroscopic force. As they observe, the gyroscopic contribution to each particle velocity drops out in the N→∞ limit.

Appendix B: Appendix: The Kac System Property

Here we write N for D to follow the notations of Carlen, Carvalho and Loss closely. Kac’ model is a random walk on the sphere $\mathbb {S}^{N-1}$, where, at (Poisson distributed) random times, two particles are picked up at random, and their velocities (V _i ,V _j ) suffer a random rotation by an angle θ, according to a probability measure ϱ(θ) dθ in the $(\hat{e}_{i}, \hat{e}_{j})$ plane, with ϱ being continuous and even. Given a ϱ _ε with support [−επ,επ], this system can be modeled pathwise by a jump process for the velocity components V _i , driven by piecewise constant noise processes ω _ij,ε =−ω _ji,ε , with increments

$$ \Delta V_i = \sum_{j \neq i} \bigl[ V_i (\cos\Delta\omega_{ij, \varepsilon} - 1) + V_j \sin\Delta\omega_{ij, \varepsilon} \bigr], $$

(51)

where V(t) in the right hand side is understood as the left limit, V(t−). In the noise $\omega_{ij,\varepsilon} (t) = \sum_{k=1}^{K_{ij}(t)} \theta_{ij,k}$, the jumps Δω _ij,ε =θ _ij,k =−θ _ji,k are independent, distributed with law ϱ _ε , and their number K _ij (t) in [0,t[ is Poisson distributed with rate λ _ε =1/ε ². In the limit ε→0, one can show that the noise becomes Brownian, and solutions to equation (51) converge to those of (46).

In the following, we show directly that our model has all features of Kac systems, which are defined in [3] as systems of probability spaces, depending on $N \in \mathbb {N}_{0}$, verifying four features. The first feature is the invariance, under particle labeling permutations, of the stationary measure μ _N which is the microcanonical measure on the sphere $X_{N} = \mathbb {S}^{N-1}_{\sqrt{N}}$. The second feature characterizes ν _N as the marginal distribution of the component j of interest obtained by the projection π _j , in our case $\pi_{j}(\mathbf{V}) = V_{j} \in Y_{N} = [-\sqrt{N}, \sqrt{N}]$:

$$ \mathrm{d}\nu_N (y) = \frac{|\mathbb {S}^{N-2}|}{|\mathbb {S}^{N-1}|} \biggl (1 - \frac{y^2}{N} \biggr)^{(N-3)/2} N^{-1/2} \,\mathrm{d}y, $$

(52)

where $|\mathbb {S}^{N-1}|$ is the measure of the unit (N−1)-dimensional sphere. The third feature introduces the lift from N−1 components in X _N−1 along with one additional component in Y _N , in a form consistent with the definition of the projection π _j , viz. for j=N

$$ \phi_N (u, y) = \bigl( s_N(y) u, y \bigr) $$

(53)

with the scaling function $s_{N}(y) := \sqrt{\frac{N-y^{2}}{N-1}}$ projecting u onto the sphere with radius $\sqrt{N-y^{2}}$, so that $(\mu_{N-1} \otimes\nu_{N})(\phi _{j}^{-1}(A)) = \mu_{N}(A)$ for any measurable A⊂X _N .

The fourth feature deals with the Markov transition operator Q _N , which occurs in the generator of the evolution for the probability distribution function, such that ∂ _t f=(1/τ _N )(Q _N −I _N )f where I _N is the identity and τ _N is a characteristic time, e.g. τ _N =1/N for Kac’ original model (see Sect. 1 in [3]). In our case, the Fokker-Planck equation yields

$$ Q_N = I_N + \frac{1}{2 N} \sum _{k=1}^N \sum_{i=1}^N \partial_{v_i} \sigma_{ik}(v) \partial_{v_k} $$

(54)

from the Stratonovich formulation, recalling that σ is symmetric and idempotent.

Then we must check that for any $f \in \mathcal {H}_{N} := L^{2}(X_{N}, \mu_{N})$

$$ \langle f, Q_N f \rangle_{\mathcal {H}_N} = \frac{1}{N} \sum _{j=1}^N \int_{Y_N} \langle f_{j,y} , Q_{N-1} f_{j,y} \rangle_{\mathcal {H}_{N-1}} \,\mathrm{d}\nu_N(y) $$

(55)

where f _j,y (u):=f(ϕ _j (u,y)) for y∈Y _N and u∈X _N−1; in our case, it simply reads f _N,y (u)=f(s _N (y)u ₁,…s _N (y)u _N−1,y), and indices j<N select similarly the j-th component instead of the N-th one.

With our Q, f must be twice differentiable, so we take f∈H ²(X _N ,μ _N ). Now, condition (55) is linear in Q, so it suffices to prove it for I and (Q−I)/τ separately. First, for I, note that

$$\begin{aligned} \int_{Y_N} \langle f_{j,y} , f_{j,y} \rangle_{\mathcal {H}_{N-1}} \,\mathrm{d}\nu_N(y) = & \int_{Y_N} \biggl[ \int_{X_{N-1}} |f \bigl( \phi_j(u,y) \bigr) |^2 \,\mathrm{d}\mu_{N-1}(u) \biggr] \,\mathrm{d}\nu_N(y) \\ = & \langle f , f \rangle_{\mathcal {H}_N} \end{aligned}$$

(56)

where the first equality follows from the definition of the Hilbert scalar product, and the second from Fubini and feature 3. Then summing for 1≤j≤N and dividing by N shows (unsurprisingly) that I fulfills feature 4.

For Q−I, we use the representation [8] of the Laplace-Beltrami operator on the sphere $X_{N} = \mathbb {S}^{N-1}_{\sqrt{N}}$

$$ 2 (Q_N - I_N) / \tau_N = (1/N) \sum _{k<l}^N (v_k \partial_{v_l} - v_l \partial_{v_k})^2. $$

(57)

Then,

$$\begin{aligned} & \bigl\langle f_{j,y} , 2 (Q_{N-1} - I_{N-1}) f_{j,y} \bigr\rangle_{\mathcal {H}_{N-1}} \\ &\quad = \frac{\tau_{N-1}}{N-1} \sum_{\substack{k < l\\ k \neq j,\ l \neq j}} \int _{X_{N-1}} \bigl( f_{j,y}(u) (u_k \partial_{u_l} - u_l \partial_{u_k})^2 f_{j,y}(u) \bigr) \,\mathrm{d}\mu_{N-1} (u), \end{aligned}$$

(58)

where we note that the operator $u_{k} \partial_{u_{l}} - u_{l} \partial _{u_{k}}$ is homogeneous, so that the factor s(y) in the change of variable (53) from u to v=ϕ _j (u,y) will be absorbed. Thus,

$$\begin{aligned} & \sum_{j=1}^N \int_{Y_N} \bigl\langle f_{j,y} , 2 (Q_{N-1} - I_{N-1}) f_{j,y} \bigr\rangle_{\mathcal {H}_{N-1}} \,\mathrm{d}\nu_N(y) \\ &\quad = \sum_{j=1}^N \frac{\tau_{N-1}}{N-1} \sum_{\substack{k < l\\ k \neq j,\ l \neq j}} \int_{Y_N} \int _{X_{N-1}} \bigl( f_{j,y}(u) (u_k \partial_{u_l} - u_l \partial_{u_k})^2 f_{j,y}(u) \bigr) \,\mathrm{d}\mu_{N-1} (u) \,\mathrm{d}\nu_N(y) \\ &\quad = \sum_{j=1}^N \frac{\tau_{N-1}}{N-1} \sum_{\substack{k < l\\ k \neq j,\ l \neq j}} \int_{X_N} \bigl( f(v) (v_k \partial_{v_l} - v_l \partial_{v_k})^2 f(v) \bigr) \,\mathrm{d}\mu_{N} (v), \end{aligned}$$

(59)

where the second equality holds from Fubini and feature 3. Note that we may interchange the sums and keep the restrictions on indices by changing

$$ \sum_{j=1}^N \sum _{\substack{k < l\\ k \neq j,\ l \neq j}}^N = \sum _{k<l}^N \sum_{\substack{j = 1\\ j \neq k,l}}^N. $$

(60)

Thus, we have

$$\begin{aligned} & \frac{1}{N}\sum_{j=1}^N \int _{Y_N} \bigl\langle f_{j,y} , 2 (Q_{N-1} - I_{N-1}) f_{j,y} \bigr\rangle_{\mathcal {H}_{N-1}} \,\mathrm{d} \nu_N(y) \\ &\quad = \sum_{k<l}^N \sum _{\substack{j = 1\\ j \neq k,l}}^N \frac{\tau _{N-1}}{N-1} \biggl[ \frac{1}{N} \int_{X_N} \bigl( f(v) (v_k \partial_{v_l} - v_l \partial_{v_k})^2 f(v) \bigr) \,\mathrm{d}\mu_{N} (v) \biggr]. \end{aligned}$$

(61)

Now, the summand depends only on k and l, so that the N−2 values of j contribute with the same result. Furthermore, we recognize the resulting (k,l) sum as $\langle f , 2 \tau_{N}^{-1} (Q_{N} - I_{N}) f \rangle_{\mathcal {H}_{N}}$. Hence,

$$ \frac{1}{N}\sum_{j=1}^N \int _{Y_N} \bigl\langle f_{j,y} , 2 (Q_{N-1} - I_{N-1}) f_{j,y} \bigr\rangle_{\mathcal {H}_{N-1}} \,\mathrm{d} \nu_N(y) = \frac{(N-2) \tau_{N-1}}{(N-1) \tau_N} \bigl\langle f , 2 (Q_N - I_N) f \bigr\rangle_{\mathcal {H}_{N}}. $$

(62)

To meet feature 4, it suffices to set τ _N =1/(N−1), which scale as O(1/N) as N→∞. Note that 1/(Nτ _N ) also describes the N(=D)-dependence of the drift term in the Itô (8).

In our pathwise formulation, feature 4 relates the stochastic differential equation (46) of the N-particle system to the equations for (N−1)-particle subsystems as follows. First, the (N−1)-particle subsystem, with particle j removed, obeys the equation for $u_{i, \not j} = V_{i} / s_{N}(V_{j})$ (i≠j)

$$ \mathrm{d}u_{i, \not j} = \tau_{N-1}^{1/2} \sum _k u_{k, \not j} \circ \,\mathrm {d}\varOmega_{ik, \not j} $$

(63)

where the $\varOmega_{ik, \not j}$’s are independent Brownian motions with variance t/(N−1) (see App. A for the denominator N−1, for the diffusion on X _N−1) for 1≤i<k≤N,i≠j,k≠j, and $\varOmega_{ki, \not j} = - \varOmega_{ik, \not j}$. Then the generator on the right hand side of (55) corresponds formally to the evolution equation

$$\begin{aligned} \mathrm{d}V_i = & \frac{1}{\sqrt{N}}\sum_{j=1}^N \sum_{l \neq j} \biggl(\frac{\partial V_i}{\partial u_{l, \not j}} \biggr)_{V_j} \circ \,\mathrm {d}u_{l, \not j} \end{aligned}$$

(64)

$$\begin{aligned} = & \frac{1}{\sqrt{N}}\sum_{j=1}^N (1 - \delta_{ij}) \, s_N(V_j) \circ \,\mathrm {d}u_{i, \not j} \end{aligned}$$

(65)

$$\begin{aligned} = & \frac{\tau_{N-1}^{1/2} }{\sqrt{N}}\sum_{j \neq i } s_N(V_j) \sum_{k \neq j} u_{k, \not j} \circ \,\mathrm {d}\varOmega_{ik, \not j} \end{aligned}$$

(66)

$$\begin{aligned} = & \frac{\tau_{N-1}^{1/2} }{\sqrt{N}}\sum_{k \neq i} \sum _{j \neq i, j \neq k} V_k \circ \,\mathrm {d}\varOmega_{ik, \not j}, \end{aligned}$$

(67)

where the first equality is our interpretation of (55), the second follows from the projection of X _N onto X _N−1 and the lift (53), the third follows from (63) and the fourth again from the lift (53) (and k≠i because Ω is antisymmetric). To obtain (46) formally, it suffices to define $\varOmega_{ik} := (\tau_{N-1} / \tau_{N})^{1/2} N^{-1/2} \sum_{j \neq i, j \neq k} \varOmega_{ik, \not j}$; for this Ω _ik to be the Brownian motion with variance t/N, we set again τ _N =1/(N−1) so that τ _N−1/τ _N =(N−1)/(N−2) since the sum over j has only N−2 terms.

Note that this latter check for feature 4 works pathwise, with an explicit construction of the new driving noises Ω for N degrees of freedom from the subsystems’ driving noises. In contrast, working with the generators (Q−I)/N ensures only equivalence in law of the processes constructed from the subsystems with the N-particle process.