Large deviations for Kac-like walks

We introduce a Kac's type walk whose rate of binary collisions preserves the total momentum but not the kinetic energy. In the limit of large number of particles we describe the dynamics in terms of empirical measure and flow, proving the corresponding large deviation principle. The associated rate function has an explicit expression. As a byproduct of this analysis, we provide a gradient flow formulation of the Boltzmann-Kac equation.


Introduction
The statistics of rarefied gas is described, at the kinetic level, by the Boltzmann equation. It has become paradigmatic since it encodes most of the conceptual and technical issues in the description of the statistical properties for out of equilibrium systems. In the spatially homogeneous case the Boltzmann equation reads of one dimensional spatially dependent Boltzmann equation with discrete velocities. A main issue behind the proof of large deviation lower bound is to establish a law of large number for a perturbed dynamics, proving in particular the uniqueness of the perturbed Boltzmann equation, which in general fails.
Regarding the Newtonian dynamics, in view of the previous discussion, the validity of a large deviation principle is a most challenging issue. The general structure of the rate function associated to the Boltzmann equation for hard sphere is discussed in [7]. A derivation from Newtonian dynamics is presented in [6], see also [5] for a comparison of the rate function derived in [6] with the one proposed in [7].
We focus on the large deviation principle for Kac-type spatially homogeneous models. Beside the empirical density, it is convenient to introduce another observable, the empirical flow, which records the incoming and outgoing velocities in the collisions, i.e., letting N the number of particles, N q t (dv, dv * , dv ′ , dv ′ * ) dt is the number of collisions in the time-window [t, t + dt] with incoming velocities in dv dv * and outgoing velocities in dv ′ dv ′ * . In particular, the mass of the empirical flow is the normalized total number of collisions. The empirical measure and flow are linked by the balance equation which expresses the conservation of the mass. The idea of considering the pair of observables empirical measure and flow has been exploited in the context of Markov processes in [4,3,12]. In this setting the rate function relative to the pair empirical measure and flow has a closed form and it is equal to I(f, q) = I 0 (f 0 )+J(f, q), where I 0 (f 0 ) takes into account the fluctuations of the initial datum and the dynamical term J is given by where q f (dv, dv * , dv ′ , dv ′ * ) = 1 2 f (v)f (v * ) dv dv * r(v, v * ; dv ′ , dv ′ * ). By projecting J on the empirical density f (v)dv one recovers the variational expression for the rate function obtained for the empirical measure in [18,15,7].
The rate function (1.3) has a simple interpretation in terms of Poisson point processes. Let {Z N } be a sequence of Poisson random variables with parameters N λ. By Stirling's formula, for N large Therefore, for N large the statistics of the collisions in the Kac's walk can be thought as sampled from a Poisson point process with intensity N q f , where now f and q are related by the balance equation (1.2).
Here we implement this program for a model of N particles, interacting via stochastic binary collisions satisfying the conservation of the momentum, but not of the kinetic energy. Such a model is relevant, as example, in the case of a molecular gas when the internal degrees of freedom are disregarded. We do not assume a detailed balance condition and the corresponding Boltzmann-Kac equation is of the form (1.1). We prove the large deviation upper bound with the rate function introduced above. The proof of the matching lower bound is achieved when q has bounded second moment. From a technical viewpoint, the advantage of momentum conservation with respect to energy conservation is the linearity of the constraint, which allows to use convolution in the approximation argument for the lower bound. We also derive the variational formula for the projection of the rate function on the empirical measure.
In the context of i.i.d. Brownians, the connection between the large deviation rate function and the gradient flow formulation of the heat equation is discussed in [1], see also [2,17] for the case of i.i.d. reversible Markov chains. Here we derive a gradient flow formulation for the Boltzmann-Kac equation from the large deviation rate function (1.3). On general grounds, a gradient flow formulation of evolution equations is based on the choice of a pair of functions in Legendre duality. In [10] it is shown how this pair can be chosen so that the Boltzmann-Kac equation is the gradient flow of the entropy with respect to a suitable distance on the set of probability measures. As we here show, the choice in [10] is not the one associated to the large deviation rate function. Instead, analogously to [2], in the formulation here presented the non-linear Dirichlet form associated to the Boltzmann-Kac equation plays the role of the slope of the entropy.

Notation and main result
Kac walk. For a Polish space X we denote by M(X) the set of positive Radon measures on X with finite mass; we consider M(X) endowed with the weak* topology and the associated Borel σ-algebra.
Given N ≥ 2, a configuration is defined by N velocities in R d . The configuration space is therefore given by Σ N := (R d ) N . Elements of Σ N are denoted by v = (v k ) k=1,...,N , with v k ∈ R d . The Kac walk that we here consider is the Markov processes on Σ N whose generator acts on continuous and bounded functions f : Σ N → R as where the sum is carried over the unordered pairs {i, j} ⊂ {1, .., N }, i = j, and Here and the collision rate r is a continuous function from Finally, the scattering rate λ : We assume that r satisfies the following conditions in which we set V : Non degeneracy of the scattering kernel. There exists a density B : We remark that we do not assume balance conditions. Observe that item (iv) implies 2) An example of a scattering kernel meeting the above conditions is We denote by (v(t)) t≥0 the Markov process generated by L N . Let Σ N,0 := v ∈ Σ N : N −1 k v k = 0 be the subset of configurations with zero average velocity, that it is invariant by the dynamics in view of the conservation of the momentum. By the positivity of the collision rate (see Assumption 2.1, item (iii)), the Kac walk is ergodic when restricted to Σ N,0 . We shall consider the Kac's walk restricted to Σ N,0 .
Fix hereafter T > 0. Given a probability ν on Σ N,0 we denote by P N ν the law of the Kac walk on the time interval [0, T ]. Observe that P N ν is a probability on the Skorokhod space D([0, T ]; Σ N,0 ). As usual if ν = δ v for some v ∈ Σ N,0 , the corresponding law is simply denoted by P N v . Empirical measure and flow. We denote by P 0 (R d ) the set of probability measures on R d with zero mean. We consider P 0 (R d ) as a closed subset of the space of probability measure with finite mean equipped with the W 1 Wasserstein distance. Then P 0 (R d ) endowed with the relative topology is a Polish space. The empirical measure is the map π N : Σ N,0 → P 0 (R d ) defined by Let D [0, T ]; P 0 (R d ) the set of P 0 (R d )-valued cádlág paths endowed with the Skorokhod topology and the corresponding Borel σ-algebra. With a slight abuse of notation we denote also by π N the map from , and F is continuous and bounded, while (τ i,j k ) k≥1 are the jump times of the pair For each v ∈ Σ N,0 with P N v probability one the pair (π N , Q N ) satisfies the following balance equation that express the conservation of probability. For each φ : [0, T ] × R d → R bounded, continuous, and continuously differentiable with respect to time In view of the conservation of the momentum, the measure Q N (dt; ·) is supported on the hyperplane V.
The rate function. Let S be the (closed) subset of D [0, T ]; P 0 (R d ) × M given by elements (π, Q) that satisfies the balance equation for each φ : [0, T ] × R d → R continuous, bounded and continuously differentiable in t, with bounded derivative. We consider S endowed with the relative topology and the corresponding Borel σ-algebra.
For π ∈ D [0, T ]; P 0 (R d ) let Q π be the measure defined by and observe that Q π (dt, ·) is supported on V.
Definition 2.2. Let S ac be the subset of S given by the elements (π, Q) that satisfy the following conditions: Observe that by item (iv) of Assumption 2.1, condition (ii) implies that if (π, Q) ∈ S ac then Q π is a finite measure. Moreover, by choosing positive functions φ not depending on t in the balance equation (2.6) and neglecting the loss term we obtain Since Q ≪ Q π and, by Assumption 2.1, item (iii), the marginal on v ′ of Q π is absolutely continuous with respect to the Lebesgue measure, we deduce that π 0 ≪ dv implies π t ≪ dv, for any t ≥ 0. As a consequence, also Q is absolutely continuous with respect to the Lebesgue measure on [0, T ] × V.
The dynamical rate function J : S → [0, +∞] is defined by (2.8) In order to obtain a large deviation principle, chaotic initial conditions are not sufficient but we need that the empirical measure at time zero satisfies a large deviation principle. Referring to [8] for a discussion on entropically chaotic initial conditions, we next provide an example of a class of allowed initial data. Assumption 2.3. Given m ∈ P 0 (R d ) set µ N = m ⊗N and choose as initial distribution of the Kac's walk the probability on Σ N,0 given by ν N = µ N ( · | i v i = 0). We assume that m is absolutely continuous with respect to the Lebesgue measure and still denote by m its density. We further more assume that there exists γ > 0 such that Given two probabilities µ 1 , µ 2 ∈ P 0 (R d ), the relative entropy H(µ 2 |µ 1 ) is defined as H(µ 2 |µ 1 ) = dµ 1 ρ log ρ, where dµ 2 = ρ dµ 1 , understanding that H(µ 2 |µ 1 ) = +∞ if µ 2 is not absolutely continuous with respect to µ 1 .
Denoting byŜ the set of paths (π, Q) ∈ S such that The proof of the upper bound does not rely on item (iii) of Assumption 2.1. In particular it holds also when the collision rate conserves the energy. Likewise, item (iii) in Assumption 2.3 is used only in the proof of the lower bound. We also remark that, if we replace item (iii) in Assumption 2.1 by the condition , or the collision r has non degenerate density on (R d ) 4 , the lower bound holds in the whole S. As we show in Proposition 5.1, the projection of I on the empirical measure coincides with the variational expression in [18,15].

Upper bound
The upper bound is achieved by an established pattern in large deviation theory. We first prove the exponential tightness, which allows us to reduce to compacts. By an exponential tilting of the measure, we prove an upper bounds for open balls and finally we use a mini-max argument to conclude. Proposition 3.1 (Exponential tightness). There exists a sequence of compacts K ℓ ⊂ S such that for any N By standard compactness criteria (Banach-Alaoglu, Prokhorov and Ascoli-Arzelà theorems), the proof follows from the bounds in the next three lemmata.
To deal with the initial conditions, as detailed in Assumption 2.3, we need the following elementary statement whose proof is omitted.
Proof of Lemma 3.2. Given γ > 0 to be chosen later, let Ψ(v) = γ k |v k | 2 and set To complete the proof we show that Assumption 2.3 implies that, possibly by redefining γ > 0, there exists a constant c such that for any N (3.5) In order to prove this bound, we apply Lemma 3.
In view of the previous lemma, it is enough to show that for each h > 0 Recall that the scattering rate λ has been defined in (2.1). Given a bounded measur- where in last inequality we have used (2.2).
Proof of Lemma 3.4. In view of the balance equation (2.6) and Lemma 3.2, it is enough to show that for any h > 0 there exists a function c : (0, 1) → R + with c(δ) ↑ +∞ as δ ↓ 0 such that, for any ε > 0 with ζ(v) = |v| 2 . By a straightforward inclusion of events, the previous bound follows from 1 δ sup Consider the super-martingale (3.7) with F = γ 1I [t,t+δ] , γ > 0. Using the same argument of the previous lemma and (2.2) we deduce The proof is concluded by choosing γ = log(1/δ).

Upper bound on compacts. Given a bounded continuous function
Recalling that Λ(v) := λ(v, v), and using Lemma 3.5, we get where in the last inequality we used (2.2). The statement is achieved by observing that E Ñ ν N (N F T ) ≤ 1, and noting that by the local central limit theorem both Recall that H(·|m) denotes the relative entropy and let J be the functional defined in (2.8).
Proposition 3.7 (Variational characterization of the rate functional). For any pair (π, Q) ∈ S satisfying (i) and (ii) in Definition 2.2 (3.10) In the first formula the supremum is carried out over the continuous and bounded φ : R d → R such that the probability m φ (as defined in Lemma 3.5) is centered. In the second formula the supremum is carried out over all continuous and bounded . Since the set of π satisfying the condition in Definition 2.2, items (i), (ii), is a closed subset of D([0, T ]; P 0 (R d )), the previous characterization of the rate functional readily implies the lower semicontinuity of I; Proof. The first statement follows from the variational characterization of the relative entropy and the observation that since π 0 is centered it is enough to consider φ satisfying the stated constraint.
To prove the second statement, recall the definition of Q π in (2.7) and observe that 1 2

This implies that if sup
The proof is now completed by a direct computation, see Lemma 4.4 in [3].

Lower bound
In order to obtain the large deviation lower bound, given (π, Q) we need to produces a perturbation of the dynamics such that the law of large number for (π N , Q N ) is (π, Q). While the compactness of (π N , Q N ) follows from the arguments of the previous section, in order to identify the limit point we need uniqueness of the perturbed Boltzmann-Kac equation that we are able to prove only if the perturbed scattering rate is bounded. Therefore, we shall first prove the lower bound for open neighborhoods of "nice" (π, Q), and then use a density argument, that will be completed with the restriction that Q has bounded second moment.
Perturbed Kac walks. We start by the following law of large numbers for a class of perturbed Kac's walks. Consider perturbed time-dependent collision ratesr t , with densityB t , i.e.r which we assume to meet condition (iv) in Assumption 2.1 uniformly for t ∈ [0, T ], and to satisfy the following extra condition. There exists C < +∞ such that for Given a probability ν on Σ N,0 we denote byP N ν the law of the perturbed Kac walk with initial datum ν. where Here we understand that (4.3) holds by integrating against continuous, bounded test functions which are continuous differentiable in time.
It remains to show that f ∈ C([0, T ]; L 1 (R d )) and that the solution to (4.3) is unique. Choosing test functions independent of time and integrating (4.3) we deduce that for each t ∈ [0, T ] and Lebesgue almost every v it actually holds Sinceλ is bounded, it is now straightforward to show that f ∈ C([0, T ]; L 1 (R d )).
The collection of "nice" (π, Q) is specified as follows.
Definition 4.2. LetS be the collection of elements (π, Q) ∈ S ac whose densities (f, q) are such that and for some η > 0.
Given (π, Q) ∈S, denote byr t the time dependent perturbed rate whose density is defined byB that meets the condition (iv) in Assumption 2.1 uniformly for t ∈ [0, T ] and the extra assumption (4.2). The next statement provides the large deviation lower bound for neighborhood of elements inS. Proposition 4.3. Let (π, Q) ∈S. Assume that π 0 satisfies items (i), (ii) in Assumption 2.3, and suppose π 0 (dv) = e φ m(dv)/m(e φ ) for some φ bounded and continuous. Moreover, denote byν N = π ⊗N 0 (·| i v i = 0) the corresponding probability on Σ N,0 . Then We premise the following lemma.
The first bound in the statement is achieved by observing that Lemma 3.2 holds also for the perturbed chain. In order to prove the second bound, let that it is aPνN martingale, with predictable quadratic variation In view of (4.7) and (4.6) in Definition 4.2, the random variable M N T is uniformly bounded in N . This completes the proof.
Proof of Proposition 4.3. We first prove that where, by the local central limit theorem, the last term on the right hand side vanishes as N → +∞. As a corollary of Lemma 4.1 we deduce that π N converges inν N -probability to π 0 . Hence, in view of assumptions on φ, we deduce We now show that In view of the assumptions onr t the super-martingale defined in (3.7) with F t = log(dr t / dr) is actually a martingale and its value at time T is the Radon-Nykodim

By definition ofS
where we used Assumption 2.1, item (iii). Now observe that, by Lemma 4.1, (π N , Q N ) converges to (π, Q) inP Ñ ν N probability. Moreover, Lemma 4.4 provides sufficient conditions for the uniform integrability of Q N (F ) and of T 0 dt π N t ⊗ π N t (λ). Finally, by the boundedness ofλ t and the absolutely continuity of π t we obtain lim N →∞ Recalling (2.10), the statement follows from (4.8) and (4.9).
By general results, see e.g. [16], the previous proposition implies the following lower bound statement. Then the sequence P N ν N •(π N , Q N ) −1 satisfies a large deviations lower bound with rate functionĨ.
The lower bound in Theorem 2.4 follows directly from Corollary 4.5 and the above theorem. In order to prove it, we premise the following lemma.
Lemma 4.7. Let (π, Q) ∈Ŝ be such that I(π, Q) < +∞. Denote by (f, q) the densities of (π, Q). Moreover, set σ : , where g is the standard Gaussian density on R d . Then Proof. We start by proving By Assumption 2.1, item (iv), there exists η > 0 such that Recall that I(π, Q) = H(π 0 |m) + J(π, Q), where J is defined in (2.8). By the variational representation of J in Proposition 3.7, for any F bounded and continuous By a truncation argument, we can choose F = η|v ′ − v ′ * | 2 , and thus deduce (4.10). Let ζ n (v) = |v| 2 ∧ n. By using the continuity equation we get By item (ii) in Definition 2.2, (2.9), and (4.10), taking the limit n → ∞ we deduce which, together with the conservation of the momentum, implies the statement (i). In view of Assumption 2.1, items (iii) and (v), the statement (ii) follows from (i).
Proof of Theorem 4.6. Observe that by the lower semicontinuity of I, for any sequence (π n , Q n ) → (π, Q) we have lim n I(π n , Q n ) ≥ I(π, Q). The converse inequality is achieved by combining steps 1 and 2 below and a standard diagonal argument.
Step 1 -Convolution. Since I(π, Q) < +∞, there exist (f, q) such that dπ t = f t (v) dv and dQ Given 0 < δ < 1, let g δ be the Gaussian kernel on R d with variance δ and define We now show that the pair (f δ , q δ ) satisfies the balance equation. Given a test function φ and denoting by * the convolution One can repeat the same argument with φ(v) replaced by φ(v * ). Moreover, since Using the balance equation for the pair (f, q) with the test function g δ * φ we deduce that (f δ , q δ ) satisfies the balance equation. Now we show that lim sup δ→0 I(f δ , q δ ) ≤ I(π, Q). To this end, we use the decomposition provided by item (ii) of Lemma (4.7). We start by observing that, in view of (2.2) and sup t π t (ζ) < +∞, ζ(v) = |v| 2 , by dominated convergence Analogously, since | log σ| ≤ C(|v| 2 + |v * | 2 + |w ′ | 2 ), in view of Lemma 4.7, item (i) By the convexity of the map [0, +∞) 2 ∋ (a, b) → a log(a/b) and Jensen's inequality By Lemma 4.7, item (i), Gathering the above statements we deduce To conclude the proof of (4.11) it remains to show that  By item (iii) in Assumption 2.3, since f 0 has bounded second moment, we have The bound (4.12) follows by using item (iii) in Assumption 2.3 and Jensen's inequality.
Step 2 -Truncation of the flux. Given a pair (f, q) that satisfies the balance equation, we denote by q (i) t , and the balance equation reads In the sequel we assume (f, q) such that I(f, q) < +∞, f strictly positive on compacts uniformly in time, and q Observe that pair (f δ , q δ ) constructed in Step 1 meets the above conditions. Indeed, by item (ii) in Definition 2.2, for each compact subset K ∈ R d and δ ∈ (0, 1), there exists c K,δ > 0 such that, for any t ∈ [0, T ] we have inf v∈K g δ * f t ≥ c K,δ . Moreover, by Young inequality, Given ℓ > 0, set Ω ℓ the subset of V given by and define (f ℓ ,q ℓ ) bỹ (4.14) The previous identity implies that if |v| ≤ ℓ, then f ℓ t ≥ c ℓ f t , while if |v| > ℓ, then f ℓ t = c ℓ f 0 . In particular, for any t, f ℓ t > 0, and f ℓ t = 1. Observe that by construction the pair (f ℓ , q ℓ ) satisfies the balance equation and it is an element ofS (see definition 4.2), since q ℓ is bounded and compactly supported and f is strictly positive on compacts, uniformly in time. Moreover, (f ℓ , q ℓ ) converges to (f, q). Now we prove that lim ℓ→+∞ I(f ℓ , q ℓ ) ≤ I(f, q).
We start by proving that s 1I |v|≤ℓ .
Since c ℓ → 1, by the convexity of H(·|m) it is enough to show Observe that Since, by assumption on q ℓ, (3) , h ℓ ∈ L 2 and it converges to zero point-wise, using item (iii) of Assumption 2.3, we deduce (4.15) by dominated convergence. It remains to show that Recalling the scattering rate λ defined in (2.1), we rewrite Since q| log(2 q/f f * B)| < +∞, using the bound f ℓ ≥ c ℓ f for |v| ≤ ℓ, by dominated convergence Recalling (2.2), item (i) in Lemma 4.7 implies the uniform integrability of λ with respect to f ℓ f ℓ * , hence which concludes the proof of (4.16).

Projection on the empirical measure
In this section we analyze the large deviation asymptotics of the empirical measure only. By contraction principle, the corresponding rate function is obtained by projecting the joint rate function I. Regarding the upper bound, we prove that this projection corresponds to the rate function in [15,18,7]. For the lower bound, we identify the projection of I only for suitable π.

Gradient flow formulation of the Boltzmann-Kac equation
Assuming the detailed balance condition, here we derive the gradient flow formulation of the Boltzmann-Kac equation (1.1) associated to the large deviation rate function (1.3). We remark that such formulation is logically independent from the validity of the large deviation principle.
Let M be the standard Maxwellian on R d . In this section we assume that the collision rate r satisfies the following detailed balance condition This implies that the Kac walk on Σ N is reversible with respect to the product measure N k=1 M (dv k ). We still consider the Kac walk restricted to Σ N,0 , then the corresponding reversible measure is the product measure N k=1 M (dv k ) conditioned to N −1 k v k = 0, that is a Gaussian measure on Σ N,0 . In this section we express the empirical measure and flow in terms of their densities with respect to Maxwellians.
Let H : P 0 (R d ) → [0, +∞] be the relative entropy with respect to M , i.e. H(π) := H(π|M ). For π ∈ P 0 (R d ) with bounded second moment, define the non linear Dirichlet form D : P 0 (R d ) → [0, +∞] as the lower semicontinuous map defined by where the supremum is carried out over the continuous and bounded functions φ : R d → R. Note that D(π) is well defined in view of (2.2). To illustrate this definition, consider the Markov generator L acting on functions ξ : By the detailed balance condition, L is reversible with respect to the product measure M (dv) M (dv * ). The variational representation (6.2) thus corresponds to the Donsker-Varadhan functional E, that it is defined on the probabilities on R d × R d by where the supremum is carried out over the continuous and bounded functions ξ : R d × R d → R. Then D(π) = E(π × π), observe indeed, as proven in Lemma 6.2 below, that for product measures Π we can restrict the class test functions ξ to functions of the form ξ(v, v * ) = φ(v) + φ(v * ). On the set of functions G : Recalling the definition of Q π in (2.7), we define the kinematic term as the lower semicontinuous functional R on the pairs (π, Q) satisfying conditions (i) and (ii) in Definition 2.2 where the supremum is carried out over the bounded and continuous F, α : , and inf α > 0. The main result of this section provides, when the detailed balance condition (6.1) holds, a gradient flow formulation of the Boltzmann-Kac equation. Recall that the functionals J and I have been introduced in (2.8) and (2.10) and thatŜ is the set of paths (π, Q) ∈ S ac that satisfy (2.9).
In particular, when the scattering rate λ is bounded, I(π, Q) = 0 if and only if π 0 = m and We start by the following characterization of the Dirichlet form D and the kinematic term R in which we recall that V is the hyperplane of ( Lemma 6.2. Let π ∈ P 0 (R d ) be such that π(dv) = h(v)M (dv), π(ζ) < +∞, ζ(v) = |v| 2 , and D(π) < +∞. Then and Proof. We first note that Indeed, by standard arguments, see e.g. [14, App. 1, Thm. 10.2], D(π) is bounded above by the right hand side in the previous displayed formula. The converse inequality is obtained by choosing as test function a sequence of continuous and bounded φ n that converges to 1 2 log h. Recalling (2.2), the proof of (6.6) is achieved by expanding the square on the right hand side of (6.9) and using the detailed balance condition. The representation (6.7) now follows directly by (6.9). Finally, by (6.6) and using that Q(1) < +∞, the representation (6.8) is achieved by a direct computation.

(6.11)
Observe in fact that the converse inequalities follows from the lower-semicontinuity of J, H, D, and R. Let (π n , Q n ) be the sequence constructed in the proof of Theorem 4.6, so that the first inequality in (6.11) holds. The proof of the others is achieved in two steps.
By the representation of D provided by Lemma 6.2, using (4.17) and Fatou's lemma we conclude that It remains to show that lim ℓ R(π ℓ , Q ℓ ) ≤ R(π, Q). (6.17) This is achieved by using the representation (6.8) and the argument in Step 2 of Theorem 4.6.