Thermal approximation of the equilibrium measure and obstacle problem

We consider the probability measure minimizing a free energy functional equal to the sum of a Coulomb interaction, a confinement potential and an entropy term, which arises in the statistical mechanics of Coulomb gases. In the limit where the inverse temperature $\beta$ tends to $\infty$ the entropy term disappears and the measure, which we call the"thermal equilibrium measure"tends to the well-known equilibrium measure, which can also be interpreted as a solution to the classical obstacle problem. We provide quantitative estimates on the convergence of the thermal equilibrium measure to the equilibrium measure in strong norms in the bulk of the latter, with a sequence of explicit correction terms in powers of $1/\beta$, as well as an analysis of the tails appearing after a boundary layer of size $\beta^{-1/2} (\log \beta)^{1/2}$.

1. Introduction 1.1.Setting of the problem.The Coulomb gas is a system of points in R d with pairwise interaction g defined by and an external (or confinining) potential (or field) V , so that the total energy of the system of N point at locations x 1 , . . ., x N is given by (1.1) Here, the strength of the external potential V has been scaled so that the potential energy is of the same order as the interaction energy.In the limit N → ∞, called the "mean field limit", one is led to minimizing among probability measures the (mean-field) energy Here µ should be thought of as the limit as N → ∞ of the empirical measures 1 N N i=1 δ x i .It is well known that if V grows sufficiently fast at infinity, problem (1.2) has a unique minimizer among probability measures, called the equilibrium measure, or the Frostman equilibrium measure, see for instance [19] for the two-dimensional case.This measure will be denoted µ ∞ .It can be shown for instance that minimizers of (1.1) converge to µ ∞ (see [9], or [20,Chap. 2]).
The equilibrium measure µ ∞ is typically compactly supported and characterized by the fact that there exists a constant c ∞ such that letting (1.3) ζ(x) := ˆRd g(x − y)dµ ∞ (y) + V (x) − c ∞ , we have ζ = 0 q.e. in supp µ ∞ and ζ ≥ 0 q.e.where q.e. is the abbreviation of "quasieverywhere" which means except on a set of zero capacity.This way we can see that µ ∞ can be interpreted in terms of the classical obstacle problem.Using the notation (1.4) h µ (x) := ˆRd g(x − y)dµ(y) the function h µ∞ satisfies −∆h µ∞ = c d µ ∞ , where (1.5) is the constant for which −∆g = c d δ 0 .By the above properties on ζ it holds that which is precisely the equation for the solution to the classical obstacle problem in whole space with obstacle c ∞ −V .For more details about this correspondance between equilibrium measure and obstacle problem, one can see for instance [20,Chap. 2], [3] and references therein.The dependence of µ ∞ in V has been previously examined in this full space context in [22].The Gibbs measure corresponding to a Coulomb gas at inverse temperature β is Different normalizations of β with respect to N can be chosen, the specific above choice with 1/N in front of the energy leads in the mean-fied limit N → ∞ to a minimization problem with an added entropy term of the form: see for instance [14,16,8,5].Again (1.8) should be minimized among probability measures, and if V grows sufficiently fast, it has a unique solution µ β which we will call the thermal equilibrium measure.The functional (1.8) can also be seen as the free energy associated to the McKean-Vlasov equation which is its Wasserstein gradient flow, see for instance [13] and references therein.
On the other hand, a natural normalization for the energy and temperature in (1.7) is shown in [15,2] to be (1.9)exp −βN it is natural as β fixed is then shown to be the temperature choice that leads to a competition at the microsopic scale between interaction energy and entropy.This is in particular the normalization most studied in dimension 2 where β = 2 then corresponds to the famous determinantal case of the Ginibre ensemble.This choice, for which β can still be considered to depend on N , then leads in the mean-field limit to minimizing in place of (1.8).In other words it leads to considering the regime where β in (1.8) tends to ∞ as N → ∞, and thus formally to minimizing just (1.2).In [2], we showed however that, compared to the usual equilibrium measure minimizing (1.2), the thermal equilibrium measure still provides a more precise description of a Coulomb gas, even for the regime with β in (1.10) of order 1, equivalently β of order N 2/d in (1.8).
In this paper we thus focus on the regime β 1 in (1.8), where one expects µ β → µ ∞ .This can also be seen as a way to smoothly approximate the obstacle problem solution.The goal of this short paper is to specify how µ β is close to µ ∞ and h µ β to h µ∞ , which we will do in C k norms.The quantitative estimates we provide are crucially used in the papers [2,21] and allow to treat possibly quite large temperature regimes in (1.9) (note that large temperature regimes for Coulomb or log gases have started to gain interest quite recently, see [18,2,12]).
We note that this question, although quite natural, does not seem to have been fully answered in the literature, the only results that we are aware of are less precise, they are those in [14] which consider the two-dimensional case with no external potential, and [18] which provide some results in the particular case V (x) = |x| 2 , and finally the work [6] motivated by Kähler geometry, which proves an L ∞ bound on the difference of h µ β and h µ∞ analogous to (1.21) but in the compact setting of a manifold.There were also explicit formulae for the one-dimensional logarithmic case (related but slightly out of our scope) and quadratic potential in [1].
By contrast with µ ∞ , µ β is not compactly supported, but always positive in R d and regular.In fact h µ β defined as in (1.4) solves the PDE (1.11) for some constant c β .Taking the Laplacian of that equation leads to a PDE on log µ β with notoriously delicate exponential nonlinearity Instead of studying this equation directly, we observe for the first time that when subtracting two such equations (with possible error term) with solutions µ and ν respectively, the quotient u = µ/ν − 1 rewrites nicely as a divergence form equation (1.13) div ∇u 1 + u = βµu + error for which elliptic regularity theory is readily applicable as soon as u is small enough.This allows to obtain corrections to arbitrary order of the approximation µ β µ ∞ , see (1.28) below.In fact our proofs only use maximum principle-based arguments and regularity theory, and does not require going through energy estimates.
Finally, we comment that the other extreme regime β → 0 is easier to treat.We can formally expect the interaction energy to become negligible and we are then led to minimizing among probability measures ˆRd V dµ + 1 β ˆRd µ log µ whose solution is µ = e −βV ´e−βV , see [18].
1.2.Assumptions and results.We let Σ := supp µ ∞ and assume that ∂Σ ∈ C 1 .In fact what we really need is a uniform interior ball condition.(If ∂Σ was irregular, we could get our results at an appropriate further distance from ∂Σ.)Note that it was very recently established in [10] that this is generic with respect to V .We assume in addition (1.17) ∆V ≥ α > 0 in a neighborhood of Σ.
Note that (1.14) and (1.15) imply that V is bounded below.In dimension The set {ζ = 0} corresponds to the contact set or coincidence set of the obstacle problem, and Σ is the set where the obstacle is active, sometimes called the droplet.The assumption (1.17) in fact ensures that they coincide.Note that in {ζ = 0}, h µ∞ = c ∞ − V hence the density Thanks to this connection, the regularity of µ V and of Σ can be known by the standard regularity theory for the classical obstacle problem [7].For the precise reformulation in the whole space one can also refer to [22].
There exists C(V, d) such that letting the following holds.Let m be an integer ≥ 2 such that V ∈ C 2m,γ for some γ ∈ (0, 1] and letting f k be defined iteratively by The functions f k provide a sequence of improving approximations to µ β defined iteratively.Spelling out the iteration we easily find the expansion in powers of 1/β (1.28) up to an order dictated by the regularity of V and the size of β.
The relation (1.21) retrieves in particular the equivalent result in [6], while (1.23) improves on the energy comparison based estimate in 1/ √ β given in [18].The estimates reveal the natural lengthscale 1/ √ β appearing in the approximation of µ β by µ ∞ .
Remark 1.1.If d ≥ 3, since h µ β and h µ∞ vanish at infinity, (1.21) implies that It is not clear how to obtain such a precise estimate from energy considerations only.
The rest of the paper is organized as follows: In the next section we obtain at once the L ∞ comparison between the solutions to (1.11) and (1.6) via a comparison principle and a uniform bound on µ β .This bound then serves to obtain a lower bound for µ β inside Σ by a barrier argument in the following section.The estimates on h µ β − h µ∞ are then eventually upgraded to C k spaces via the iterative approximation sequence f k thanks to DeGiorgi-Schauder elliptic regularity theory applied to (1.13).
Acknowledgements: SA was supported by NSF grant DMS-1700329 and a grant of the NYU-PSL Global Alliance and SS was supported by NSF grant DMS-1700278 and by the Simons Investigator program.

The comparison principle and upper bound on µ β
In all the rest of the paper, C will denote a generic positive constant which depends only on V and d.
We observe that by definition, the functions h µ β = g * µ β and h µ∞ = g * µ ∞ satisfy the following asymptotics and We will use these facts repeatedly.
2.1.A preliminary lemma.We will use the following comparison principle for the obstacle problem in the whole plane.
Lemma 2.1.Suppose that v, w are two continuous function in R 2 which satisfy as well as Proof of Lemma 2.1.Let φ = c ∞ − V be the obstacle function.We may assume without loss of generality that φ ≤ 0 (otherwise we may subtract a constant).Then v ≤ 0 by the maximum principle, since the zero function is a harmonic function which, due to (2.4), is larger than v in the complement of a bounded set.Moreover, min{tw, 0}, with 0 < t ≤ 1, satisfies the same assumptions as w, and thus it suffices to show that v ≤ tw for every 0 < t < 1.In light of this, we may assume that lim sup In particular, {v > w} is bounded.Observe also that {v > w} ⊆ {v > φ}.Since v is subharmonic in the latter and w is superharmonic in R 2 , we deduce that v−w is subharmonic in {v − w > 0}.Assume that this set is nonempty, to get a contradiction.Let x 0 be the point at which v − w attains its global maximum, say Then, since v − w is subharmonic at x 0 , we deduce that it is constant in a neighborhood of x 0 .In fact, this argument shows that the set hence in view of (1.15), we have lim and thus µ β must achieve a positive maximum, denoted Proof.To compare h µ β − c β and h µ∞ − c ∞ we recall that h µ β satisfies (1.11) while h µ∞ satisfies (1.6).We may write from (1.11) that In dimension d = 2, applying the comparison principle of Lemma 2.
and h µ∞ we deduce that which is the desired result.For dimension d ≥ 3, we first show that we have decays at infinity like |x| 2−d , and the fact that ϕ ≥ ψ bring a contradiction, which shows that lim inf |x|→∞ ϕ ≥ 0. Since (2.11) holds in any case, we then deduce by the maximum principle that ϕ ≥ 0 in all R d , which is the desired result.
We deduce the following bounds on µ β .Lemma 2.3.For every x ∈ R d and β ≥ 1, we have Proof.With the result of (2.7) and bounds on h µ∞ , we have Inserting into (1.11)we deduce that In view of (1.15), this implies that x β remains in a fixed ball B R independent of β.On the other hand, we must have ∆ log µ β (x β ) ≤ 0 by local maximality of x β , hence by (1.12) We may then deduce that Proof.The lower bound is now an immediate consequence of (2.7) and (2.12).Let us turn to the upper bound.We know that min(h We would like the right-hand side to be ≤ 0, so we need to modify our test function slightly.Define Let us estimate µ β (E): using (2.12) and (1.16), we find that Let now w be (2.17) This way w decays like |x| 1−d in all dimensions d ≥ 2, and in view of (2.15) we have Let us then set for the C of (2.18).Observe that By choice of E, (1.11) and (2.18), we have in In Arguing as in the proof of Lemma 2.2, let ψ be a harmonic function equal to 0 on ∂E and β log β + Cβ −1 < 0, ψ tends to its limit from above at speed |x| 2−d .On the other hand ´Rd ∆ϕ = 0.As in the proof of Lemma 2.2, we get a contradiction and conclude that c β − c ∞ + 2 β log β + Cβ −1 ≥ 0. We then conclude from (2.21) and the maximum principle that ϕ ≥ 0 everywhere, which yields the desired result.
Proof.Taking the exponential of (1.11) and using (1.3), we find (2.25) It follows from (2.14) that If we assume (1.15), by standard results on the obstacle problem [7], we have Combining with (2.15) it follows that (2.24) holds.

Study of the radial case and barrier argument
We now specialize to V (x) = λ 2 |x| 2 with λ ≥ 1 which will provide a barrier function for the general case.The problem is then radial and the solution µ β (x) = e u(|x|) with u solving in place of (1.11) the ODE The coincidence set Σ is then a ball of radius R d λ −1/d .The next lemma shows that except in a layer of size O β −1/2 (log β) 1/2 near the boundary of B(0, R d λ −1/d ), the density µ β is well bounded below.
Lemma 3.1.Let m > 0. In the case V (x) = λ 2 |x| 2 , there exists a radius with C positive depending only on d and m, such that Moreover, there exists a constant M depending only on d and m such that for every x such , we have Proof.We note that 1 λ µ β (λ 1/d x) is the solution of the same problem with V replaced by 1 2 |x| 2 and β replaced by βλ 1− 2 d .We thus reduce to studying the case λ = 1 by rescaling the estimates.
One may then check that µ ∞ = 1 c d 1 B(0,R d ) .We first note that at a point of local maximum of µ β we have ∆ log µ β ≤ 0 hence µ β ≤ ∆V c d = 1 c d .We thus know that c d µ β ≤ 1 everywhere and thus (ru ) ≤ 0 and ru ≤ 0 hence u is nonincreasing.
Integrating (3.1) we also have The analogue of (2.23) in this context is + + C where (•) + denotes the positive part of a number, hence there is for some appropriate C, then u(r) ≥ log 1 2c d .Indeed, assume not, then by monotonicity of u we have for every r in if M is chosen large enough (independent of β), a contradiction.We may thus write that if , then u(r) ≥ log 1 2c d .The result follows by rescaling.

3.1.
A bound from below for µ β inside Σ.Let Σ be as in (1.25) for C large enough.
We may now use the radial solution as a barrier for the solution in the general case.
Proof.Since we assumed a uniform interior ball condition for Σ, we have the same for Σ with a ball of radius which can be chosen independently of the point, say of radius ε.
We then choose λ large enough that λ ≥ α, Cλ −1/d ≤ ε and that the r β in Lemma 3.1 is smaller than ε.Given this λ, we consider ν to be α/λ times µβα λ of Lemma 3.1.We may then check that in B r where r is the r βα/λ , a ball included in Σ.
In view of (3.2) and λ ≥ α, we also note that ν ≤ µ β on ∂B r if m is chosen large enough, in view of the lower bound in (2.23) and the definition of r β .We now substract (1.12) and (3.5) and test it against (log ν − log µ β ) + which is 0 on ∂B r .We obtain ˆBr Using that ∆V ≥ α in B r by (1.15) and an integration by parts, we are led to It follows that ν ≤ µ β a.e. in B r , thus ν is a barrier for µ β .In view of the result of Lemma 3.1, we deduce that µ β ≥ α 2c d as soon as x ∈ B r and dist( 1/2 .The result follows. Remark 3.3.Up to increasing the constant C in the definition of Σ (1.25), we may now assume that (3.4) holds in Σ.

Regularity theory
Once µ β is bounded below, the PDE (1.13) becomes uniformly elliptic and we may apply regularity theory tools to compare µ β to the expected solution.In the case that ∆V is constant, then we can show that µ β is very close to the constant µ ∞ inside Σ, however in the case where ∆V is not constant, there are corrections to arbitrary order that need to be added to µ ∞ .
Assuming that V ∈ C 2m,γ for some m ∈ N and exponent γ > 0, we recursively define f k by (1.26).We note that, for β sufficiently large depending on the norms of V and on k, and by (1.17), (4.1) We also define (4.2) and check that Thus f k is a good approximate solution to (1.12).In view of (4.2) and (4.3), if V ∈ C ∞ then f k converges as k → ∞ in all C m spaces to f ∞ , an exact solution of (1.12).We recall that we may assume that (3.4) holds in Σ.
Proposition 4.1.Assume m ∈ N, m ≥ 2, and γ ∈ (0, 1] are such that V ∈ C 2m,γ .Then for every n even integer with n ≤ 2(m − 1), there exists C > 0 depending only on V, d, n such that This applies to m possibly infinite.
Proof.Define u β := In view of (4.2), we get This equation is uniformly elliptic in Σ since, by Lemma 2. In particular, keeping only the second term on the left side, we obtain Note that the function c d f k is bounded.Applying the De Giorgi-Nash Hölder estimate (see [11,Theorem 8.24]) for uniformly elliptic equations, we obtain, for some σ > 0 and again for k ≤ m − 1, Remark 4.2.The estimates above apply in the same way to all solutions of relations of the form (4.5).This allows to handle questions of stability of the solutions with respect to V : if V is changed into V + tξ with ξ supported in Σ, then letting µ t β be the corresponding thermal equilibrium measure, the function u t =  which is of the same form as (4.5).The same method then allows to estimate u t hence µ t β /µ β .
that m β is bounded independently of β.The first bounds in the right-hand side of (2.12) follow.The bound in (2.13) then gets improved to log µ β ≤ β max(1, log |x|)1 d=2 + βC − βV which yields the second set of bounds in (2.12).