arXiv subjects

Sylvia Serfaty

Publications and source records attributed to Sylvia Serfaty.

At least 19 recordsLinked to original sources

arXiv2026

Singular mean-field limits via a multiscale mollification metric

We consider a general class of first order ODE systems for the evolution of $N$ interacting particles (in Euclidean space $\mathbb{R}^d$) in a mean-field regime. The class of interactions treated includes singular interactions of inverse power type up to power $d+1$, attractive or repulsive, and not necessarily deriving from a potential -- unlike, for instance, the modulated energy method. We introduce a new method to prove quantitative convergence of the discrete system to solutions of the mean-field equation. It relies on studying the evolution of a metric encoding a multiscale control of the difference between the empirical measure and its limit, via mollification by heat kernels. We prove that the desired convergence holds (i) up to the maximal time of existence of the smooth solution to the limiting equation if the singularity is sub-coulombic in any dimension, or coulombic in dimensions 1 and 2 (where, to do so, we introduce a notion of weak solution to the ODE system), or (ii) for short time in the case of Coulomb singularity in dimension 3 and above and (iii) up to a short $N$-dependent timescale for super-coulombic interactions in all dimensions. The latter two results are demonstrated to be optimal as we prove that collisions occur within the same timescale for a class of attractive interactions.

math.AP↗

arXiv2026

Multipole and Berezinskii-Kosterlitz-Thouless Transitions in the Two-component Plasma

We study the two-dimensional two-component Coulomb gas in the canonical ensemble and at inverse temperature $β>2$. In this regime, the partition function diverges and the interaction needs to be cut off at a length scale $λ\in (0,1)$. Particles of opposite charges tend to pair into dipoles of length scale comparable to $λ$, which themselves can aggregate into multipoles. Despite the slow decay of dipole--dipole interactions, we construct a convergent cluster expansion around a hierarchical reference model that retains only intra-multipole interactions. This yields a large deviations result for the number of $2p$-poles as well as a sharp free energy expansion as $N\to\infty$ and $λ\to0$ with three contributions: (i) the free energy of $N$ independent dipoles, (ii) a perturbative correction, and (iii) the contribution of a non-dilute subsystem. The perturbative term has two equivalent characterizations: (a) a convergent Mayer series obtained by expanding around an i.i.d.\ dipole model; and (b) a variational formula as the minimum of a large-deviation rate function for the empirical counts of $2p$-poles. The Mayer coefficients exhibit transitions at $β_p=4-\tfrac{2}{p}$, that accumulate at $β=4$, which corresponds to the Berezinskii-Kosterlitz-Thouless transition in the low-dipole-density limit. At $β=β_p$ the $p$-dipole cluster integrals switch from non-integrable to integrable tails. The non-dilute system corresponds to the contribution of large dipoles: we exhibit a new critical length scale $R_{β, λ}$ which transitions from $λ^{-(β-2)/(4-β)}$ to $+\infty$ as $β$ crosses the critical inverse temperature $β=4$, and which can be interpreted as the maximal scale such that the dipoles of that scale form a dilute set.

math-ph↗

arXiv2026

Another look at regularity in transport-commutator estimates

We are interested in how regular a transport velocity field must be in order to control Riesz-type commutators. Estimates for these commutators play a central role in the analysis of the mean-field limit and fluctuations for systems of particles with pairwise Riesz interactions, which we start by reviewing. Our first new result shows that the usual $L^\infty$ assumption on the gradient of the velocity field cannot, in general, be relaxed to a BMO assumption. We construct counterexamples in all dimensions and all Riesz singularities $-2< s<d$, except for the one-dimensional logarithmic endpoint $s=0$. At this exceptional endpoint, such a relaxation is possible, a fact related to the classical Coifman-Rochberg-Weiss commutator bound for the Hilbert transform. Our second result identifies a trade-off between the singularity of the interaction potential and the required regularity of the velocity field. Roughly speaking, smoother (less singular) interactions require stronger velocity control if one wants a commutator estimate in the natural energy seminorm determined by the potential. We formulate this principle for a broad class of potentials and show that, in the sub-Coulomb Riesz regime, the velocity regularity appearing in the known commutator inequality is sharp. Despite these negative findings, we show as our third result that a defective commutator estimate holds for almost-Lipschitz transport fields. Such a defective estimate, which is a consequence of the celebrated Brezis-Wainger-Hansson inequality, allows us to prove rates of convergence when the mean-field density belongs to the scaling-critical Sobolev space.

math.AP↗

arXiv2025

Local Laws and Fluctuations for Super-Coulombic Riesz Gases

We study the local statistical behavior of the super-Coulombic Riesz gas of particles in Euclidean space of arbitrary dimension, with inverse power distance repulsion integrable near $0$, and with a general confinement potential, in a certain regime of inverse temperature. Using a bootstrap procedure, we prove local laws on the next order energy and control on fluctuations of linear statistics that are valid down to the microscopic lengthscale, and provide controls for instance, on the number of particles in a (mesoscopic or microscopic) box, and the existence of a limit point process up to subsequences. As a consequence of the local laws, we derive an almost additivity of the free energy that allows us to exhibit for the first time a CLT for Riesz gases corresponding to small enough inverse powers, at small mesoscopic length scales, which can be interpreted as the convergence of the associated potential to a fractional Gaussian field. Compared to the Coulomb interaction case, the main new issues arise from the nonlocal aspect of the Riesz kernel. This manifests in (i) a novel technical difficulty in generalizing the transport approach of Leblé and the second author to the Riesz gas which now requires analyzing a degenerate and singular elliptic PDE, (ii) the fact that the transport map is not localized, which makes it more delicate to localize the estimates, (iii) the need for coupling the local laws and the fluctuations control inside the same bootstrap procedure.

math-ph↗

arXiv2025

A sharp commutator estimate for all Riesz modulated energies

We prove a functional inequality in any dimension controlling the derivative along a transport of the Riesz modulated energy in terms of the modulated energy itself. This modulated energy was introduced by the third author and collaborators in the study of mean-field limits and statistical mechanics of Coulomb/Riesz gases, where this control is an essential ingredient. Previous work of the last two authors and Q.H. Nguyen arXiv:2107.02592 showed a similar functional inequality but with an additive $N$-dependent error (where $N$ is the number of particles, $\mathsf{d}$ the dimension, and $\mathsf{s}$ the inverse power of the Riesz potential) which was not sharp. In this paper, we obtain the optimal $N^{\frac{\mathsf{s}}{\mathsf{d}}-1}$ error, for all cases, including the sub-Coulomb case. Our method is conceptually simple and, like previous work, relies on the observation that the derivative along a transport of the modulated energy is the quadratic form of a commutator. Through a new potential truncation scheme based on a wavelet-type representation of the Riesz potential to handle its singularity, the proof reduces to averaging over a family of Kato-Ponce type estimates. The commutator estimate has applications to sharp rates of convergence for mean-field limits, quasi-neutral limits, and central limit theorems for the fluctuations of Coulomb/Riesz gases both at and out of thermal equilibrium. In particular, we show here for $\mathsf{s}<\mathsf{d}-2$ the expected $N^{\frac{\mathsf{s}}{\mathsf{d}}-1}$-rate in the modulated energy distance for the mean-field convergence of first-order Hamiltonian and gradient flows. This complements the recent work arXiv:2407.15650 on the optimal rate for the (super-)Coulomb case $\mathsf{d}-2\le \mathsf{s}<\mathsf{d}$ and therefore resolves the entire potential Riesz case.

math.AP↗

arXiv2025

Sharp commutator estimates of all order for Coulomb and Riesz modulated energies

We prove functional inequalities in any dimension controlling the iterated derivatives along a transport of the Coulomb or super-Coulomb Riesz modulated energy in terms of the modulated energy itself. This modulated energy was introduced by the second author and collaborators in the study of mean-field limits and statistical mechanics of Coulomb/Riesz gases, where control of such derivatives by the energy itself is an essential ingredient. In this paper, we extend and improve such functional inequalities, proving estimates which are now sharp in their additive error term, in their density dependence, valid at arbitrary order of differentiation, and localizable to the support of the transport. Our method relies on the observation that these iterated derivatives are the quadratic form of a commutator. Taking advantage of the Riesz nature of the interaction, we identify these commutators as solutions to a degenerate elliptic equation with a right-hand side exhibiting a recursive structure in terms of lower-order commutators and develop a local regularity theory for the commutators, which may be of independent interest. These estimates have applications to obtaining sharp rates of convergence for mean-field limits, quasi-neutral limits, and in proving central limit theorems for the fluctuations of Coulomb/Riesz gases. In particular, we show here the expected $N^{\frac{\mathsf{s}}{\mathsf{d}}-1}$-rate in the modulated energy distance for the mean-field convergence of first-order Hamiltonian and gradient flows.

math.AP↗

arXiv2025

Vortex lines interaction in the three-dimensional magnetic Ginzburg--Landau model

We complete our study of the three dimensional Ginzburg--Landau functional with magnetic field, in the asymptotic regime of a small inverse Ginzburg--Landau parameter $\varepsilon$, and near the first critical field $H_{c_1}$ for which the first vortex filaments appear in energy minimizers. Under a nondegeneracy condition, we show a next order asymptotic expansion of $H_{c_1}$ as $\varepsilon \to 0$, and exhibit a sequence of transitions, with vortex lines appearing one by one as the intensity of the applied magnetic field is increased: passing $H_{c_1}$ there is one vortex, then increasing $H_{c_1}$ by an increment of order $\log |\log\varepsilon|$ a second vortex line appears, etc. These vortex lines accumulate near a special curve $Γ_0$, solution to an isoflux problem. We derive a next order energy that the vortex lines must minimize in the asymptotic limit, after a suitable horizontal blow-up around $Γ_0$. This energy is the sum of terms where penalizations of the length of the lines, logarithmic repulsion between the lines and magnetic confinement near $Γ_0$ compete. This elucidates the shape of vortex lines in superconductors.

math.AP↗

arXiv2024

Dipole Formation in the Two-Component Plasma

We consider the two-dimensional two-component plasma, or Coulomb gas, consisting of $N$ positive and $N$ negative charges with logarithmic interaction. We introduce a suitable regularization of the interaction by smearing the charges over a small length scale $λ$, which allows us to give meaning to the system in the continuum at any temperature. We provide an expansion of the free energy in terms of the inverse temperature $β$, as $N \to \infty$ and $λ\to 0$. Doing so allows us to show that, for $β\geq 2$, the charges (for the most part) pair into neutral dipoles of very small size as $λ\to 0$. This complements the prior work of Leblé, Zeitouni, and the second author, which proved that this does not happen for $β< 2$, thereby implying a transition at $β= 2$ from free charges to dipole pairs. Moreover, we obtain an estimate on the size of linear statistics. The description in terms of dipoles is made via a decomposition into nearest-neighbor graphs of the point configurations, à la Gunson-Panta. This is combined with new energy estimates obtained via an electric reformulation of the interaction energy and a ball-growth method, which are expressed in terms of the nearest-neighbor graph distances only. In this way, the model is compared to a reduced nearest-neighbor interaction model, by showing the relative smallness of the dipole-dipole interactions.

math-ph↗

arXiv2024

The Lake equation as a supercritical mean-field limit

We study so-called supercritical mean-field limits of systems of trapped particles moving according to Newton's second law with either Coulomb/super-Coulomb or regular interactions, from which we derive a $\mathsf{d}$-dimensional generalization of the Lake equation, which coincides with the incompressible Euler equation in the simplest setting, for monokinetic data. This supercritical mean-field limit may also be interpreted as a combined mean-field and quasineutral limit, and our assumptions on the rates of these respective limits are shown to be optimal. Our work provides a mathematical basis for the universality of the Lake equation in this scaling limit -- a new observation -- in the sense that the dependence on the interaction and confinement is only through the limiting spatial density of the particles. Our proof is based on a modulated-energy method and takes advantage of regularity theory for the obstacle problem for the fractional Laplacian.

math.AP↗

arXiv2024

Lectures on Coulomb and Riesz gases

This is the preliminary version of a book on Coulomb and Riesz gases that grew out of a set of notes written for graduate courses taught at the Courant Institute, at the Ecole Normale Supérieure, and at the 2024 Saint-Flour Probability summer school.

math-ph↗

arXiv2024

Relative entropy and modulated free energy without confinement via self-similar transformation

This note extends the modulated entropy and free energy methods for proving mean-field limits/propagation of chaos to the whole space without any confining potential, in contrast to previous work limited to the torus or requiring confinement in the whole space, for all log/Riesz flows. Our novel idea is a scale transformation, sometimes called self-similar coordinates in the PDE literature, which converts the problem to one with a quadratic confining potential, up to a time-dependent renormalization of the interaction potential. In these self-similar coordinates, one can then establish a Grönwall relation for the relative entropy or modulated free energy, conditional on bounds for the Hessian of the mean-field log density. This generalizes recent work of Feng-Wang arXiv:2310.05156, which extended the Jabin-Wang relative entropy method to the whole space for the viscous vortex model. Moreover, in contrast to previous work, our approach allows to obtain uniform-in-time propagation of chaos and even polynomial-in-time generation of chaos in the whole space without confinement, provided one has suitable decay estimates for the mean-field log density. The desired regularity bounds and decay estimates are the subject of a companion paper.

math.AP↗

arXiv2023

The attractive log gas: stability, uniqueness, and propagation of chaos

We consider overdamped Langevin dynamics for the attractive log gas on the torus $\mathbb{T}^\mathsf{d}$, for $\mathsf{d}\geq 1$. In dimension $\mathsf{d}=2$, this model coincides with a periodic version of the parabolic-elliptic Patlak-Keller-Segel model of chemotaxis. The attractive log gas (for our choice of units) is well-known to have a critical inverse temperature $β_{\mathrm{c}}={2\mathsf{d}}$ corresponding to when the free energy is bounded from below. Moreover, it is well-known that the uniform distribution is always a stationary state regardless of the temperature. We identify another temperature threshold $β_{\mathrm{s}}$ sharply corresponding to the nonlinear stability of the uniform distribution. We show that for $β>β_{\mathrm{s}}$, the uniform distribution does not minimize the free energy and moreover is nonlinearly unstable, while for $β<β_{\mathrm{s}}$, it is stable. We also show that there exists $β_{\mathrm{u}}$ for which uniqueness of equilibria holds for $β<β_{\mathrm{u}}$. We establish a uniform-in-time rate for entropic propagation of chaos for a range of $β<β_{\mathrm{s}}$. To our knowledge, this is the first such result for singular attractive interactions and affirmatively answers a question of Bresch et al. arXiv:2011.08022. The proof of the convergence is through the modulated free energy method, relying on a modulated logarithmic Hardy-Littlewood-Sobolev (mLHLS) inequality. Unlike Bresch et al., we show that such an inequality holds without truncation of the potential -- the avoidance of the truncation being essential to a uniform-in-time result -- for sufficiently small $β$ and provide a counterexample to the mLHLS inequality when $β>β_{\mathrm{s}}$. As a byproduct, we show that it is impossible to have a uniform-in-time rate of propagation of chaos if $β>β_{\mathrm{s}}$.

math.AP↗

arXiv2023

Modulated logarithmic Sobolev inequalities and generation of chaos

We consider mean-field limits for overdamped Langevin dynamics of $N$ particles with possibly singular interactions. It has been shown that a modulated free energy method can be used to prove the mean-field convergence or propagation of chaos for a certain class of interactions, including Riesz kernels. We show here that generation of chaos, i.e. exponential-in-time convergence to a tensorized (or iid) state starting from a nontensorized one, can be deduced from the modulated free energy method provided a uniform-in-$N$ "modulated logarithmic Sobolev inequality" holds. Proving such an inequality is a question of independent interest, which is generally difficult. As an illustration, we show that uniform modulated logarithmic Sobolev inequalities can be proven for a class of situations in one dimension.

math.PR↗

arXiv2023

Sharp uniform-in-time mean-field convergence for singular periodic Riesz flows

We consider conservative and gradient flows for $N$-particle Riesz energies with mean-field scaling on the torus $\mathbb{T}^d$, for $d\geq 1$, and with thermal noise of McKean-Vlasov type. We prove global well-posedness and relaxation to equilibrium rates for the limiting PDE. Combining these relaxation rates with the modulated free energy of Bresch et al. and recent sharp functional inequalities of the last two named authors for variations of Riesz modulated energies along a transport, we prove uniform-in-time mean-field convergence in the gradient case with a rate which is sharp for the modulated energy pseudo-distance. For gradient dynamics, this completes in the periodic case the range $d-2\leq s<d$ not addressed by previous work of the second two authors. We also combine our relaxation estimates with the relative entropy approach of Jabin and Wang for so-called $\dot{W}^{-1,\infty}$ kernels, giving a proof of uniform-in-time propagation of chaos alternative to Guillin et al.

math.AP↗

arXiv2022

Gaussian Fluctuations and Free Energy Expansion for Coulomb Gases at Any Temperature

We obtain concentration estimates for the fluctuations of Coulomb gases in any dimension and in a broad temperature regime, including very small and very large temperature regimes which may depend on the number of points. We obtain a full Central Limit Theorem for the fluctuations of linear statistics in dimension 2, valid for the first time down to microscales and for possibly very small or very large temperatures. We show that a similar CLT can also be obtained in any larger dimension conditional on a "no phase-transition" assumption, as soon as one can obtain a precise enough error rate for the expansion of the free energy -- an expansion is obtained in any dimension, but the obtained rate is so far not good enough to conclude. These CLTs can be interpreted as a convergence to the Gaussian Free Field. All the results are valid as soon as the test-function lives on a larger scale than the temperature-dependent minimal scale $ρ_β$ introduced in our previous work \cite{as}.

math-ph↗

arXiv2022

Bounded vorticity for the 3D Ginzburg-Landau model and an isoflux problem

We consider the full three-dimensional Ginzburg-Landau model of superconductivity with applied magnetic field, in the regime where the intensity of the applied field is close to the "first critical field" $H_{c_1}$ at which vortex filaments appear, and in the asymptotics of a small inverse Ginzburg-Landau parameter $\varepsilon$. This onset of vorticity is directly related to an "isoflux problem" on curves (finding a curve that maximizes the ratio of a magnetic flux by its length), whose study was initiated in [Román, C. On the First Critical Field in the Three Dimensional Ginzburg-Landau Model of Superconductivity. Commun. Math. Phys. 367, 317-349 (2019). https://doi.org/10.1007/s00220-019-03306-w] and which we continue here. By assuming a nondegeneracy condition for this isoflux problem, which we show holds at least for instance in the case of a ball, we prove that if the intensity of the applied field remains below $H_{c_1}+ C \log |\log \varepsilon|$, the total vorticity remains bounded independently of $\varepsilon$, with vortex lines concentrating near the maximizer of the isoflux problem, thus extending to the three-dimensional setting a two-dimensional result of [Sandier, E., Serfaty, S. Ginzburg-Landau minimizers near the first critical field have bounded vorticity. Cal Var 17, 17-28 (2003). https://doi.org/10.1007/s00526-002-0158-9]. We finish by showing an improved estimate on the value of $H_{c_1}$ in some specific simple geometries.

math.AP↗

arXiv2022

Mean-field limits of Riesz-type singular flows

We provide a proof of mean-field convergence of first-order dissipative or conservative dynamics of particles with Riesz-type singular interaction (the model interaction is an inverse power $s$ of the distance for any $0<s<d$) when assuming a certain regularity of the solutions to the limiting evolution equations. It relies on a modulated-energy approach, as introduced in previous works where it was restricted to the Coulomb and super-Coulombic cases. The method is also capable of incorporating multiplicative noise of transport type into the dynamics. It relies in extending functional inequalities of arXiv:1803.08345, arXiv:2011.12180, arXiv:2003.11704 to more general interactions, via a new, robust proof that exploits a certain commutator structure.

math.AP↗

arXiv2021

Compactness and structure of zero-states for unoriented Aviles-Giga functionals

Motivated by some models of pattern formation involving an unoriented director field in the plane, we study a family of unoriented counterparts to the Aviles-Giga functional. We introduce a nonlinear curl operator for such unoriented vector fields as well as a family of even entropies which we call "trigonometric entropies". Using these tools we show two main theorems which parallel some results in the literature on the classical Aviles-Giga energy. The first is a compactness result for sequences of configurations with uniformly bounded energies. The second is a complete characterization of zero-states, that is, the limit configurations when the energies go to 0. These are Lipschitz continuous away from a locally finite set of points, near which they form either a vortex pattern or a disclination with degree 1/2. The proof is based on a combination of regularity theory together with techniques coming from the study of the Ginzburg-Landau energy. Our methods provide alternative proofs in the classical Aviles-Giga context.

math.AP↗