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NOTE-146: Phase transitions, logarithmic Sobolev inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions

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PaperLIT-310 — Phase transitions, logarithmic Sobolev inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions

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Contribution

The paper establishes precise links between (i) the non-degeneracy of the log-Sobolev inequality (LSI) constant of the N-particle Gibbs measure as N→∞, (ii) absence of phase transitions for the McKean-Vlasov mean-field limit, and (iii) uniform-in-time propagation of chaos plus Gaussianity of equilibrium fluctuations. It provides gradient-flow / Wasserstein-calculus proofs of a generalized Talagrand inequality and quantitative propagation of chaos, and proves uniform LSI at high temperature for weakly-interacting, possibly non-convex potentials via the two-scale approach.

Key insight

Both the N-particle Fokker-Planck equation and its McKean-Vlasov mean-field limit are 2-Wasserstein gradient flows of related free energies (E^N and E^MF), and E^N Gamma-converges to E^MF. Consequently, the large-N behavior of the N-particle LSI constant λ_LS^N governs all important features of the mean-field system: a non-minimising critical point of E^MF forces λ_LS^N→0, while λ_LS^N bounded away from 0 yields a unique steady state, uniform-in- time propagation of chaos (with O(1/√N) Wasserstein distance) and Gaussian equilibrium fluctuations. Phase transitions of the mean-field PDE and degeneracy of the uniform LSI are thus conjecturally equivalent.

Assumptions

  • V lower semicontinuous, bounded below, K_V-convex (semi-convex, possibly non-convex), with growth V(x) ≥ |x|^δ at infinity
  • W is K_W-convex on R^{2d} (semi-convex pair potential)
  • Doubling-type bound: ||∇_1 W(x,y)|| ≤ C(1+|W(x,y)|+V(x)+V(y))
  • Exchangeable IID initial data ρ_in^{⊗N}; symmetric N-particle laws (IID across workers does NOT apply—this is a mean-field physics model)
  • Far-field convexity D^2 H^N ≥ λ I for ||x||>R (used for LSI)
  • For fluctuation theorem: Ω = T^d, V and W smooth, liminf λ_LS^N > 0

Key results

  • Theorem 3.2 (generalised Talagrand). E^N[ρ^N] - E^N[M_N] ≥ (λ_LS^N/2) d̄_2^2(ρ^N, M_N); analogously E^MF[ρ] - inf E^MF ≥ (λ_LS^∞/2) d_2^2(ρ, K) Holds when: Holds for gradient-flow-regular energies; proof via Otto calculus on (P(Ω), d_2)
  • Theorem 3.3. limsup_{N→∞} λ_LS^N ≤ λ_LS^∞ Holds when: Under Assumptions 2.1–2.2; mean-field LSI upper-bounds the large-N particle LSI
  • Theorem 3.6 (Gibbs approximation). d̄_2^2(ρ_β^{⊗N}, M_N) ≤ (2/λ_LS^N) Ē(ρ_β^{⊗N}||M_N) ≤ C/N Holds when: Requires λ_LS^N bounded away from 0 uniformly in N
  • Theorem 3.7 (uniform-in-time propagation of chaos). sup_{t≥0} d̄_2(ρ^N(t), ρ(t)^{⊗N}) ≤ C N^{-θ} for some θ>0 Holds when: Requires uniform LSI (λ_LS^N bounded below) and E^MF[ρ_in] < ∞; θ not sharp when K_V+K_W(1-1/N)<0
  • Theorem 3.11 (equilibrium fluctuations). η^N = √N(μ^N - ρ_β) converges in law to Gaussian field η^∞ solving linear SPDE with covariance Q_G Holds when: Ω = T^d, V,W smooth, liminf λ_LS^N > 0; convergence in C([0,T]; H^{-m}) for m > d/2+3
  • Theorem 3.14 (uniform LSI at high temperature). λ_LS^N ≥ c > 0 uniformly in N Holds when: Compact Ω, or bounded W with small ||W||∞ and ||D^2{xy} W||_∞ relative to temperature (β small); proved via two-scale LSI
  • Classical propagation of chaos (Theorem 10.1). d̄_2(ρ^N(t), ρ(t)^{⊗N}) ≤ [(1-e^{-Kt/2})/K] · C/√N where K = K_V+K_W(1-1/N) Holds when: Constants blow up exponentially if K<0 (non-convex)

Claims

idclaimstrengthsupport
C1Non-degenerate uniform LSI for the N-particle Gibbs measure implies uniform-in-time propagation of chaos at rate N^{-θ}.strongTheorem 3.7
C2Phase transition (non-unique mean-field critical points) forces the N-particle LSI constant to degenerate as N→∞.strongTheorem 3.4 and the Talagrand inequality structure
C3Under uniform LSI, equilibrium fluctuations around ρ_β are Gaussian with explicit covariance operator.strongTheorem 3.11
C4At high temperature (or small interaction) with bounded W, the uniform LSI holds.strongTheorem 3.14 via two-scale criterion of Otto-Reznikoff
C5lim_{N→∞} λ_LS^N = λ_LS^∞ (mean-field dissipation constant).weakConjecture 1; only upper bound proved

Method

Wasserstein gradient flow / Otto calculus approach to N-particle LSI.

Treat both the N-particle Fokker-Planck equation and the McKean-Vlasov PDE as 2-Wasserstein gradient flows of E^N and E^MF respectively. Use Gamma- convergence of E^N → E^MF (via de Finetti / Hewitt-Savage) to transfer properties between levels. Prove Talagrand-type inequalities using Otto calculus on geodesics in (P(Ω), d_2). For uniform LSI, apply the two-scale Otto-Reznikoff criterion to marginal and conditional measures of M_N. For fluctuations, linearize around ρ_β and identify the limiting Ornstein- Uhlenbeck SPDE.

  • Gamma-convergence of free energies E^N → E
  • de Finetti/Hewitt-Savage limit for exchangeable measures
  • Scaled 2-Wasserstein distance d̄_2 = d_2/√N
  • Two-scale LSI criterion (marginal + conditional)
  • HWI-type inequality via Otto calculus
  • Linearised McKean-Vlasov operator L_{ρ_β}

Concepts

  • Logarithmic Sobolev Inequality (LSI) constant λ_LS^N — Largest constant such that β^{-1} Ī(ρ^N||M_N) ≥ λ_LS^N Ē(ρ^N||M_N) for all ρ^N on Ω^N, where Ē is the scaled relative entropy and Ī the scaled relative Fisher information.
  • Mean-field LSI constant λ_LS^∞ — inf over non-minimizers of D(ρ)/(E^MF[ρ] - min E^MF), where D(ρ) is the mean-field dissipation.
  • Scaled 2-Wasserstein distance — d̄_2(ρ^N_1, ρ^N_2) = inf (E[||X-Y||^2]/N)^{1/2}; makes ρ → ρ^{⊗N} an isometry.
  • Phase transition (flat case) — On T^d with V≡0, W(x-y), existence of β_c>0 beyond which local stability (Property A), self-consistency uniqueness (Property B), or dissipation inequality (Property C) fails.
  • Gibbs measure M_N — M_N = Z_N^{-1} exp(-β H_N) dx, the invariant measure of the N-particle Langevin system.
  • Equilibrium fluctuation field — η^N(t) = √N(N^{-1} Σ δ_{X_t^i} - ρ_β), an H^{-s}-valued process.

Connections

Builds on.

  • Ambrosio-Gigli-Savaré, Gradient flows in metric spaces (2008) — Foundational Wasserstein gradient flow framework used throughout.
  • Otto-Reznikoff, A new criterion for the LSI (2007) — Two-scale LSI criterion used to prove Theorem 3.14.
  • Hauray-Mischler, On Kac’s chaos (2014) — Provides the gradient-flow/Gamma- convergence approach to propagation of chaos.
  • Carrillo-Gvalani-Pavliotis-Schlichting, Long-time behaviour and phase transitions for McKean-Vlasov on the torus — Phase transition classification and linearisation machinery reused here.
  • Bauerschmidt-Bodineau, Very simple proof of LSI for high-temperature spin systems (2019) — Yields sharp uniform LSI up to β_c for the Brownian mean- field / XY model.
  • Malrieu, Convergence to equilibrium for granular media (2003) — Classical convex-case uniform-in-time propagation of chaos extended here to non- convex regime.

Recommendations

  • R1 — When analyzing convergence of interacting-particle / SGD-like systems with pair-wise interactions, check whether the N-particle Gibbs measure satisfies a log-Sobolev inequality uniformly in N — it controls mixing, equilibrium accuracy, and fluctuation behaviour. [not filed as a practice: advice on proof technique rather than on training] Topic: log-Sobolev diagnostics · Strength: strong · When: Applies to reversible (Langevin/SDE) dynamics with a Gibbs stationary measure.
  • R2 —

Bearing on the record

Uniform-in-N log-Sobolev inequalities, which is what makes propagation of chaos hold uniformly in time. Filed for the mean-field line; no practice here rests on it.

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