Modulated Free Energy Functional

• Modulated free energy functional is a mathematical framework combining fluctuation energy and relative entropy to measure differences between N-body systems and their mean-field limits.
• It leverages Fourier analysis and truncation techniques to handle singular kernels like Riesz and Coulomb, enabling sharp convergence estimates with controlled error terms.
• The approach extends to both repulsive and attractive interactions, such as the Patlak–Keller–Segel model, offering rigorous bounds on convergence rates and propagation of chaos.

A modulated free energy functional is a mathematical construct that unifies multiple measures of distance between an N-body particle system and its mean-field or macroscopic limit, serving as a key analytical tool in quantifying convergence rates and propagation of chaos in nonlinear stochastic and deterministic systems. This concept, as introduced in the modern literature on mean-field limits, systematically combines two central components: (1) the modulated (or "fluctuation") energy, capturing deviations in interaction energies, and (2) the relative entropy, quantifying statistical disorder with respect to a reference (typically the limiting product) measure. The modulated free energy formalism enables sharp convergence analysis (even for systems with strongly singular kernels such as Riesz and Coulomb interactions), facilitates the extension to broader classes of interaction potentials through Fourier techniques, and creates a framework for tackling both repulsive and attractive (e.g., Patlak–Keller–Segel) interactions.

1. Conceptual Foundation and Motivation

The modulated free energy functional arises from the need to quantify how close an N-particle probability distribution $p_N$ is to a mean-field product state in both statistical (entropic) and energetic terms. Earlier works focused on either the modulated potential energy (Serfaty) or relative entropy (Jabin, Wang). The introduction of the modulated free energy combines these to provide a comprehensive, distance-like functional that encapsulates both measures: $$E_N(t) = \frac{1}{N} \int_{(T^d)^N} p_N(t,X^N) \log\left(\frac{p_N(t,X^N)}{G_{N,*}(t,X^N)}\right)dX^N,$$ where $G_N$ is the full Gibbs equilibrium (with exact interaction) and $G_{N,*}$ is a mean-field-modulated reference equilibrium. The functional serves as a Lyapunov-type functional, whose time evolution under the N-particle dynamics can be rigorously estimated.

This approach not only captures the difference in energy (via a potential energy modulation term) but also the entropy/distance between the N-body distribution and a mean-field candidate, which is crucial for kinetic regimes and viscous systems.

2. Mathematical Formulation and Structure

Explicitly, the modulated free energy is decomposed as: $$E_N = H_N(p_N \mid p^{\otimes N}) + K_N(G_N \mid G_{N,*}),$$ with:

• $$H_N(p_N \mid p^{\otimes N}) = \frac{1}{N} \int p_N \log\left(\frac{p_N}{p^{\otimes N}}\right) dX^N$$ (normalized relative entropy to the product law of the one-particle marginal $p$),
• $K_N(G_N \mid G_{N,*})$ penalizing the discrepancy in the interaction energies between the true N-body law and its factorized mean-field version (explicit formulas via integration with respect to $p_N$ and $G_N$ in the paper).

For singular or nontrivial interaction potentials $V$ (including Riesz and Coulomb cases), $G_N$ and $G_{N,*}$ are defined as: $$G_N(t,X^N) = \exp\left( \sum_{i\neq j} V(x_i - x_j) \right), \quad G_{N,*}(t,X^N) = \exp\left( -\frac{1}{\sigma}\sum_{i=1}^N (V * p)(x_i) \right),$$ with $\sigma$ a scaling parameter (small in some regimes).

The functional $E_N$ is then evolved under the appropriate particle (or mean-field) dynamics, yielding a differential inequality of the form: $$E_N(t) + \int_0^t I_N(s) ds \leq E_N(0) + \text{error terms},$$ where $I_N$ is a modulated Fisher information and the error terms can be controlled via precise analysis, leading to explicit propagation estimates.

3. Rigorous Convergence Rates and Propagation of Chaos

A notable outcome of this framework is the possibility to derive explicit rates of convergence between finite-N systems and their mean-field limits, especially for repulsive interactions of Riesz and Coulomb type, even in the presence of viscous terms. The argument proceeds by controlling the evolution of $E_N$ (using Grönwall-type inequalities) and integrating the dissipation of both entropy and modulated energy. In particular, under reasonable regularity conditions on $V$, for any fixed marginal of rank $k$,

$$W_1(p_{N,k}(t,\cdot), p(t,\cdot)^{\otimes k}) \leq C N^{-\alpha},$$

for some $C, \alpha > 0$, is established—where $W_1$ is the Wasserstein-1 distance, thus quantifying chaos propagation at a sharp algebraic rate in $N$ for all fixed times.

The analysis hinges on fine control at short distances (near the singularity of $V$), with truncation arguments and compactness leveraged through the dual energetic and entropic structure of $E_N$.

4. Fourier Transform Approach and Extension to General Kernels

A technical advancement enabled by the modulated free energy formulation is the use of Fourier analysis to generalize beyond specific potential classes. Specifically, rather than working in physical space (where singularities or non-integrabilities may obstruct the analysis), the interaction energy terms are recast in Fourier space. Key hypotheses include: $$\widehat{V}(\xi) \geq 0, \qquad |V(\xi) - V(\eta)| \leq C \text{ (etc.)}$$ This enables the main energy controlling term: $\int V(x-y) (du_N - dp)^2 dx\,dy,$ to be controlled in terms of weighted $L^2$ norms of the Fourier transform of the difference between empirical and mean-field measures. This approach streamlines the proof and extends results to more general repulsive kernels, beyond Riesz/Coulomb.

5. Application to Attractive Interactions: Patlak–Keller–Segel (PKS) System

For attractive interactions exemplified by the 2D PKS model (notably with logarithmic kernel), the modulated free energy is adapted to control deviations between the particle system and the PKS PDE. In continuous space, the functional controlling the dynamics is: $$F(p) = \int p \log p\,dx + \lambda \iint \log|x-y|\,p(x)p(y)\,dx\,dy.$$ In the N-particle context, the modulated free energy analogously combines the relative entropy and a truncated modulated interaction term to ensure the system remains below mass-thresholds where aggregation or blow-up may occur. Large deviation and entropy inequalities (e.g., logarithmic Hardy-Littlewood-Sobolev) assist in capturing the necessary concentration behaviors.

6. Core Formulas and Quantitative Characterization

Key formulas from the modulated free energy framework include: $$\begin{align*} E_N(t) &= \frac{1}{N} \int p_N(t,X^N) \log \frac{p_N(t,X^N)}{G_{N,*}(t,X^N)} dX^N \ H_N(p_N | p^{\otimes N}) &= \frac{1}{N} \int p_N \log \frac{p_N}{p^{\otimes N}} dX^N \ G_N(t,X^N) &= \exp\left( \sum_{i\neq j} V(x_i-x_j) \right), \quad G_{N,*}(t,X^N) = \exp\left( -\frac{1}{\sigma} \sum_{i=1}^N (V*p)(x_i) \right) \ \text{Convergence Rate:} &\quad W_1(p_{N,k}(t, \cdot), p(t, \cdot)^{\otimes k}) \leq C (E_N(0) + \text{err}) e^{Ct} N^{-\alpha} \end{align*}$$ and the primary control of Fourier norms: $$\int |\widehat{V}(\xi)|\, |\mathcal{F}[u_N - u]|^2 d\xi \leq \text{(modulated energy terms)}.$$

7. Synthesis and Broader Significance

By synthesizing the modulated potential energy with relative entropy, the modulated free energy functional has become an essential functional-analytic object for understanding the quantitative mean-field convergence of large interacting systems. It allows precise, general, and robust results in regimes with both strong repulsion and attraction, including viscous and non-Markovian cases. Its flexibility in accommodating Fourier-based arguments and singular potentials demonstrates its power relative to earlier methods based solely on energy or entropy. The blend of tools from statistical mechanics, information theory, and harmonic analysis ensures applicability across diverse models, ranging from particle approximations of PDEs (e.g., PKS, McKean–Vlasov) to high-dimensional stochastic systems.

The modulated free energy functional is now central in the toolkit for rigorous derivation of macroscopic limits, quantifying rates of propagation/generation of chaos, and understanding how microscopic randomness translates into deterministic continuum evolution in large systems (Bresch et al., 2019).