Rates of convergence in long time asymptotics of an alignment model with symmetry breaking

1. Problem Statement

The paper investigates the long-time asymptotic behavior of a nonlinear Fokker-Planck equation derived from a Cucker-Smale model for flocking with Brownian noise and a self-propulsion force. The system describes the evolution of the velocity density $f(t, v)$ of individuals in $\mathbb{R}^d$:
$$\frac{\partial f}{\partial t} = D \Delta f + \nabla \cdot \left[ (\nabla \psi_\alpha(v) + v - u_f) f \right]$$
where $D$ is the noise intensity, $\alpha > 0$ is a potential parameter, and $u_f$ is the average velocity. The potential $\psi_\alpha(v) = \frac{\alpha}{4}|v|^4 - \frac{\alpha}{2}|v|^2$ creates a double-well structure in velocity space.

The Core Challenge:
The model exhibits a phase transition dependent on the noise level $D$ relative to a critical threshold $D^*$:

• Symmetric Regime ($D \ge D^*$): The system converges to a unique, isotropic stationary state $G_0$ (zero average velocity).
• Symmetry-Breaking Regime ($D < D^*$): The isotropic state $G_0$ becomes unstable. Instead, there exists a continuum of stable, polarized stationary states $\{G_u\}_{u \in S}$, where $S$ is a sphere of radius $r(D)$ representing non-zero average velocities.

Open Question:
While previous work (e.g., [11]) established that solutions converge to the set of polarized states in terms of relative entropy, it remained an open question whether the solution converges to a unique specific polarized state (i.e., does the average velocity $u_f(t)$ converge to a specific limit $u_\infty \in S$?) and what the rate of convergence is.

2. Methodology

The author employs a combination of entropy methods, spectral analysis, and geometric projection techniques to address the convergence issues.

• Free Energy and Lyapunov Structure: The analysis relies on the free energy functional $F[f]$, which acts as a Lyapunov function. The decay of $F[f]$ is linked to the Fisher information $I[f]$ via the relation $\frac{d}{dt}F[f] = -I[f]$.
• Relative Entropy and Log-Sobolev Inequalities: The convergence is measured using the relative entropy $H[f|g]$. The author utilizes generalized Log-Sobolev and Poincaré inequalities adapted to the Gibbs states $G_u$ to bound the entropy decay.
• Polyak-Łojasiewicz (PL) Inequalities: A crucial step involves analyzing the potential function $V(u) = -D \log Z(u)$ associated with the Gibbs states. The author establishes PL-type inequalities relating the difference in free energy to the gradient of the potential, distinguishing between the cases $D < D^*$, $D > D^*$, and the critical case $D = D^*$.
• Projection onto the Manifold of Stationary States: To handle the lack of a spectral gap when the average velocity $u_f(t)$ approaches the sphere $S$ tangentially, the author introduces a time-dependent projection $u^*(t)$ of the average velocity $u_f(t)$ onto the sphere $S$. The analysis is performed by linearizing the evolution operator around the projected state $G_{u^*(t)}$ rather than a fixed limit state.
• Coercivity and Nonlinear Control: The proof involves establishing coercivity estimates for the linearized operator around the projected state and controlling the nonlinear remainder terms using weighted $L^2$ norms and moment bounds.

3. Key Contributions and Results

A. Identification of a Unique Limiting State (Theorem 1.1)

The paper resolves the long-standing open question regarding the uniqueness of the limit in the symmetry-breaking regime.

• Result: If the initial data $f_{in}$ has finite free energy strictly less than that of the isotropic state ($F[f_{in}] < F[G_0]$), the average velocity $u_f(t)$ converges to a unique limit $u_\infty \in S$.
• Convergence Rate: The convergence is algebraic. For any $\beta > 0$, the quantity $t^\beta (H[f(t)|G_{u_\infty}] + |u_f(t) - u_\infty|^2)$ tends to zero as $t \to \infty$. This implies that the solution converges to a specific polarized state, not just the set of such states.

B. Exponential Rate of Convergence (Theorem 1.2)

Under stronger assumptions on the initial data (specifically, that $f_{in}$ belongs to a weighted $L^2$ space for all $u \in S$), the author proves exponential convergence.

• Result: There exists a constant $\beta > 0$ such that:
  $$H[f(t, \cdot)|G_{u_\infty}] + |u_f(t) - u_\infty|^2 = O(e^{-\beta t})$$
• Significance: This establishes that the system forgets its initial conditions exponentially fast and settles into a single, stable polarized configuration.

C. Analysis of the Critical Case ($D = D^*$)

The paper provides a novel proof for the critical case $D = D^*$, showing that the solution still converges to the unique isotropic state $G_0$, albeit with an algebraic rate of convergence (slower than exponential). This extends previous results which were limited to the non-critical case.

D. Technical Innovations

• Handling Tangential Convergence: By projecting the dynamics onto the manifold $S$ and analyzing the perturbation around the moving projected state $G_{u^*(t)}$, the author circumvents the spectral gap issue that arises when $u_f(t)$ approaches $S$ tangentially.
• Moment Estimates: The proof utilizes detailed moment lemmas to ensure that the moments of the solution remain uniformly bounded, which is essential for controlling the nonlinear terms in the linearized equations.

4. Significance

• Mathematical Biology and Physics: The results provide a rigorous mathematical foundation for understanding how collective behavior (flocking) emerges from individual stochastic interactions. It confirms that in the low-noise regime, the system spontaneously breaks symmetry and selects a specific direction of motion, rather than oscillating between different directions.
• Phase Transition Theory: The work offers a refined understanding of the phase transition in kinetic models, specifically characterizing the stability of the symmetry-broken states and the rates at which the system relaxes to equilibrium.
• Methodological Advancement: The technique of linearizing around a time-dependent projected state to recover exponential decay rates in systems with continuous symmetries (manifolds of equilibria) is a powerful tool that can be applied to other alignment models and non-linear Fokker-Planck equations with similar phase transition structures.

In summary, Surin's paper bridges the gap between qualitative convergence (convergence to a set) and quantitative convergence (convergence to a unique point with a specific rate) for a class of alignment models exhibiting symmetry breaking, providing both algebraic and exponential convergence guarantees depending on the initial data regularity.