Separation of Time Scales in Weakly Interacting Diffusions

Arch. Rational Mech. Anal. (2026) 250:33 Digital Object Identifier (DOI) https://doi.org/10.1007/s00205-026-02180-w Separation of Time Scales in Weakly Interacting Diffusions Zachary P. Adams, Maximilian Engel & Rishabh S. Gvalani Communicated by M. Hairer Abstract We study metastable behaviour in systems of weakly interacting Brownian particles with localised, attractive potentials which are smooth and globally bounded. In this particular setting, numerical evidence suggests that the particles converge on a short time scale to a “droplet state” which is metastable, i.e. persists on a much longer time scale than the time scale of convergence, before eventually diffusing to 0. In this article, we provide rigorous evidence and a quantitative characterisation of this separation of time scales. Working at the level of the empirical measure, we show that (after quotienting out the motion of the centre of mass) the rate of convergence to the quasi-stationary distribution, which corresponds with the droplet state, is O(1) as the inverse temperature β → ∞. Meanwhile the rate of leakage away from its centre of mass is O(e−β ). Furthermore, the quasi-stationary distribution is 1 localised on a length scale of order O(β − 2 ). Our proofs rely on understanding the large β-asymptotics of the first two eigenvalues of the generator, which we study using techniques from semiclassical analysis. We thus provide a partial answer to a question posed by Carrillo et al. (see Aggregation–diffusion equations: dynamics, asymptotics, and singular limits. Active particles. Advances in theory, models, and applications, modeling and simulation in science, engineering and technology, vol 2, pp 65–108, Birkhäuser/Springer, Cham, 2019, Section 3.2.2) in the microscopic setting. 1. Introduction Consider the following system of weakly interacting diffusions dX ti = − N    1  j ∇W X ti − X t dt + 2β −1 dBti , N i=1 (1.1) 33 Page 2 of 36 Arch. Rational Mech. Anal. (2026) 250:33 N , N ∈ N, represent the positions of N exchangeable particles (or where (X ti )i=1 agents or spins), taking values in some smooth manifold , W is a smooth even N are N ininteraction potential with well-behaved growth properties, and (Bti )i=1 dependent Brownian motions. Depending on the choice of the interaction potential W , the system in (1.1) can exhibit a wide variety of interesting dynamical phenomena. We are particularly interested in the case in which W is attractive (i.e. ∇W (x)·x  0), globally bounded, and has smooth and globally bounded derivatives of all orders. In this situation the pairwise attraction mediated by W competes with the diffusive behaviour produced by the independent Brownian motions. This competition is most easily observed at the level of the thermodynamic limit N → ∞ when  is bounded. Consider the empirical measure μtN of the system (1.1), which is defined as μtN := N 1  δXi . t N i=1 Then, assuming W is sufficiently well-behaved, it is well-known (see, for example, [45]) that we have the estimate    1 sup E d22 μt , μtN  , N t∈[0,T ] for any T < ∞, where d2 (·, ·) is the 2-Wasserstein distance and μt is the unique distributional solution of the nonlocal parabolic PDE ∂t μt = β −1 μt + ∇ · (μt ∇W ∗ μt ) , (1.2) often referred to as the McKean–Vlasov equation. The above PDE can exhibit the phenomenon of phase transitions (see [14,18]): for β sufficiently small, Eq. (1.2) has a unique steady state (in the space of probability measures) while for β sufficiently large, Eq. (1.2) possesses multiple steady states. This switch between one and multiple steady states for (1.2) is caused exactly by the aforementioned competition between the diffusive behaviour produced by the Laplacian, which dominates in the β  1 regime and the tendency to aggregate due to the nonlocal drift, which prevails in the β  1 regime. The Setting: Bounded Potential on Unbounded Domain In this article, we are interested in the setting of  = Rd . Since W and all its derivatives are assumed to be bounded, solutions of (1.2) are not uniformly tight and the equation has no steady states which are probability measures. A proof of this can be found in [13, Theorem 3.1]. Even though (1.2) has no steady states, the competition between the diffusive and attractive terms manifests itself in a different manner. Numerical experiments (see [12, Figure 8]) suggest that if the potential W is sufficiently localised, β  1, and the initial datum is well-prepared, then solutions of (1.2) appear to converge to a localised droplet state and stay close to it for a very long time before eventually Arch. Rational Mech. Anal. (2026) 250:33 Page 3 of 36 33 Fig. 1. A schematic depiction of dynamical metastability: the coloured paths represent trajectories of a dynamical system which can be divided into a regime of fast convergence towards the slow submanifold M (red), and slow motion along M (blue) converging to 0, as expected. Thus, there is a separation of time scales between convergence to the droplet state and the eventual escape of mass to infinity and convergence to 0. This separation is an instance of what is commonly referred to as dynamical metastability, wherein solutions of a dissipative dynamical system converge on a fast time scale to a submanifold of its state space, along which the time evolution is slow. We refer the reader to Fig. 1, where we provide a simple schematic sketch of this phenomenon, which has been studied in the context of other equations as well, such as reaction–diffusion equations including the Allen–Cahn equation (see [3,10,17,28,36]) or the Cahn–Hilliard equation (see [8,26,44]). We also point the reader to [12, Section 3.2.2] for a more detailed discussion of this phenomenon in the context of (1.2). Absence of Invariant Probability Measure for The Particle System We are studying this separation of time scales as it shows up in the corresponding particle system. In analogy to the absence of steady states for the thermodynamic limit, the particle system (1.1) does not have an invariant probability measure under our assumptions. To see this, consider the Fokker–Planck equation associated to the law ρt (x) dx of (1.1), ∂t ρt = β −1 ρt + ∇ · (ρt ∇ H N ), 33 Page 4 of 36 Arch. Rational Mech. Anal. (2026) 250:33 where the Hamiltonian H N : (Rd ) N → R is given by H N (x) := N 1  W (xi − x j ). 2N i, j=1 We define f t := ρt eβ HN and observe that it solves the following equation ∂t f t = β −1  f t − ∇ H N · ∇ f t = β −1 eβ HN ∇ · (e−β HN ∇ f t ). Multiplying by f t and integrating by parts against the measure e−β HN (x) dx, we obtain the estimate   d | f t |2 e−β HN (x) dx = −2β −1 |∇ f t |2 e−β HN (x) dx. dt (Rd ) N (Rd ) N We now use the fact that W is globally bounded by some constant K < ∞ along with the Nash inequality [35] to assert that there exists a constant C = C(N , d, K , β) such that N d+2   Nd d | f t |2 e−β HN (x) dx  −C | f t |2 e−β HN (x) dx . dt (Rd ) N (Rd ) N Appl (...truncated)