A Tale of Three Approaches: Dynamical Phase Transitions for Weakly Bound Brownian Particles

We investigate a system of Brownian particles weakly bound by attractive parity-symmetric potentials that grow at large distances as \(V(x) \sim |x|^\alpha \), with \(0< \alpha < 1\). The probability density function P(x, t) at long times reaches the Boltzmann–Gibbs equilibrium state, with all moments finite. However, the system’s relaxation is not exponential, as is usual for a confining system with a well-defined equilibrium, but instead follows a stretched exponential \(e^{- \textrm{const} \, t^\nu }\) with exponent \(\nu =\alpha /(2+\alpha )\), as we announced recently in a short letter. In turn, the stretched exponential relaxation is related to large-deviation theory, which is studied from three perspectives. First, we propose a straightforward and general scaling rate-function solution for P(x, t). This rate function displays anomalous time scaling and a dynamical phase transition. Second, through the eigenfunctions of the Fokker–Planck operator, we obtain, using the WKB method, more complete solutions that reproduce the rate function approach and provide important pre-exponential corrections. Finally, we show how the alternative path-integral formalism allows us to recover the same results, with the above rate function being the solution of the classical Hamilton–Jacobi equation describing the most probable path. Properties such as parity, the role of initial conditions, and the dynamical phase transition are thoroughly studied in all three approaches.