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 $e^{- \mathrm{const} \, t^\nu}$ with exponent $\nu=\alpha/(2+\alpha)$. This problem is studied from three perspectives. First, we propose a straightforward and general scaling rate-function solution for $P(x,t)$. This rate-function, which is an important tool from large deviation theory, also 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. 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.