A minimal model of dynamical phase transition

Abstract

We calculate the large deviation functions characterizing the long-time fluctuations of the occupation of drifted Brownian motion and show that these functions have non-analytic points. This provides the first example of dynamical phase transition that appears in a simple, homogeneous Markov process without an additional low-noise, large-volume or hydrodynamic scaling limit.Keywords: Brownian motion, large deviations, dynamical phase transitions Dynamical phase transitions are phase transitions in the fluctuations of physical observables that give rise, similarly to equilibrium phase transitions, to non-analytic points in generalized potentials or large deviation functions characterizing the likelihood of fluctuations. Such transitions are known to arise in many physical systems and scaling limits, including the low-noise limit of diffusion equations modeling noise-perturbed dynamical systems [1-6], thermodynamic-like limits of chaotic systems [7], and the hydrodynamic limit of interacting particles systems, which corresponds, via the macroscopic fluctuation theory [8][9][10], to a low-noise limit [11][12][13][14][15].Similar transitions also appear in the long-time fluctuations of time-integrated quantities, such as Lyapunov exponents [16][17][18], dynamical activities [19][20][21][22][23][24][25][26][27][28][29][30], currents [31][32][33][34][35][36][37][38][39], and the entropy production [40][41][42], which now play a central role in studies of nonequilibrium processes. In this case, the large deviation functions are found to be smooth in the long-time limit; singularities start to appear only when a low-noise or a scaling (hydrodynamic, particle or mean-field) limit is taken in addition to the long-time limit [43], leading many to believe and claim that these additional limits are necessary for dynamical phase transitions to appear in Markov processes.The study in [44] of non-homogeneous random walks that are reset in time has just shown, by a mapping to DNA models, that this is not always true -a dynamical phase transition can occur in the long-time limit without a low-noise or scaling limit. Here, we present a simple, minimal model based on a one-dimensional and homogeneous diffusion process that confirms this. The process has no reset and so also shows that random resetting is not needed for such transitions to arise.The process that we consider is the drifted Brownian motion, defined aswhere µ is the drift, σ is the noise power, and W t is the simple, one-dimensional Brownian motion (BM) started at W 0 = 0. Physically, X t may represent the position of a small particle evolving in a fluid moving at slow, constant velocity or in a static fluid but with additional forces (created, e.g., by laser tweezers [45] or an AC trap [46]) that pull the particle at constant velocity. By analogy with electrical circuits perturbed by Nyquist noise [47], X t can also represent the charge dissipated in a resistor upon the application of a ramped voltage. In both cases, we are interested in the fluctuations of the fracti...

Dynamical phase transition in drifted Brownian motion

“…as ρ → 1. This confirms that the probability that dBM stays in [−a, a] for a time T (or, equivalently, that its exit time from [−a, a] is greater than T ) scales asymptotically as e −T I (1) where…”

“…We continue in this paper our study of the occupation fluctuations of drifted Brownian motion (dBM) [1]. The motivation for studying this model is that it shows a dynamical phase transition (DPT), that is, a sudden change in the way that fluctuations are created in the long-time limit, leading to singularities in large deviation functions, the nonequilibrium analogs of thermodynamic potentials [2].…”

“…We define in the next sections the dBM model and present a complete account of its occupation large deviations and of its DPT, first announced in [1], which is fundamentally related to the non-Hermitian nature of the spectral problem underlying long-time large deviations [23]. We complement these results by studying in detail the so-called driven or auxiliary process [24][25][26][27], which explains in our case how fluctuations of the occupation are created in the long-time limit [28], and by presenting simulation results that confirm the confinement and escape regimes.…”

Effective driven dynamics for one-dimensional conditioned Langevin processes in the weak-noise limit

“…The appearance of singularities (as non-differentiabilities) is known to occur in the quasi-potential of non-equilibrium dynamics in the weak-noise limit [24,25,26], when considering the large-deviation scaling of the steady-state distribution. In the context of time-integrated observables, the occurrence of another type of singularities has been reported during the last decade in varied systems [27,10,28,29,30,12,31], and corresponds to the type of dynamical phase transitions we are interested in in this paper. These describe how the trajectories that lead to an atypical value of the time-integrated observable can change from one class to another when varying the value of this observable.…”

Finite-size and finite-time effects in large deviation functions near dynamical symmetry breaking transitions

“…In particular, we are able to characterize in detail the different finite-T scaling regimes. We 1 See [50,60,61] for exceptions. 2 As we will see, the LDF in the infinite-time limit is given by the maximum eigenvalue of a well-defined operator, while the finite-time behavior of the LDF involves more eigenvalues.…”