Dynamic fluctuations in unfrustrated systems: random walks, scalar fields and the Kosterlitz–Thouless phase

Abstract

We study analytically the distribution of fluctuations of the quantities whose average yield the usual two-point correlation and linear response functions in three unfrustrated models: the random walk, the d dimensional scalar field and the 2d XY model. In particular we consider the time dependence of ratios between composite operators formed with these fluctuating quantities which generalize the largely studied fluctuation-dissipation ratio, allowing us to discuss the relevance of the effective temperature notion beyond linear order. The behavior of fluctuations in the aforementioned solvable cases is compared to numerical simulations of the 2d clock model with p = 6, 12 states. Contents

Probability Distributions with Singularities

“…Beyond the cases discussed before, other examples of singular behavior include the probability distribution of the work done by active particles [ 38 ], of the heat exchanged by harmonic oscillators during a quench with a thermal bath [ 39 ], of the magnetization in the spherical model [ 6 , 7 ], of the displacement of a Brownian walker with memory [ 10 ], of the work done in a quantum quench [ 12 ], and many others [ 4 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 ].…”

“…It has been found that, in many cases, exhibits a singular behavior, in that it is non-differentiable around some value (or values) of the fluctuating variable [ 3 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 ]. Such singularities have an origin akin to those observed in the thermodynamic potentials of systems at criticality.…”

Large deviations, condensation and giant response in a statistical system

“…In this case, a fluctuation N = N well above the typical value can be associated to a condensed configuration of the system [22][23][24][25][26][27][28][29][30][31][32]. This phenomenon, referred to as condensation of fluctuations, is not restricted to the particle number N but was observed for quantities as diverse as energy, exchanged heats, particles currents etc... [22][23][24][25][26][27][28][29][30][31][32]. It was shown [22,23] that in some systems condensation of fluctuations may occur because, from the mathematical point of view, asking for a specific value N = N constraints the system similarly to what a conservation law does.…”

Condensation of fluctuations in and out of equilibrium

“…In the usual contexts mentioned above, condensation is a phenomenon observed in the average behavior of the system. Instead, we shall be concerned with condensation occurring in the fluctuations, namely with condensation as a rare event [7][8][9][10][11]. The conceptual and substantial difference is that condensation of fluctuations may occurr even in systems which cannot sustain condensation on average, such as non interacting systems.…”