Distribution of Time-Averaged Observables for Weak Ergodicity Breaking

Abstract

We find a general formula for the distribution of time-averaged observables for systems modeled according to the subdiffusive continuous time random walk. For Gaussian random walks coupled to a thermal bath we recover ergodicity and Boltzmann's statistics, while for the anomalous subdiffusive case a weakly nonergodic statistical mechanical framework is constructed, which is based on Lévy's generalized central limit theorem. As an example we calculate the distribution of X, the time average of the position of the particle, for unbiased and uniformly biased particles, and show that X exhibits large fluctuations compared with the ensemble average <X>.

Pesin-Type Identity for Intermittent Dynamics with a Zero Lyaponov Exponent

“…separation of trajectories is sub-exponential, we have a strong indication that the usual Boltzmann-Gibbs statistical mechanics is not valid. Indeed it was found that certain systems with zero Lyapunov exponents break ergodicity [8,9]. Classical entropy theory is also not applicable in this case [6,7], particularly the entropy and average algorithmic complexity grow non linearly in time [7], while for a system with a positive Lyapunov exponent they increase linearly in time.…”

“…Its generalizations attracted vast research using different methods such as continuous time random walks [20,21] and periodic orbit theory [22] to name a few. Sojourn times of trajectories in the vicinity of the unstable fixed point are described by power law statistics leading to aging [23] and non Gaussian fluctuations [7], which are related to weak ergodicity breaking [8,9]. In the non ergodic phase z > 2 the density function of the map is concentrated on the unstable fixed point in the long time limit.…”

“…Classical entropy theory is also not applicable in this case [6,7], particularly the entropy and average algorithmic complexity grow non linearly in time [7], while for a system with a positive Lyapunov exponent they increase linearly in time. Still the situation is not hopeless from the point of view of statistical mechanics and one may consider distributions of time average observables [8,9,10,11,12].…”

Stationarity, ergodicity, and entropy in relativistic systems

“…Ergodicity -the equivalence of the two averaging procedures-is a commonly employed assumption in statistical mechanics [1], albeit difficult to prove for realistic systems. During the past decades, the ergodicity hypothesis was intensely examined for nonrelativistic classical [2][3][4][5] and quantum models [6][7][8]. However, much less is known about its meaning and validity in relativistic settings [9], when even more basic concepts like "stationarity" may become ambigous as time becomes relative [10][11][12].…”

The effects of distributed life cycles on the dynamics of viral infections

“…We stress that the implementation of delays into dynamical models is sometimes tricky, as memory effects can lead to the breakdown of hypothesis that are well established for Markovian processes. In fact, there is currently a very active research on this subject from the point of view of statistical mechanics (see, for example, Allegrini et al, 2003, Allegrini et al, 2007, Rebenshtok and Barkai, 2007 and the references therein).…”