Non-Poisson renewal events and memory

Abstract

We study two different forms of fluctuation-dissipation processes generating anomalous relaxations to equilibrium of an initial out of equilibrium condition, the former being based on a stationary although very slow correlation function and the latter characterized by the occurrence of crucial events, namely, non-Poisson renewal events, incompatible with the stationary condition. Both forms of regression to equilibrium have the same non-exponential Mittag-Leffler structure. We analyze the single trajectories of the two processes by recording the time distances between two consecutive origin re-crossings and establishing the corresponding waiting time probability density function (PDF), ψ(t). In the former case, with no crucial events, ψ(t) is exponential and in the latter case, with crucial events, ψ(t) is an inverse power law PDF with a diverging first moment. We discuss the consequences that this result is expected to have for the correct interpretation of some anomalous relaxation processes.

Meditation-Induced Coherence and Crucial Events

“…where ξ( t ) is a correlated fluctuation adopted to make the diffusion process x identical to FBM. The rate equation thereby provides a dynamical origin for FBM (Cakir et al, 2006 ; Tuladhar et al, 2017 ). This approach is, in fact, based on a rigorous Hamiltonian model for a Generalized Langevin Equation (GLE), which can be made equivalent to the generator of Fractional Gaussian Noise (FGN) used by Kou and Xie ( 2004 ) to derive a diffusive process equivalent to FBM.…”

Diffusion Entropy vs. Multiscale and Rényi Entropy to Detect Progression of Autonomic Neuropathy

“…To understand the significance of the conclusion of this paper that only Type I 1/f -noise is left in the definite CAN condition of the processed RR data, we stress that in the original work on crucial events (Allegrini et al, 2002) the non-crucial events were assumed to be Poisson events. In a sequel (Tuladhar et al, 2017) to that early work, it was established that ǫ represents the concentration of all non-crucial events including the Type I 1/f -noise (FBM). In fact, Tuladhar et al (2017) noticed that the RR-trajectory crossings of adjacent stripes signify events characterized by exponential waiting-time PDFs not only in the Poisson case, but more generally by Gaussian processes, including FBM diffusion.…”

“…In a sequel (Tuladhar et al, 2017) to that early work, it was established that ǫ represents the concentration of all non-crucial events including the Type I 1/f -noise (FBM). In fact, Tuladhar et al (2017) noticed that the RR-trajectory crossings of adjacent stripes signify events characterized by exponential waiting-time PDFs not only in the Poisson case, but more generally by Gaussian processes, including FBM diffusion. This result is based on a theorem presented in Sinn and Keller (2011).…”

Distribution with a simple Laplace transform and its applications to non-Poissonian stochastic processes

“…with ψ n (t) defined as in equation ( 2). This function plays a central role in several stochastic processes [10][11][12][13][14][15][16][17] and it is studied in detail in section 3 (for a detailed explanation see reference [18]). The expressions containing R(t) as part of a more complicated expression can be found in subordination processes, such as in the Montroll-Weiss formalism [1,[19][20][21] or statistics of rare events in renewal theory [22].…”