# Conflict between trajectories and density description: the statistical source of disagreement

### Description

Paper discussing the statistical source of disagreement between trajectories and density description.

8 p.

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### Authors

• Allegrini, Paolo Istituto di Linguistica Computazionale del Consiglio Nazionale delle Ricerche
• Grigolini, Paolo University of North Texas; Università di Pisa; Istituto di Biofisica del Consiglio Nazionale delle Ricerche
• Palatella, Luigi Università di Pisa
• Rosa, Angelo International School for Advanced Studies

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### Description

Paper discussing the statistical source of disagreement between trajectories and density description.

8 p.

### Notes

This is the pre-print version of the paper.

Abstract: We study an idealized version of intermittent process leading the fluctuations of a stochastic dichotomous variable. It consists of an overdamped and symmetric potential well with a cusp-like minimum. The right-hand and left-hand portions of the potential corresponds to = W and = W, respectively. When the particle reaches this minimum is injected back to a different and randomly chosen position, still within the potential well. We build up the corresponding Frobenius-Perron equation and we evaluate the correlation function of the stochastic variable, called (t). We assign the potential well a form yielding (t) = (T = (t=T)), with > 0. Thanks to the symmetry of potential, there are no biases, and we limit ourselves to considering correlation functions with an even number of times, indicated for concision, by h12i, h1234i, and more, in general, by h1:::2ni. The adoption of a formal treatment, based on density, and thus of the operator driving the density time evolution, establishes a prescription for the evaluation of the correlation functions, yielding h1::2ni - h12i:::h(2n 1)2ni. We study the same dynamic problem using trajectories, and we establish that the resulting two-time correlation function coincides with that ordered by the density picture, as it should. We then study the four-times correlation function and we prove that in the non-Poisson case it departs from the density prescription, namely, from h1234i=h12ih34i. We conclude that this is the main reason why the two pictures yield two different diffusion processes, as noticed in an earlier work. [M. Bologna, P. Grigolini, B. J. West, Chem. Phys. 284, (1-2) 115-128 (2002)].

### Source

• arXiv: cond-mat/0212614

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#### UNT Scholarly Works

Materials from the UNT community's research, creative, and scholarly activities and UNT's Open Access Repository. Access to some items in this collection may be restricted.

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## When

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### Creation Date

• December 2002

### Added to The UNT Digital Library

• July 24, 2013, 1:20 p.m.

### Description Last Updated

• March 27, 2014, 3:54 p.m.

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