Linear instability and statistical laws of physics

G. Casati; C. Tsallis; F. Baldovin

doi:10.1209/epl/i2005-10248-2

All issues

Volume 72 / No 3 (November 2005)

Europhys. Lett., 72 3 (2005) 355-361

Abstract

Issue		Europhys. Lett. Volume 72, Number 3, November 2005


Page(s)		355 - 361
Section		General
DOI		https://doi.org/10.1209/epl/i2005-10248-2
Published online		30 September 2005

Europhys. Lett., 72 (3), pp. 355-361 (2005)

Linear instability and statistical laws of physics

G. Casati¹^,2^,3, C. Tsallis⁴^,5 and F. Baldovin⁶^,7

¹ Center for Nonlinear and Complex Systems, Università degli Studi dell'Insubria Via Valleggio 11, 22100 Como, Italy
² CNR-INFM, Istituto Nazionale di Fisica Nucleare, Sezione di Milano - Milano, Italy
³ Department of Physics, National University of Singapore Singapore 117542, Republic of Singapore
⁴ Santa Fe Institute - 1399 Hyde Park Road, Santa Fe, NM 87501, USA
⁵ Centro Brasileiro de Pesquisas Físicas - Rua Xavier Sigaud 150 22290-180 Rio de Janeiro-RJ, Brazil
⁶ INFM-Dipartimento di Fisica, Università di Padova Via Marzolo 8, I-35131 Padova, Italy
⁷ Sezione INFN, Università di Padova - Via Marzolo 8, I-35131 Padova, Italy

Received: 29 July 2005
Accepted: 8 September 2005

Abstract

We show that a meaningful statistical description is possible in conservative and mixing systems with zero Lyapunov exponent in which the dynamical instability is only linear in time. More specifically, i) the sensitivity to initial conditions is given by $\xi =[1+(1-q)\lambda_q t]^{1/(1-q)}$ with $q=0$ ; ii) the statistical entropy $S_q=(1-\sum_i p_i^q)/(q-1)\ (S_1=-\sum_i p_i \ln p_i)$ in the infinitely fine graining limit (i.e., $W\equiv$ number of cells into which the phase space has been partitioned $\to\infty$ ), increases linearly with time only for $q=0$ ; iii) a nontrivial, q-generalized, Pesin-like identity is satisfied, namely the $\lim_{t \to \infty} \lim_{W \to \infty} S_0(t)/t=\max\{\lambda_0\}$ . These facts (which are in analogy to the usual behaviour of strongly chaotic systems with $q=1$ ), seem to open the door for a statistical description of conservative many-body nonlinear systems whose Lyapunov spectrum vanishes.

PACS: 05.45.Ac – Low-dimensional chaos / 05.45.Mt – Quantum chaos; semiclassical methods / 03.67.Lx – Quantum computation

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