Linear instability and statistical laws of physics

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., $W\equiv$ {\it 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.