  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.
