  We study out of equilibrium dynamics and aging for a particle diffusing in
one dimensional environments, such as the random force Sinai model, as a toy
model for low dimensional systems. We study fluctuations of two times $(t_w,
t)$ quantities from the probability distribution $Q(z,t,t_w)$ of the relative
displacement $z = x(t) - x(t_w)$ in the limit of large waiting time $t_w \to
\infty$ using numerical and analytical techniques. We find three generic large
time regimes: (i) a quasi-equilibrium regime (finite $\tau=t-t_w$) where
$Q(z,\tau)$ satisfies a general FDT equation (ii) an asymptotic diffusion
regime for large time separation where $Q(z) dz \sim \bar{Q}[L(t)/L(t_w)]
dz/L(t)$ (iii) an intermediate ``aging'' regime for intermediate time
separation ($h(t)/h(t_w)$ finite), with $Q(z,t,t') = f(z,h(t)/h(t')) $. In the
unbiased Sinai model we find numerical evidence for regime (i) and (ii), and
for (iii) with $\bar{Q(z,t,t')} = Q_0(z) f(h(t)/h(t'))$ and $h(t) \sim \ln t$.
Since $h(t) \sim L(t)$ in Sinai's model there is a singularity in the diffusion
regime to allow for regime (iii). A directed model, related to the biased Sinai
model is solved and shows (ii) and (iii) with strong non self-averaging
properties. Similarities and differences with mean field results are discussed.
A general approach using scaling of next highest encountered barriers is
proposed to predict aging properties, $h(t)$ and $f(x)$ in landscapes with fast
growing barriers. We introduce a new exactly solvable model, with barriers and
wells, which shows clearly diffusion and aging regimes with a rich variety of
functions $h(t)$.
