  The effect of finite temperature $T$ and finite strain rate $\dot\gamma$ on
the statistical physics of plastic deformations in amorphous solids made of $N$
particles is investigated. We recognize three regimes of temperature where the
statistics are qualitatively different. In the first regime the temperature is
very low, $T<T_{\rm cross}(N)$, and the strain is quasi-static. In this regime
the elasto-plastic steady state exhibits highly correlated plastic events whose
statistics are characterized by anomalous exponents. In the second regime
$T_{\rm cross}(N)<T<T_{\rm max}(\dot\gamma)$ the system-size dependence of the
stress fluctuations becomes normal, but the variance depends on the strain
rate. The physical mechanism of the cross-over is different for increasing
temperature and increasing strain rate, since the plastic events are still
dominated by the mechanical instabilities (seen as an eigenvalue of the Hessian
matrix going to zero), and the effect of temperature is only to facilitate the
transition. A third regime occurs above the second cross-over temperature
$T_{\rm max}(\dot\gamma)$ where stress fluctuations become dominated by thermal
noise. Throughout the paper we demonstrate that scaling concepts are highly
relevant for the problem at hand, and finally we present a scaling theory that
is able to collapse the data for all the values of temperatures and strain
rates, providing us with a high degree of predictability.
