In this paper, we present an extension of the Ghirardi-Rimini-Weber
model for the spontaneous collapse of the wave function. Through the
inclusion of dissipation, we avoid the divergence of the energy on the
long-time scale, which affects the original model. In particular, we
define jump operators, which depend on the momentum of the system and
lead to an exponential relaxation of the energy to a finite value. The
finite asymptotic energy is naturally associated to a collapse noise
with a finite temperature, which is a basic realistic feature of our
extended model. Remarkably, even in the presence of a low-temperature
noise, the collapse model is effective. The action of the jump
operators still localizes the wave function and the relevance of the
localization increases with the size of the system, according to the
so-called amplification mechanism, which guarantees a unified
description of the evolution of microscopic and macroscopic systems. We
study in detail the features of our model, at the level of both the
trajectories in the Hilbert space and the master equation for the
average state of the system. In addition, we show that the dissipative
Ghirardi-Rimini-Weber model, as well as the original one, can be fully
characterized in a compact way by means of a proper stochastic
differential equation.