In this paper we present an extension of the Ghirardi-Rimini-Weber model for
the spontaneous collapse of the wavefunction. 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 new 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
new jump operators still localizes the wavefunction 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, both at the level of the trajectories in the Hilbert
space and at the level of 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.Comment: 25 pages, 2 figures; v2: close to the published versio
