  Penrose has suggested that large fluctuations of the gravitational energy of
quantum systems, resulting from fluctuations of its density in space, may
induce a quantum collapse mechanism \cite{Penrose-1996}, but he did not propose
a precise dynamics for this process. We use the GBC (Gravitational Bohmian
Collapse) model \cite{GBC}, which provides such a dynamics. The effects of
collapse in dilute quantum systems are investigated, both in ordinary 3D space
and in configuration space.
  We first discuss how a single result appears during a quantum measurement.
The GBC model predicts a continuous but very fast evolution of the state vector
that, at the end of the measurement, reproduces the von Neumann projection
postulate. This ensures that the model remains compatible with the relativistic
nosignaling constraint. In the absence of any measurement, we study the
spontaneous effects of the GBC process, which depend on the quantum correlation
function of observables with the spatial density operator. If the selected
observable is the local current of the density fluid, we show that the collapse
term leads to modifications of the Newton force, in a scalar or tensor form.
