We use the system-plus-reservoir approach to study the dynamics of a system composed of two independent Brownian particles. We present an extension of the well-known model of a bath of oscillators which is capable of inducing an effective coupling between the two particles depending on the choice made for the spectral function of the bath oscillators. The coupling is non-linear in the variables of interest and an exponential dependence on these variables is imposed in order to guarantee the translational invariance of the model if the two particles are not subject to any external potential. The effective equations of motion for the particles are obtained by the Laplace transform method and besides recovering all the local dynamical properties for each particle we end up with an effective interaction potential between them. We explicitly analyze one of its possible forms.PACS numbers: 05.40.JcUsually the Brownian motion of a given dynamical variable is modelled by considering the system it describes -the system of interest -coupled to a thermal bath responsible for its energy loss. Assuming that any degree of freedom of this environment is only weakly perturbed by the system of interest, we can describe it as a set of independent harmonic oscillators with a coupling which is bilinear in the reservoir and system variables and endowed with a particular spectral function [1]. This model has been successfully used to describe general properties, classical or quantum mechanical, of dissipative systems with only one degree of freedom subject to arbitrary potentials [1,2,3,4,5]. Indeed, it has been extensively shown in the literature that, within the range of interest, other approaches to dealing with dissipative systems described by a single dynamical variable always furnish us with the the same results as those obtained by the bath of oscillators with a properly chosen spectral function. It is the case , for example, of the application of the collective coordinate method to describing the damped motion of quantum solitons [6] or microscopic attempts to describe more realistic systems such as the electron gas of a metallic environment [7,8,9].However, despite all its success there are certain dissipative systems for which the usual model of the bath of oscillators can be shown to be inappropriate to account for the physics we expect from them. Here it should be stressed that by the usual model we mean independent oscillators coupled bilinearly in coordinates to the system of interest.Suppose one immerses two independent particles in the same medium where each of them would separately behave as a Brownian particle. Since for each individual particle we could mimic the effect of the medium by the bath of oscillators it would be very natural to try to generalize the model to cope with the presence of those two particles. This generalization is quite straightforward and the only point where one should be a bit cautious is when introducing the well-known counter-term [1,2] in the generalized model. In order to do that in an...

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Inspired by the notion that environmental noise is in principle observable, while fundamental noise due to spontaneous localization would not be, we study the estimation of the diffusion parameter induced by wave function collapse models under continuous monitoring of the environment. We take into account finite measurement efficiencies and, in order to quantify the advantage granted by monitoring, we analyse the quantum Fisher information associated with such a diffusion parameter, identify optimal measurements in limiting cases, and assess the performance of such measurements in more realistic conditions.