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We develop a theory of fluctuations for Brownian systems with weak long-rangeinteractions. For these systems, there exists a critical point separating ahomogeneous phase from an inhomogeneous phase. Starting from the stochasticSmoluchowski equation governing the evolution of the fluctuating density field,we determine the expression of the correlation function of the densityfluctuations around a spatially homogeneous equilibrium distribution. In thestable regime, we find that the temporal correlation function of the Fouriercomponents of the density fluctuations decays exponentially rapidly with thesame rate as the one characterizing the damping of a perturbation governed bythe mean field Smoluchowski equation (without noise). On the other hand, theamplitude of the spatial correlation function in Fourier space diverges at thecritical point $T=T_{c}$ (or at the instability threshold $k=k_{m}$) implyingthat the mean field approximation breaks down close to the critical point andthat the phase transition from the homogeneous phase to the inhomogeneous phaseoccurs sooner. By contrast, the correlations of the velocity fluctuationsremain finite at the critical point (or at the instability threshold). We giveexplicit examples for the Brownian Mean Field (BMF) model and for Brownianparticles interacting via the gravitational potential and via the attractiveYukawa potential. We also introduce a stochastic model of chemotaxis forbacterial populations generalizing the mean field Keller-Segel model by takinginto account fluctuations.

Hamiltonian and Brownian systems with long-range interactions: V. Stochastic kinetic equations and theory of fluctuations