Bosonic and fermionic holographic fluctuation and dissipation

In this paper we investigate some general aspects of fluctuation and dissipation in the holographic scenario at zero and finite density. We model this situation with a probe string in a diagonal metric representing a black brane. The string stretches from the black brane to a probe brane, thus simulating a stochastic driven particle. In this scenario, we compute the admittance, the diffusion coefficient, the correlation functions, and the regularized mean square displacement, for bosons and fermions, all from the metric components. We check these calculations with the fluctuation-dissipation theorem. Further, we show that at finite temperature and density, the mean square displacement in the limit of short times reproduces the usual quadratic (ballistic) behavior, for bosons and fermions. For large times, we find ultraslow diffusive processes in various cases, except for bosons at the zero chemical potential. We apply this general analysis in two different models: hyperscaling violation at finite temperature and a charged dilatonic anti-de Sitter black hole, both for bosons and fermions. This is important because we found the fermionic diffusion in systems that allow the appearance of Fermi surfaces and Fermi liquids.