We study non-interacting particles in a small subsystem which is weakly coupled to a reservoir. We show that this class of systems can be mapped into an extended form of the Friedrichs model. We derive from the Hamiltonian dynamics that the number fluctuation in a subsystem is 1/f or 1/fβ noise. We show that this effect comes from the sum of resonances.
We study the connection between Hamiltonian dynamics and irreversible, stochastic equations, such as the Langevin equation. We consider a simple model of a harmonic oscillator (Brownian particle) coupled to a field (heat bath). We introduce an invertible transformation operator Λ that brings us to a new representation where dynamics is decomposed into independent Markovian components, including Brownian motion. The effects of Gaussian white noise are obtained by the non-distributive property of Λ with respect to products of dynamical variables. In this way we obtain an exact formulation of white noise effects. Our method leads to a direct link between dynamics of Poincaré nonintegrable systems, probability and stochasticity.