Dissipation Functions and Brownian Oscillators

We develop a general framework for response theory in Markovian diffusion processes governed by Fokker–Planck equations. Our formalism, based on the notion of the dissipation function, is capable of handling dynamics that are driven by an external perturbation arbitrarily far from a given reference process. Using the analytically solvable Brownian oscillator model, we derive exact response formulae for both overdamped and underdamped dynamics of harmonically bound Brownian particles. We also demonstrate that for certain observables and under suitable time scaling, the operations of model reduction and response formula extraction commute, which highlights a relevant symmetry of the adopted mathematical formalism. Notably, the time reversal invariance symmetry, which is manifested as detailed balance in stochastic processes and is often required in statistical mechanics, is not necessary in our response framework.