Non-isothermal Brownian motion
Final Report Abstract
The theory of isothermal Brownian motion relies on fundamental principles of equilibrium classical statistical mechanics, such as the equipartition theorem embedded in the stochastic framework of the Langevin equation. In the presence of a non-isothermal solvent, it becomes questionable whether a Langevin-like description still applies and, if so, no general criterion exists which uniquely determines friction and thermal fluctuations. Starting from the fluctuating hydrodynamics of a solvent in local equilibrium, we have constructively shown that a generalized Langevin description does hold and derived the statistics of the corresponding thermal noise. The coupling between the hydrodynamic modes excited by the particle itself and the solvent temperature gradient turns the Langevin noise energy spectrum into a frequency-dependent tensor. We have derived an explicit expression for this energy spectrum and a generalized non-isothermal Fluctuation-Dissipation Theorem (FDT) in the analytically tractable case of hot Brownian motion, i.e. a constantly heated particle generating a co-moving radial temperature field. This allowed us to explain the breaking of energy equipartition and to express the energy content of the particle velocity and position in terms of effective temperatures. Furthermore, we demonstrated that the generalized FDT and Langevin equations provide a quantitative metric for violations of the FDT and the equipartition theorem in terms of a frequency-dependent noise temperature. The theory was verified by non-equilibrium molecular-dynamics simulations of a Brownian thermospectrometer. In an effort going beyond a thorough description of a single nonequilibrium Brownian particle to collective behavior, we set up a mesoscopic theory for interacting Brownian particles embedded in a non-equilibrium environment, starting from the microscopic interacting many-body theory. Using non-equilibrium linear-response theory, we characterized the effective dynamical interactions on the mesoscopic scale and the statistics of the non-equilibrium environmental noise, arising upon integrating out the fast degrees of freedom. As hallmarks of non-equilibrium, the breakdown of the fluctuation-dissipation and action-reaction relations for Brownian degrees of freedom was exemplified with two prototypical models for the environment, namely active Brownian particles and stirred colloids. In the former we supposed linear coupling between the Brownian particles and the active particle of the environment. In the latter we considered a one-dimensional system consisting of two probes under harmonic confinement embedded in a fluid of particles moving freely in a periodic domain and driven out of equilibrium by a persistent external force, and inducing a breaking of action-reaction relations. Altogether, we thus achieved a quantitative characterization of the violation of a number of classical equilibrium symmetries in non-equilibrium baths.
