Stochastic Transport: Impedance and a Brownian Translator

Abstract

This thesis studies transport properties of microscopic systems in a stochastic framework. Abstractly, transport arises in any system whenever it is brought out of equilibrium as a result of the system’s natural tendency to move back towards said equilibrium. The main focus of this thesis is on transport properties when the driving force which causes the non-equilibrium conditions is time-dependent. We will build on previous work done on the linearized stochastic thermodynamics of periodically driven systems to arrive at a stochastic impedance which exactly characterizes the properties of the resulting time-dependent probability currents. We will start this thesis with a brief introduction about the framework of stochastic thermodynamics and the previous work that allows us to formulate the stochastic impedance framework immediately afterwards. In the following chapters, the concept of stochastic impedance is explored through examples and potential applications. Example systems include analytical solutions, as well as numerical solutions where analytical calculations are not feasible. The analytical solutions cover some small systems and some arbitrarily large systems where parameters are chosen such that the expressions simplify enough to be solvable. Numerical solutions are only limited by available computing power and can explore more complicated systems. Potential applications come from attempting to measure the stochastic impedance as a response function of some applied driving frequency. It is shown that such measurements have the potential to detect hidden states or transitions in the system of interest. In the last part of the thesis, an excursion is made into a time-independent world where a translational analogue of the classical Brownian gyrator, dubbed the Brownian translator, is discussed. It consists of an asymmetric object embedded in a nonequilibrium gas, characterized by the velocity distribution being almost Maxwellian but with different effective temperatures for each velocity component. It is shown that this results in the object gaining an average velocity, permitting a heat flow between the two components.