Optimal processes in stochastic thermodynamics

This is the authors' abstract. We don't add key points for this paper.

The concept of Stochastic Thermodynamics deals with the question how to define thermodynamic quantities for nonequilibrium mesoscopic systems. Here, thermal fluctuations must be considered. The main objective of this thesis is the analysis of optimization problems in the context of Stochastic Thermodynamcis. A quite natural optimization principle for nonequilibrium processes is the requirement that a defined result should be achieved with the smallest possible amount of dissipation. For a transition between two given equilibrium states in a given finite time, this is directly linked to a process schedule which leads to a minimal (mean) work. For systems in an externally controllable time-dependent potential, the optimal protocol minimizes the mean work spent in a finite-time transition between two given equilibrium states. Surprisingly, the optimal protocol involves jumps for overdamped Langevin dynamics and even delta-type singularities for underdamped Langevin dynamics. For purely Hamiltonian and Schrödinger dynamics in harmonic potentials, we show that the optimal protocol is highly degenerate and that even in the limit of short transition times, the optimal work is given by the adiabatic work which is substantially smaller than the work for an instantaneous jump. These optimal protocols significantly improve free energy calculations via the Jarzynski equality. Most processes in the biological cell, however, cannot be described by a nonequilibrium transition between equilibrium states. Rather, these systems are permanently driven out of equilibrium, e.g. by chemical potential differences. An important model class of such dynamics are Brownian motors which transfer either chemical or thermal energy into mechanical work leading to directed transport against a load force. It is meaningful to characterize such thermodynamic machines by their performance at maximum power output rather than at maximum efficiency. The efficiency at this maximum power then is a relevant quantity. We consider a Carnot engine on the mesoscale which can be constructed by using a Brownian particle instead of the working gas and a time-dependent trapping potential instead of the confining vessel. The efficiency at maximum power output can be calculated analytically. Surprisingly, it is given by a quite universal expression which does only depend on the viscosity (or more generally on the mobility matrices) at the two temperatures. This result is independent of the shape of the potential used to trap the particle. In contrast to heat engines, molecular motors in the biological cell are mostly driven by chemical potential differences. For two simple motor models, the efficiency of the molecular motor at maximum power shows two unexpected features: (i) Both the power output and the efficiency increase when the transition state position is moved closer to the initial motor position and (ii) for appropriate parameters, the efficiency increases when the system is driven further out of equilibrium by a higher chemical potential difference. Beyond their relevance for directed transport within the cell, molecular motors are also important for the synthesis of proteins. We study the protein production rate at a given error rate for the second stage of gene expression (translation). We find that for a given error rate equivalent to the experimentally observed value, the protein production rate is not at its theoretical maximum. We therefore conjecture that other evolutionary goals or structural reasons are responsible for the observed rate constants.