Speed and dissipation in the paths to dynamic function

Physical systems that generate work and assemble into three-dimensional structures often accomplish these dynamic functions transiently and away from steady-state. To analyze these processes, we demonstrate a path-integral formalism for stochastic paths that occur in a fixed amount of time. We illustrate the theory with four models: a clock, a ratchet, a self-assembling tetramer, and a copier. Central to the theory is an analytical expression for path probabilities that we use to determine the speed, through the mean path occurrence time, the efficiency, through the entropy flow and energy dissipation, and the effectiveness of the dynamical functions, such as the work and assembly yield. Our results confirm a recent thermodynamic uncertainty relation and establish a method for characterizing the efficiency of functioning on finite timescales.