Engineered swift equilibration for arbitrary geometries

Engineered swift equilibration (ESE) is a class of driving protocols that enforce an equilibrium distribution with respect to external control parameters at the beginning and end of rapid state transformations of open, classical nonequilibrium systems. ESE protocols have previously been derived and experimentally realized for Brownian particles in simple, one-dimensional, time-varying trapping potentials; one recent study considered ESE in two-dimensional Euclidean configuration space. Here we extend the ESE framework to generic, overdamped Brownian systems in arbitrary curved configuration space and illustrate our results with specific examples not amenable to previous techniques. Our approach may be used to impose the necessary dynamics to control the full temporal configurational distribution in a wide variety of experimentally realizable settings.

Figure 1

Simulations demonstrate that ESE protocols produce densities closely tracking the equilibrium distribution in configuration space corresponding to the control parameters at each moment in time. (a) Schematic diagram of the ensemble considered in subplots [(b) and (c)]. An ensemble of electric dipoles are placed in a uniform, time-varying electric field. Due to these constraints, the state of a dipole is specified by a polar coordinate $\theta$ and an azimuthal angle $\varphi$ such that the system's configuration space is a (2D) sphere. (b) The electric field is varied sigmoidally in time, as shown in the inset. In the top panel, we show the average values of $\theta$ over the entire ensemble for a system undergoing Langevin dynamics (dashed blue), ESE dynamics (solid orange), and for a Boltzmann distibution for the given electric field $E\left(t\right)$, denoted ${\rho }_{\text{eq}}$ (dotted green). The three lower panels are the corresponding full distributions for $\theta$ at all times. (c) Same as panel b for the sinusoidally varying electric field shown in the inset. (d) Schematic diagram of the ensemble considered in subplot (e). An ensemble of two coupled pendula with time-varying coupling constant. The state of a coupled pendulum is specified by the two angular coordinates of the pendula, ${\theta }_1,{\theta }_2\in [-\pi ,\pi )$, such that the full system's configuration space is a torus. (e) The probability distributions of an ensemble of coupled pairs of pendula undergoing Langevin dynamics (top) and ESE dynamics (middle) plotted against the instantaneous Boltzmann distribution (bottom) for a coupling constant that changes from zero at small times to a negative value. See supplemental material for a movie of this process [28].