Solution to the Fokker-Planck equation for slowly driven Brownian motion: Emergent geometry and a formula for the corresponding thermodynamic metric.

Considerable progress has recently been made with geometrical approaches to understanding and controlling small out-of-equilibrium systems, but a mathematically rigorous foundation for these methods has been lacking. Towards this end, we develop a perturbative solution to the Fokker-Planck equation for one-dimensional driven Brownian motion in the overdamped limit enabled by the spectral properties of the corresponding single-particle Schrödinger operator. The perturbation theory is in powers of the inverse characteristic timescale of variation of the fastest varying control parameter, measured in units of the system timescale, which is set by the smallest eigenvalue of the corresponding Schrödinger operator. It applies to any Brownian system for which the Schrödinger operator has a confining potential. We use the theory to rigorously derive an exact formula for a Riemannian "thermodynamic" metric in the space of control parameters of the system. We show that up to second-order terms in the perturbation theory, optimal dissipation-minimizing driving protocols minimize the length defined by this metric. We also show that a previously proposed metric is calculable from our exact formula with corrections that are exponentially suppressed in a characteristic length scale. We illustrate our formula using the two-dimensional example of a harmonic oscillator with time-dependent spring constant in a time-dependent electric field. Lastly, we demonstrate that the Riemannian geometric structure of the optimal control problem is emergent; it derives from the form of the perturbative expansion for the probability density and persists to all orders of the expansion.

Author Correction: Analog Coding in Emerging Memory Systems.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Analog Coding in Emerging Memory Systems.

Exponential growth in data generation and large-scale data science has created an unprecedented need for inexpensive, low-power, low-latency, high-density information storage. This need has motivated significant research into multi-level memory devices that are capable of storing multiple bits of information per device. The memory state of these devices is intrinsically analog. Furthermore, much of the data they will store, along with the subsequent operations on the majority of this data, are all intrinsically analog-valued. Ironically though, in the current storage paradigm, both the devices and data are quantized for use with digital systems and digital error-correcting codes. Here, we recast the storage problem as a communication problem. This then allows us to use ideas from analog coding and show, using phase change memory as a prototypical multi-level storage technology, that analog-valued emerging memory devices can achieve higher capacities when paired with analog codes. Further, we show that storing analog signals directly through joint coding can achieve low distortion with reduced coding complexity. Specifically, by jointly optimizing for signal statistics, device statistics, and a distortion metric, we demonstrate that single-symbol analog codings can perform comparably to digital codings with asymptotically large code lengths. These results show that end-to-end analog memory systems have the potential to not only reach higher storage capacities than discrete systems but also to significantly lower coding complexity, leading to faster and more energy efficient data storage.