James Anglin

Project A6 continues to explore the microscopic limits of thermodynamics in the practical OSCAR spirit of open-system control. In OSCAR’s final funding period we will combine A6 ideas from both previous periods by theoretically investigating the smallest possible heat engine. Heat engines with microscopic working parts have already been considered and even realized experimentally, as in OSCAR Projects A4 and B5; we will go further by making even the hot and cold reservoirs themselves microscopic. While a microscopic heat reservoir may seem to be a contradiction in terms, careful reading of textbooks on statistical mechanics shows that the only essential ingredient for a reservoir is dynamical ergodicity, which should be attained in small systems as long as they are dynamically chaotic. Testing whether chaotic ergodization can be harnessed to lift a weight is thus a practical way to probe microscopic thermodynamics. 

We will build on our previous experience in OSCAR with Bose-Hubbard models, which offer tractable quantum many-body systems, having finite-dimensional Hilbert spaces that can be kept small enough for efficient numerical solution, and which also admit an accurate mean-field approximation with a rich classical dynamics. Bose-Hubbard models with as few as three bosonic modes can be chaotic. We will therefore generalize the “Hamiltonian daemon” systems that we studied in OSCAR’s second period, by replacing the daemon’s two oscillators with two three-mode Bose-Hubbard systems, which are predicted to behave as hot and cold reservoirs. As in the simpler daemon system, we couple these two reservoirs with a kind of quantum turbine, such that bosons can hop between the two Bose-Hubbard systems, but only while giving momentum kicks to a weight. Ergodic filling of the available energy shell should then ensure a secular flow of bosons from one set of modes to the other—and steadily lift the weight. The process could have been described by Carnot, if Carnot had known about quantum chaos. 

The twelve-dimensional phase space of the model will be difficult to visualize and understand in full detail, making it hard to explore parameter space efficiently. We will therefore rely crucially on concepts of open-system control, regarding each of the Bose-Hubbard systems as a reservoir for the other, and applying a Fokker-Planck-like description of the dynamics. Our approach is thus to use open-system control to harness chaos itself as a way of starting and stabilizing a useful process, and thereby understand theoretically the smallest and simplest device which could, like the earliest steam engines, “raise water with fire”.