Prediction and Retrodiction in Statistical Mechanical Systems Out of Equilibrium

Statistical mechanics attempts to describe the collective behavior of systems composed of many small interacting parts. Systems in equilibrium are particularly simple to analyze since their behavior is stationary in time. For systems out of equilibrium, the past and future states and behavior of the system may be radically different then that of the present.

In this thesis, we analyze how information about the past or future can be best used.

We start by establishing a theoretical framework for quantifying how well the future and past of systems can be inferred, which we call the prediction and retrodiction entropies. We use our measures to quantify retrodiction and prediction for systems of diffusing particles and for the Logistic map.

We then develop a formalism that allows us to decide how to make small changes to a discrete Markov process such that it becomes more susceptible to inference - either prediction or retrodiction. We test this procedure on several Markov systems: an ensemble of random transition matrices, and a semi-classical quantum system, showing that we are in fact able to get moderate changes in predictability from small changes to the transition matrix.

Following this, we study two systems that attempt to use information about the future to improve their performance in the present. The first system is a collection of agents that attempt to predict the future and use that information to compete for a scarce resource. The second system is a Maxwell demon that has constraints on how quickly it can open and close its gate. Its uses knowledge of when particles collide with the gate it controls to schedule a sequence of gate openings and closures, and maximize its net rate of particle or energy transfer.