Memory as Bayesian inference: On the connection between memory and the second law of thermodynamics

A recent theoretical paper by Leonard Mlodinow and Todd Brun suggests that the functioning of physical records or memories is never accompanied by a decrease in entropy, meaning that all memories align with the thermodynamic arrow of time. In this thesis, we characterize a class of physical systems as memories in terms of inferences that can be made about the state of the world, given certain information about these systems. Tools from Bayesian probability theory are used to quantify the informativeness and reliability associated with such inferences. Based on consideration of two model systems, one classical and one quantum, we argue in favor of Mlodinow and Brun\u27s claim that the functioning of memory systems is conditioned by thermodynamic constraints. For the classical model, we show that a memory which operates against the thermodynamic arrow, and thus remembers a relatively high-entropy state, is much less informative than a similar memory which aligns with the thermodynamic arrow. Our analysis of the quantum model, expressed in the density matrix formalism of quantum mechanics, allows us to consider the inferences that can be made when a quantum system is coupled to a simple type of quantum memory system. We ultimately show that these inferences can be expressed in terms of a probabilistic matrix completion problem

Energy absorption and diffusion in chaotic systems under rapid periodic driving

In this thesis, we study energy absorption in classical chaotic, ergodic systems subject to rapid periodic driving, and in related systems. Under a rapid periodic drive, we find that the energy evolution of chaotic systems appears as a random walk in energy space, which can be described as a process of energy diffusion. We characterize this process, and show that it generally predicts three stages of energy evolution: Initial relaxation to a prethermal state, followed by slow evolution of the system’s energy probability distribution in accordance with a Fokker-Planck equation, followed by either unbounded energy absorption or relaxation to an infinite temperature state. We then study the energy diffusion model in detail in driven billiard systems specifically; in particular, we obtain numerical results which corroborate the energy diffusion description for a specific choice of billiard. This is followed by an analysis of energy diffusion in one-dimensional oscillator systems subject to weak, correlated noise. Finally, we begin to investigate energy absorption in periodically driven quantum chaotic systems, i.e., quantum systems with a classical chaotic analogue. We invoke tools from Floquet theory and random matrix theory to investigate whether the classical energy diffusion framework can be applied to quantum systems, and under what conditions. We conclude with a discussion of potential models for energy absorption in quantum chaotic systems, and with an overview of open questions and directions for future work