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Non-equilibrium Statistical Mechanics and Curvature


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Major Goals: The motivation for the work has been to obtain Landauer-type limits for information processing by real-life (engineering or biological) systems. Additional motivation from an engineering standpoint is the desire to design optimal and robust control schemes to effect transition of thermodynamic systems (possibly of interacting particles) between target states. The project aimed to quantify the cost of transitions in geometric terms, and obtain bounds for least amount of work required for such transitions. Accomplishments: The goal has been to obtain quantitative estimates for the dissipation in the transference of a thermodynamic systems between states, in finite time. In particular, it is of interest to determine the least amount of energy that needs to be dissipated for such a transition. The research concluded with a very precise and elegant analytical results for the sought problem. More specifically, the main conclusion is that the work dissipated during a transition of thermodynamic system between two states depends on the control protocol (e.g., time-varying controlling potential) and it is precisely the length of a curve in the infinite dimensional space of probability distributions that the state (i.e., distribution) of the thermodynamic system traverses during the transition. The derivation is quite general, for multivariable systems, and underscores the geometric nature non-equilibrium thermodynamic transitions. In particular, it shows that dissipation is measured by curve length in the so-called Wasserstein space, where the metric (Wasserstein metric) between probability distributions is the one inherited by Monge-Kantorovich optimal mass transport. The accomplished goal, in particular, provides a precise formula for the least amount of ``wasted energy'' that is required for a thermodynamic transition over a specified time window.



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