Explaining Autonomy: Mesoscale Models and Many-Body Systems

Robert Batterman
University of Pittsburgh

Many-body systems often display different behaviors at different scales. The behavior of a fluid flowing past an obstacle is adequately captured by the continuum Navier-Stokes equations. But it's behavior at molecular and atomic scales are not well-modeled using continuum equations. It is remarkable that the continuum equations work at all, as they posit no structure below the continuum scale. How can one explain the (relative) autonomy of the continuum equations from the "more fundamental" lowest scale physics? I argue that to understand this autonomy, it is essential to appeal to mesoscale structures. Various parameters appearing in continuum models are best understood as coding for such mesoscale structures. I outline the nature of the explanation, connect it with homogenization theory and renormalization group arguments. In addition, I hope to hint at the role of mesoscale structures in understanding certain aspects of "learning" in deep neural networks.

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