Dannie Heineman Prize for Mathematical Physics (2021): Microscopic Origin of Macroscopic Behavior

Abstract: I will discuss the mathematical framework which statistical mechanics provides for explaining how the laws governing the microscopic constituents of matter, the atoms, determine the behavior of macroscopic (and mesoscopic) variables involving very many atoms. While some macroscopic properties, such as the pressure of a gas, can be readily understood as arising from the addition of effects produced by the individual atoms, others are clearly emergent phenomena: they have no counterpart in the behavior of individual atoms. It is the latter which are the most fascinating and most difficult to derive from the microscopic dynamics. Important examples of emergent phenomena include phase transitions and the time asymmetric behavior of macroscopic systems. The latter is encoded in the second law, stating the increase of entropy, itself an emergent property. Fortunately, many aspects of these phenomena can be understood from simplified models, beginning with the treatment of atoms as "little particles that move around in perpetual motion, attracting each other when they are a little distance apart and repelling upon being squeezed into one another" (Feynman). It is this "universality" which statistical mechanics exploits and tries to explain. Happily, there are still many open problems left for physicists and mathematicians to work upon, including the understanding of phenomena associated with living matter.