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To illustrate Boltzmann’s construction of an entropy function that is defined for a microstate of a macroscopic system, I will discuss the simple example of the free expansion of a one dimensional gas of point particles both interacting and non-interacting. The construction requires one to define macrostates, corresponding to macroscopic variables and we will discuss two specific macrostate constructions. Our results illustrate that concepts such as that of ergodicity and chaos are not as relevant as sometimes claimed, for observing macroscopic irreversibility and entropy increase. Rather, the notions of initial conditions, typicality, large numbers and coarse-graining are the important factors. We will also discuss results for an interacting gas and the important differences.
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