Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 59 / 1 / Solution

A limiting argument starts with a family of microscopic models $T_N$, a specified macroscopic quantity $M_N$ and a topology or operational criterion for convergence as $N\to \infty$. It is not enough to say that a large system has many constituents: one must identify what is held fixed, what observables are compared and whether predictions converge uniformly. A thermodynamic limit, for example, normally holds density and intensive parameters fixed while size increases. Other limits remove a resolution cutoff or enlarge an observable algebra.

A limiting theory can be both deduced from microscopic models and qualitatively novel. The compatibility depends on distinguishing an exact property at the mathematical limit from the approximate collective behaviour of actual finite systems. The conceptual distinction is developed below, followed by three complementary examples rather than a claim that every form of emergence has the same mechanism.