A particle in equilibrium with a bath realizes worldline supersymmetry

48 pages, many PNG figures, LaTeX2e; utphys.bst for the references. Extensively rewritten, errors about the fermionic action corrected and presentation clarified and sharpened.We study the relation between the partition function of a non--relativistic particle, in one spatial dimension, that describes the equilibrium fluctuations implicitly, and the partition function of the same system, deduced from the Langevin equation, that describes the fluctuations explicitly, of a bath with additive white--noise properties using Monte Carlo simulations for computing the correlation functions that satisfy the corresponding identities. We show that both can be related to the partition function of the corresponding, maximally supersymmetric, theory with one--dimensional bosonic worldvolume, by appropriate analytic continuation, from Euclidian to real time, and that they can all describe the same physics, since the correlation functions of the observables satisfy the same identities for all systems.The supersymmetric theory provides the consistent closure for describing the fluctuations. Therefore supersymmetry is relevant at the scale in which equilibrium with the bath is meaningful. At scales when the ``true'' degrees of freedom of the bath can be resolved (e.g. atoms and molecules for the case of thermal fluctuations) the superpartners become ``hidden''. They can be, always, revealed through the identities satisfied by the correlation functions of the appropriate noise field, however. In fact, the same formalism applies whatever the ``microscopic'' origin of the fluctuations. Therefore, all consistently closed physical systems are supersymmetric--and any system that is explicitly not invariant under supersymmetric transformations, is, in fact, open and, therefore, incomplete