Reference-Measure Freedom of the Effective Hamiltonian: From Stochastic Thermodynamics to the Macroscopic Limit

Abstract

The effective Hamiltonian (EH), also referred to as the potential of mean force or free-energy landscape, is widely used to describe equilibrium properties and slow dynamics in chemical and soft-matter systems. However, the conventional EH has been argued to be ill-defined because it does not transform as a scalar under coordinate transformations. We identify an implicit change of reference measure (gauge), rather than the coordinate transformation itself, as the origin of this apparent problem. We formulate the EH as a scalar relative to a chosen gauge and derive the intrinsic, gauge-independent Fokker--Planck equation (FPE) on the manifold of slow variables. The conventional EH and the diffusion-dependent EH introduced in a Riemannian formulation, together with their corresponding FPEs, are thus different gauge representations of the same equilibrium measure and intrinsic stochastic dynamics. We further demonstrate that the gauge choice has an operational meaning: different restraint protocols select different reference measures. In the macroscopic limit, the leading EH becomes gauge independent within a class of subleading gauges and is determined by the large-deviation rate function. In the same limit, the stochastic dynamics reduces to gauge-independent deterministic dynamics.