  We revisit the Ornstein-Uhlenbeck (OU) process as the fundamental
mathematical description of linear irreversible phenomena, with fluctuations,
near an equilibrium. By identifying the underlying circulating dynamics in a
stationary process as the natural generalization of classical conservative
mechanics, a bridge between a family of OU processes with equilibrium
fluctuations and thermodynamics is established through the celebrated Helmholtz
theorem. The Helmholtz theorem provides an emergent macroscopic "equation of
state" of the entire system, which exhibits a universal ideal thermodynamic
behavior. Fluctuating macroscopic quantities are studied from the stochastic
thermodynamic point of view and a non-equilibrium work relation is obtained in
the macroscopic picture, which may facilitate experimental study and
application of the equalities due to Jarzynski, Crooks, and Hatano and Sasa.
