LAUSR

By examining the deterministic limit of a general ε-dependent generator for Markovian dynamics, which includes the continuous Fokker-Planck equations and discrete chemical master equations as two special cases, the intrinsic connections among mesoscopic stochastic dynamics, deterministic ordinary differential equations or partial differential equations, large deviation rate functions, and macroscopic thermodynamic potentials are established. Our result not only solves the long-lasting question of the origin of the entropy function in classical irreversible thermodynamics, but also reveals an emergent feature that arises automatically during the deterministic limit, through its large deviation rate function, with both time-reversible dynamics equipped with a Hamiltonian function and time-irreversible dynamics equipped with an entropy function.