Covariant nonequilibrium thermodynamics from Ito-Langevin dynamics

Using the recently developed covariant Ito-Langevin dynamics, we develop a nonequilibrium thermodynamic theory for small systems coupled to multiplicative noises. The theory is based on Ito calculus, and is fully co-variant under time-independent nonlinear transformation of variables. Assuming instantaneous detailed balance, we derive expressions for various thermodynamic functions, including work, heat, entropy production, and free energy, both at ensemble level and at trajectory level, and prove the second law of thermodynamics for arbitrary nonequilibrium processes. We relate time-reversal asymmetry of path probability to entropy production, and derive its consequences such as fluctuation theorem and nonequilibrium work relation. For Langevin systems with additive noises, our theory is equivalent to the common theories of stochastic energetics and stochastic thermodynamics. Using concrete examples, we demonstrate that whenever kinetic coefficients or metric tensor depend on system variables, the common theories of stochastic thermodynamics and stochastic energetics should be replaced by our theory.