Nonequilibrium Equation of State for Open Hamiltonian Systems Maintained in Nonequilibrium Steady States

We establish the nonequilibrium equation of state in the stochastic thermodynamics framework for open Hamiltonian systems in contact with multiple heat baths governed by the Langevin equation and the Fokker-Planck equation. The derived nonequilibrium equation of state is an integral part of the nonequilibrium steady-state thermodynamics, and it reduces to the equilibrium equation of state when all the heat baths have the same temperature. We find that the nonequilibrium equation of state can be separated into two parts. One part has the form of the equilibrium equation of state, with the equilibrium temperature replaced by the average temperature of the heat baths. The other part depends on the temperature differences of the heat baths and represents nonequilibrium corrections. For systems with harmonic potentials, the nonequilibrium correction part is linear in the temperature differences of the heat baths. For more general open Hamiltonian systems, it may contain higher powers of the temperature differences. Another important feature of the nonequilibrium equation of state is that, in addition to the temperatures of the heat baths, it also depends on the friction coefficients arising from system-bath interactions in general. This suggests that it represents a relational condition between the system and the heat baths instead of the intrinsic properties of the system itself. In addition, when the potential energy is a homogeneous function, we find that the product of the generalized force and the generalized coordinate is proportional to the nonequilibrium internal energy, which is an alternative form of the nonequilibrium equation of state. Our findings may provide insights into the nonequilibrium equation of state for more general nonequilibrium open systems, including active matter and living organisms. Results in this work with suitable extension may find potential applications in nanoelectronics and biophysics including molecular motors and cells.