Complementary relations in non-equilibrium stochastic processes

• Novel complementary relations in non-equilibrium stochastic processes.
• Dependence of statistical measures (entropy, information, and work) on variables, reference frames, and time.
• Equilibrium maximises simultaneous information while minimising simultaneous disorder/uncertainty.
• Difference between Eulerian and Lagrangian entropy and its related concepts.
• Hamilton–Jacobi relation for forced-dissipative system.

We present novel complementary relations in non-equilibrium stochastic processes. Specifically, by utilising path integral formulation, we derive statistical measures (entropy, information, and work) and investigate their dependence on variables (x, v), reference frames, and time. In particular, we show that the equilibrium state maximises the simultaneous information quantified by the product of the Fisher information based on x and v while minimising the simultaneous disorder/uncertainty quantified by the sum of the entropy based on x and v as well as by the product of the variances of the PDFs of x and v. We also elucidate the difference between Eulerian and Lagrangian entropy. Our theory naturally leads to Hamilton–Jacobi relation for forced-dissipative systems.