How the arrow of time emerges from Hamiltonian systems by our incomplete knowledge

How does the arrow of time (dissipative, irreversible behavior) emerge from time-reversible Hamiltonian mechanics? Two ingredients are needed: the underlying system must be ergodic or phase-mixing, and our knowledge of the system must be incomplete. When the detailed dynamics explores its phase space and stays close to a submanifold parametrized by a reduced set of state variables, the lack-of-fit reduction method reveals that the effective equations for those reduced variables are necessarily irreversible. To make this precise, we present a path-integral formulation of the lack-of-fit reduction in non-equilibrium thermodynamics, which shows how the GENERIC framework (reversible Hamiltonian part plus irreversible gradient flow) emerges from purely Hamiltonian mechanics without any fitting parameters. The formulation is based on the Onsager-Machlup variational principle, and it yields reduced dynamical equations by minimizing the information discrepancy between the detailed and reduced evolutions. Subsequently, the reduction method is illustrated on the Kac--Zwanzig model, confirming that dissipation emerges from ignoring degrees of freedom, and on diffusion, where diffusion equation for the hydrodynamic mass density emerges from Vlasov equation. We also show how to generalize the Fisher information matrix and Kullback--Leibler divergence to arbitrary concave entropies via the principle of maximum entropy, including non-Boltzmann-Gibbs cases such as the Tsallis--Havrda--Charv\'{a}t entropy.