Deriving GENERIC from a Generalized Fluctuation Symmetry

Abstract

Much of the structure of macroscopic evolution equations for relaxation to equilibrium can be derived from symmetries in the dynamical fluctuations around the most typical trajectory. For example, detailed balance as expressed in terms of the Lagrangian for the path-space action leads to gradient zero-cost flow. We find a new such fluctuation symmetry that implies GENERIC, an extension of gradient flow where a Hamiltonian part is added to the dissipative term in such a way as to retain the free energy as Lyapunov function. I. HISTORY AND OUTLINE OF THE PAPERWhile macroscopic equations describing the return to equilibrium have been conceived and applied even before the atomistic picture of matter was widely accepted, their derivation shows important mathematical and conceptual difficulties. After all, hydrodynamic and thermodynamic behavior is described autonomously in only a few macroscopic variables and it must be understood how these variables get effectively decoupled from the many microscopic degrees of freedom. Moreover, in modifying the scale of description, the character of the dynamics could drastically change, from a unitary or a Hamiltonian to a dissipative evolution as possibly one of the most remarkable features 1 .One of the very first and still much studied examples in transiting from microscopic 1 Perhaps the earliest example where that question became manifest is through d'Alembert's paradox (1752) for reconciling, what we call today, Euler's equation with that of Navier-Stokes. As d'Alembert wrote indeed, "It seems to me that the theory (potential flow), developed in all possible rigor, gives, at least in several cases, a strictly vanishing resistance, a singular paradox which I leave to future Geometers to elucidate."

Symmetries and Geometrical Properties of Dynamical Fluctuations in Molecular Dynamics

“…The splitting J = J S + J A recovers the recently proposed (pre-)GENERIC splitting of [37], at least for conservative systems. In this case, two currents J S and J A can be identified straightforwardly, since J A corresponds to the Hamiltonian evolution and the J S to the action of the thermostat.…”

“…In this case, the antisymmetric force F A p contains the Hamiltonian evolution, while the symmetric force contains the coupling to the thermostat. That is the essence of the GENERIC formalism [34,35]: see also [36,37]. This connection is clearer at the level of probability currents, as we discuss in the next section.…”

“…We will show that there are two possible extensions of the overdamped case, which corresponds to two different splittings of the current J. One of the choices yields a geometrical structure analogous to (60), which is related to the GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) formalism [34,35]: see [36,37]. However, there is no connection between this splitting and the housekeeping heat.…”

“…This is the setting that has been named pre-GENERIC [37]. However, we emphasise that (71) and (72) apply also in the non-conservative setting.…”

Frenesy: Time-symmetric dynamical activity in nonequilibria

“…In a macroscopic dynamical fluctuation theory, [52][53][54] we examine the probability of empirically possible trajectories. Pioneered in [55,56] one tries to understand the structure of fluctuations as inherited from microscopic constituents and laws, and as relevant for typical macroscopic behavior, [13,57]. Mathematical references include [58,59], spanning almost half a century.…”