Generalization of the Dynamical Lack-of-Fit Reduction from GENERIC to GENERIC
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Generalization of the Dynamical Lack-of-Fit Reduction from GENERIC to GENERIC Michal Pavelka1
· Václav Klika2 · Miroslav Grmela3
Received: 24 September 2019 / Accepted: 10 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The lack-of-fit statistical reduction, developed and formulated first by Bruce Turkington, is a general method taking Liouville equation for probability density (detailed level) and transforming it to reduced dynamics of projected quantities (less detailed level). In this paper the method is generalized. The Hamiltonian Liouville equation is replaced by an arbitrary Hamiltonian evolution combined with gradient dynamics (GENERIC), the Boltzmann entropy is replaced by an arbitrary entropy, and the kinetic energy by an arbitrary energy. The gradient part is a generalized gradient dynamics generated by a dissipation potential. The reduced evolution of the projected state variables is shown to preserve the GENERIC structure of the original (detailed level) evolution. The dissipation potential is obtained by solving a Hamilton–Jacobi equation. In summary, the lack-of-fit reduction can start with GENERIC and obtain GENERIC for the reduced state variables. Keywords Best fit Lagrangian reduction · GENERIC · Hamiltonian mechanics · Gradient dynamics · Hamilton–Jacobi equation
With thermodynamics, one can calculate almost everything crudely; with kinetic theory, one can calculate fewer things, but more accurately; and with statistical mechanics, one can calculate almost nothing exactly. — Eugene Wigner
Communicated by Eric A. Carlen.
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Michal Pavelka [email protected]
1
Faculty of Mathematics and Physics, Mathematical Institute, Charles University, Sokolovská 83, 186 75 Prague, Czech Republic
2
Department of Mathematics – FNSPE, Czech Technical University in Prague, Trojanova 13, 120 00 Prague, Czech Republic
3
École Polytechnique de Montréal, suc. Centre-ville, C.P.6079, Montréal, QC H3C 3A7, Canada
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M. Pavelka et al.
1 Introduction Imagine a very complex detailed dynamics of state variables x on manifold (or vector space) M given by vector field X ∈ X(M ), i.e. with evolution equations x˙ = X|x .
(1)
Assume now that this detailed (upper-level) dynamics is too complex to be solved while keeping all the details, and that experimental observations indicate the existence of an autonomous lower-dimensional dynamics that displays important features of the detailed dynamics. An investigation of this type of reductions is the primary objective of thermodynamics and statistical physics. We shall focus on evolution equations in non-equilibrium thermodynamics like classical irreversible thermodynamics [1], extended irreversible thermodynamics [2], thermodynamics with internal variables [3] and the General Equation of Non-Equilibrium ReversibleIrreversible Coupling (GENERIC) [4–7]. In particular, we assume that the detailed evolution possess the structure of GENERIC, i.e. the vector field consists of a reversible Hamiltonian part (generated by a Poiss
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