Emergence of irreversible dynamics by the lack-of-fit reduction

• Název práce v češtině: Vznik nevratné dynamiky pomocí lack-of-fit redukce
• Název v anglickém jazyce: Emergence of irreversible dynamics by the lack-of-fit reduction
• Klíčová slova: Nevratnost|disipace|hamiltonovská evoluce
• Klíčová slova anglicky: Irreversibility|dissipation|Hamiltonian evolution
• Akademický rok vypsání: 2021/2022
• Typ práce: diplomová práce
• Jazyk práce: angličtina
• Ústav: Matematický ústav UK (32-MUUK)
• Vedoucí / školitel: doc. RNDr. Michal Pavelka, Ph.D.
• Řešitel: skrytý - zadáno a potvrzeno stud. odd.
• Datum přihlášení: 15.11.2021
• Datum zadání: 15.11.2021
• Datum potvrzení stud. oddělením: 07.01.2022
• Datum a čas obhajoby: 09.06.2023 10:00
• Datum odevzdání elektronické podoby: 06.05.2023
• Datum odevzdání tištěné podoby: 09.05.2023
• Datum proběhlé obhajoby: 09.06.2023
• Oponenti: doc. Ing. Václav Klika, Ph.D.
• Konzultanti: Mgr. Martin Šípka

Zásady pro vypracování

1) Review of the lack-of-fit reduction
2) Review of the Kac-Zwanzig model, numerical implementation, and reduction of the model
3) Demonstration of the lack-of-fit reduction on the Kac-Zwanzig model
4) If possible, further applications like modeling of turbulence or reductions in the kinetic theory

Seznam odborné literatury

[1] B. Turkington, An Optimization Principle for Deriving Nonequilibrium, Statistical Models of Hamiltonian Dynamics, J Stat Phys (2013), 152:569–597
[2] M. Pavelka, V. Klika and M. Grmela. Generalization of the dynamical lack-of-fit reduction, Journal of Statistical Physics, 181(1), 19-52, 2020.
[3] G. Ariel and E. Vanden-Eijnden, Testing Transition State Theory on Kac-Zwanzig Model, Journal of Statistical Physics, Vol. 126, No. 1, January 2007
[4] M. Pavelka, V. Klika, and M. Grmela, Multiscale Thermo-Dynamics, de Gruyter 2018
[5] Francisco Chinesta, Elias Cueto, Miroslav Grmela, Beatriz Moya, Michal Pavelka, Martin Šípka. Learning Physics from Data: a Thermodynamic Interpretation, chapter in Geometric Structures of Statistical Physics, Information Geometry, and Learning (SPIGL'20, Les Houches, France, July 27–31, Eds. Frédéric Barbaresco & Frank Nielsen, Springer Proceedings in Mathematics & Statistics, v. 361, 2021, Springer)

Předběžná náplň práce v anglickém jazyce

Hamilton canonical equations, which describe motion of classical particles, are purely reversible. On the other hand, macroscopic processes, which are obtained by averaging of the detailed Hamiltonian evolution, are irreversible. How does the irreversibility emerge? And how to reduce a detailed dynamics to a less detailed evolution? The lack-of-fit reduction provides answers to those questions.

The lack-of-fit reduction was introduced by B. Turkington [1] and its key idea is to construct a reduced set of evolution equations of a detailed canonical Hamiltonian system by minimizing the discrepancy (lack-of-fit) between the detailed and reduced evolutions. A lack-of-fit Lagrangian is set up and the corresponding action is then minimized with one free end, which leads to a Hamilton-Jacobi equation. Eventually, the reduced evolution is determined as the solution to the Hamilton-Jacobi equation. The method was then generalized to non-canonical Hamiltonian systems and to detailed systems with dissipation in [2].

Although the lack-of-fit reduction has already been successfully applied in a few test cases like 2D canonical turbulence, many questions remain open. For instance, the role of boundary conditions when the detailed dynamics is a Hamiltonian field theory is unclear. To address this question, it is, however, necessary to first test the generalization [2] on finite-dimensional systems like the Kac-Zwanzig model [3]. In order to carry out the numerical test, the Kac-Zwanzig model has to be first analyzed using methods of statistical physics, showing the reduced for of the evolution (a stochastic differential equation and its deterministic average). The lack-of-fit reduction has to be then applied to the detailed and less detailed deterministic evolutions of the Kac-Zwanzig model. Another problem is that the Hamilton-Jacobi equation is typically difficult to solve. However, since the evolution equations in the lack-of-fit reduction possess the structure of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) [4], one could alternatively employ the techniques of machine learning in order to find an approximate solution [5].

The generalized lack-of-fit reduction is a general reduction method that, for instance, can lead to irreversible reduced dynamics of purely reversible systems, and it does not need any fitting parameters. It has the potential to become a breakthrough in non-equilibrium thermodynamics because it could provide a general structure share among various levels of description (for instance, reduction of the kinetic theory to hydrodynamics), which would have a wide range of applications (for instance, modelling of complex fluids, many-particle systems, or in the kinetic theory).