Dinamika neautonomnoga Frenkel-Kontorovina modela

doctoral thesis

• Title (english): Dynamics of non-autonomous Frenkel-Kontorova model
• Author: Braslav Rabar
• Mentor: Siniša Slijepčević (mentor)
• Committee member: Vesna Županović (predsjednik povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Siniša Slijepčević (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-06-11, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

U radu promatramo disipativnu dinamiku Frenkel-Kontorovina (FK) modela, kao jednog od najbitnijih fizikalnih modela, primjerice u fizici čvrstog stanja. FK model je poopćenje jednodimenzionalnog niza elastično povezanih čestica u periodičnom potencijalu s konstantnom ili periodičnom uniformnom silom. Rad se usredotočuje na razvoj teorije za neautonomni FK, tj. za slučaj gdje jednadžbe ovise o vremenu. Posebno promatramo slučaj tresne dinamike (neautonomna dinamika bez vanjske sile), s nizom otvorenih pitanja poput postojanja transporta. U radu prvo pokazujemo postojanje rješenja u odgovarajućim prostorima funkcija. Pokazujemo i postojanje polutoka, te glatkoću i analitičnost rješenja ovisno o početnim uvjetima i vektorskom polju. Zatim definiramo sinkronizirano rješenje, te pokazajemo da za svaki prosječan razmak postoji bar jedno sinkronizirano rješenje. Ključan pojam za opis dinamike su nule razlike rješenja. Razlikujemo singularnu odnosno regularnu nulu (netransverzalni odnosno transverzalni presjek rješenja), te pokazujemo da je broj nula razlike dvaju rješnja neautonomnog FK model nerastuć. Posebno promatramo prostorno-vremenski invarijantne mjere, te slabi $\omega$-granični skup i prostorno vremenski atraktor kao uniju svih slabih $\omega$-graničnih skupova. Dokazujemo da se prostorno vremenski atraktor podudara sa unijom nosača svih prostorno vremenski invarijantnih mjera. Uvodimo pojam transverzalnog prostorno-vremenskog atraktora, kao atraktora takvog da se dvije konfiguracije u atraktoru ne mogu sijeći netransverzalno. Ključan rezultat su dovoljni, provjerljivi uvjeti za transverzalnost atraktora, primjerice realna analitičnost za tresni sustav. Razlikujemo dvije faze dinamike: slobodnu i zapinjajuću fazu, te strogo uvodimo pojam transporta (za tresni sustav). Za transverzalne prostorno-vremenske atraktore dajemo slab, općenit dovoljan uvjet za postojanje transporta. Slutnja da transport postoji za određene sustave ostaje otvoren problem; no uveden dovoljni uvjet daje mogućnost brze numeričke provjere za konkretan sustav. Konačno, pokazujemo da su za tresni sustav sinkronizirana rješenja stabilna u ergodsko-teoretskom smislu.

Abstract (english)

In this thesis we consider dissipative dynamics of Frenkel-Kontorova (FK) models, one of the most important physical models, for example in the solid state physics. FK model generalizes one-dimensional elastically connected chains of particles in a periodic potential, with a constant or periodic uniform force. The thesis focuses on development of the theory for non-autonomous FK model, that means in the case when the equations depend on time. In particular we consider the case of Ratchet dynamics (non-autonomous dynamics without an external force); with a number of open problems, for example existence of transport. We first show in the thesis existence of solutions on appropriate function spaces. We demonstrate existence of a semi-flow, smoothness and analyticity of the solution depending on the initial condition and the vector field. We then define a synchronized solution, and show that for every mean spacing there exist at least one synchronized solution. The key idea needed to describe the dynamics is zeroes of a difference of two solutions. We distinguish regular and singular zeroes (transversal and non-transversal intersections of solutions), and show that the number of zeroes of a difference of two solutions of a non-autonomous FK model is non-increasing. In particular we consider space-time invariant measures, weak $\omega$-limit sets, and space time attractors as unions of weak $\omega$-limit sets. We show that the space-time attractor is equal to the union of supports of space-time invariant measures. We introduce the notion of a transversal space-time attractor, as the attractor for which any two configurations in the attractor can not intersect non-transversally. The key result are sufficient, verifiable conditions for an attractor to be transversal, for example analyticity for the Ratchet system. We distinguish two dynamical phase: depinned and pinned phase, and rigorously introduce the notion of transport (for the Ratchet system). For transversal space-time attractors, we give a weak, general sufficient condition for the existence of transport. The conjecture that transport exists for specific systems remains open; however the sufficient condition gives a possibility of fast numerical verification of it for a specific system. Finally, we show that for the Ratchet system, the synchronized solutions are stable in an ergodic-theoretical sense.

• Keywords: neautonoman; transport; sinkronizirana rješenja; atraktor; invarijantne mjere
• Keywords (english): nonautonomous; transport; synchronized solutions; attractor; invariant measures
• Language: croatian
• URN:NBN: urn:nbn:hr:217:478180
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: 80 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-03-21 12:20:04