Statistical properties of time-dependent systems

doctoral dissertation

Diego Fregolente Mendes de Oliveira (Avtor), Marko Robnik (Mentor)

Povzetek

In the dissertation I have dealt with time-dependent (nonautonomous) systems, the conservative (Hamiltonian) as well as dissipative, and investigated their dynamical and statistical properties. In conservative (Hamiltonian) time-dependent systems the energy is not conserved, whilst the Liouville theorem about the conservation of the phase space volume still applies. We are interested to know, whether the system can gain energy, and whether this energy can grow unbounded, up to infinity, and we are interested in the system's behaviour in the mean, as well as its statistical properties. An example of such a system goes back to the 1940s, when Fermi proposed the acceleration of cosmic rays (in the first place protons) upon the collisions with moving magnetic domains in the interstellar medium of our Galaxy, and in other galaxies. He then proposed a simple mechanical one-dimensional model, the so-called Fermi-Ulam Model (FUM), where a point particle is moving between two rigid walls, one being at rest and the other one oscillating. If the oscillation is periodic and smooth, it turned out in a nontrivial way, which is, in the modern era of understanding the chaotic dynamical systems, well understood, namely that the unbounded increasing of the energy (the so-called Fermi acceleration) is not possible, due to the barriers in form of invariant tori, which partition the phase space into regions, between which the transitions are not possible. The research has then been extended to other simple dyanamical systems, which have complex dynamics. The first was so-called bouncer model, in which a point particle bounces off the oscillating platform in a gravitational field. In this simple system the Fermi acceleration is possible. Later the research was directed towards two-dimensional billiard systems. It turned out that the Fermi acceleration is possible in all such systems, which are at least partially chaotic (of the mixed type), or even in a system that is integrable as static, namely in case of the elliptic billiard. (The circle billiard is an exception, because it is always integrable, as the angular momentum is conserved even in time-dependent case.) The study of time-dependent systems has developed strongly worldwide around the 1990s, in particular in 2000s, and became one of the central topics in nonlinear dynamics. It turned out, quite generally, but formal and implicit, in the sense of mathematical existence theorems, that in nonautonomous Hamilton systems the energy can grow unbounded, meaning that the system ``pumps" the energy from the environment with which it interacts. There are many open questions: how does the energy increase with time, in particular in the mean of some representative ensemble of initial conditions (typically the phase space of two-dimensional time-dependent billiards is four-dimensional.) It turned out that almost everywhere the power laws apply, empirically, based on the numerical calculations, but with various acceleration exponents. If the Fermi acceleration is not posssible, like e.g. in the FUM, due to the invariant tori, then after a certain time of acceleration stage the crossover into the regime of saturation takes place, whose characteristics also follow the power laws. One of the central themes in the dissertation is the study of these power laws, their critical exponents, analytical relationships among them, using the scaling analysis (Leonel, McClintock and Silva, Phys. Rev. Lett. 2004). Furthermore, the central theme is the question, what happens, if, in a nonautonomous Hamilton system which exhibits Fermi acceleration, we introduce dissipation, either at the collisions with the walls (collisional dissipation) or during the free motion (in-flight dissipation, due to the viscosity of the fluid or the drag force etc.). Dissipation typically transforms the periodic points into point attractors and chaotic components into chaotic attractors. The Fermi acceleration is always suppressed. We are interested in the phase portraits.

Ključne besede

nonlinear dynamics;dynamical systems;conservative systems;dissipative systems;time-dependent systems;Fermi acceleration;billiards;kicked systems;chaos;chaotic attractors;periodic attractors;bifurcations;boundary crisis;dissertations;

Podatki

Jezik:	Angleški jezik
Leto izida:	2012
Izvor:	[Maribor
Tipologija:	2.08 - Doktorska disertacija
Organizacija:	UM FNM - Fakulteta za naravoslovje in matematiko
Založnik:	D. F. M. de Oliveira]
UDK:	530.182(043.3)
COBISS:	19304200
Št. ogledov:	2528
Št. prenosov:	112
Ocena:	0 (0 glasov)
Metapodatki:

Ostali podatki

Sekundarni jezik:	Slovenski jezik
Sekundarni naslov:	Statistične lastnosti časovno odvisnih sistemov
Sekundarni povzetek:	V doktorski disertaciji sem obravnaval časovno odvisne (neavtonomne) sisteme, tako konzervativne (hamiltonske), kakor tudi disipativne, in raziskoval njihove dinamične ter statistične lastnosti. V konzervativnih časovno odvisnih sistemih se energija ne ohranja, a Liouvilleov izrek o ohranitvi faznega volumna še vedno velja. Zanima nas, ali lahko sistem pridobi energijo, in ali lahko ta energija neomenejo narašča, do neskončnosti, in zanima nas vedenje sistema v povprečju, kakor tudi njegove statistične lastnosti. Primer takšnega problema sega v 1940. leta, ko je Fermi predlagal pospeševanje kozmičnih žarkov (predvsem protonov) ob trkanju s časovno gibljivimi magnetnimi domenami v interstelarnem mediju naše galaksije, ali v drugih galaksijah. Predlagal je preprost mehanski enodimenzionalni model, t.i. Fermi-Ulamov model (FUM), kjer se točkast delec giblje med dvema stenama, ena mirujoča, druga pa oscilira. Če je oscilacija periodična, se je povsem netrivialno izkazalo, kar v moderni dobi poznavanja kaotičnih dinamičnih sistemov tudi dobro razumemo, da namreč neomejeno naraščanje energije (t.i. Fermijevo pospeševanje; angl.: Fermi acceleration) ni možno, in sicer zaradi barier v obliki invariantnih torusov, ki fazni prostor razcepijo na območja, med katerimi prehod ni možen. Raziskave so se potem razširile na druge preproste dinamične sisteme, ki pa imajo kompleksno dinamiko. Najprej t.i. model odskakovanja (angl.: bouncer model), kjer točkast delec odskakuje od oscilirajoče trdne podlage v gravitacijskem polju. V tem preprostem modelu je Fermijevo pospeševanje možno. Raziskave so se potem razširile na dvodimenzionalne biljardne sisteme. Izkazalo se je, da je Fermijevo pospeševanje možno v vseh sistemih, ki so vsaj delno kaotični (t.i. mešanega tipa), ali pa celo v sistemu, ki je integrabilen kot statičen sistem, namreč v primeru eliptičnega biljarda. (Krožni biljard je izjema, saj je vedno integrabilen, ker se ohranja vrtilna količina tudi če je časovno odvisen.) Študij časovno odvisnih hamiltonskih sistemov se je močno razvil v svetovnem okviru nekako v 1990. letih, še prav posebej v 2000. letih, in je postal ena osrednjih tem nelinearne dinamike. Pokazalo se je precej splošno a formalno in implicitno, v smislu matematičnih eksistenčnih izrekov, da v neavtonomnih hamiltonskih sistemih energija lahko neomejeno narašča, se pravi, da sistem neomejeno ''črpa" energijo iz okolja, s katerim interagira. Veliko je odprtih vprašanj: kako narašča energija s časom, predvsem v povprečju za neke reprezentativne ansamble začetnih pogojev (tipično je fazni prostor dvodimenzionalnih časovno odvisnih biljardnih sistemov štiridimenzionalen). Izkaže se, da veljajo skoraj povsod potenčni zakoni, empirično, na osnovi numeričnih računov, a z zelo različnimi eksponenti pospeševanja (angl.: acceleration exponents). Če pa Fermijevo pospeševanje ni možno, kot npr. v FUM, zaradi invariantnih torusov, potem po določenem času faze pospeševanja nastopi faza prehoda (angl.: crossover) v nasičenje (angl.: saturation), katerih karakteristike prav tako sledijo potenčnim zakonitostim. Ena osrednjih tem v doktorskem delu je študij teh potenčnih zakonov, njihovih kritičnih eksponentov, ter analitičnih povezav med njimi, uporabljajoč skalirno analizo (Leonel, McClintock in Silva, Phys. Rev. Lett., 2004). Nadalje je centralna tema vprašanje, kaj se zgodi, če v neavtonomnem hamiltonskem sistemu, kjer obstaja Fermijevo pospeševanje, vključimo disipacijo, bodisi v trkih ob stenah (trkovna disipacija) ali pa med ''letom" (angl.: in-flight) (npr. zaradi viskoznosti tekocine ali zaradi sile upora, itd.). Disipacija tipično spremeni periodične točke v točkaste atraktorje, kaotične komponente pa v kaotične atraktorje. Fermijevo pospeševanje ni več možno. Zanimajo nas fazni portreti sistema, in tudi kako se njihova struktura spreminja s spreminjanjem parametrov sistema (bifukarcijska analiza).
Sekundarne ključne besede:	nelinearna dinamika;dinamični sistemi;konservativni sistemi;disipativni sistemi;časovno odvisni sistemi;Fermijevo pospeševanje;biljardi;brcani sistemi;kaos;kaotični atraktorji;periodični atraktorji;bifurkacije;kriza roba;disertacije;
URN:	URN:SI:UM:
Vrsta dela (COBISS):	Doktorska disertacija
Komentar na gradivo:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za fiziko
Strani:	VIII, 305 str.
Ključne besede (UDK):	mathematics;natural sciences;naravoslovne vede;matematika;physics;fizika;
ID:	8762231