Dinamične in statistične lastnosti časovno odvisnih enodimenzionalnih nelinearnih Hamiltonskih sistemov

doktorska disertacija

Dimitris Andresas (Author), Marko Robnik (Mentor)

Abstract

We study the one-dimensional time-dependent Hamiltonian systems and their statistical behaviour, assuming the microcanonical ensemble of initial conditions and describing the evolution of the energy distribution in three characteristic cases: 1) parametric kick, which by definition means a discontinuous jump of a control parameter of the system, 2) linear driving, and 3) periodic driving. For the first case we specifically analyze the change of the adiabatic invariant (the canonical action) of the system under a parametric kick: A conjecture has been put forward by Papamikos and Robnik (2011) that the action at the mean energy always increases, which means, for the given statistical ensemble, that the Gibbs entropy in the mean increases (PR property). By means of a detailed rigorous analysis of a great number of case studies we show that the conjecture largely is satisfied, except if either the potential is not smooth enough (e.g. has discontinuous first derivative), or if the energy is too close to a stationary point of the potential (separatrix in the phase space). We formulate the conjecture in full generality, and perform the local theoretical analysis by introducing the ABR property. For the linear driving we study first 1D Hamilton systems with homogeneous power law potential and their statistical behaviour under monotonically increasing time-dependent function A(t) (prefactor of the potential). We used the nonlinear WKB-like method by Papamikos and Robnik J. Phys. A: Math. Theor., 44:315102, (2012) and following a previous work by Papamikos G and Robnik M J. Phys. A: Math. Theor., 45:015206, (2011) we specifically analyze the mean energy, the variance and the adiabatic invariant (action) of the system for large time. We also show analytically that the mean energy and the variance increase as powers of A(t), while the action oscillates and finally remains constant. By means of a number of detailed case studies we show that the theoretical prediction is correct. For the periodic driving cases we study the 1D periodic quartic oscillator and its statistical behaviour under periodic time-dependent function A(t) (prefactor of the potential). We compare the results for three different drivings, the periodic parametrically kicked case (discontinuous jumps of A (t), the piecewise linear case (sawtooth), and the smooth case (harmonic). Considering the Floquet map and the energy distribution we perform careful numerical analysis using the 8th order symplectic integrator and present the phase portraits for each case, the evolution of the average energy and the distribution function of the final energies. In the case where we see a large region of chaos connected to infinity, we indeed find escape orbits going to infinity, meaning that the energy growth can be unbounded, and is typically exponential in time. The main results are published in two papers: Andresas, Batistić and Robnik Phys. Rev. E, 89:062927, (2014) and Andresas and Robnik J. Phys. A: Math. Theor., 47:355102, (2014).

Keywords

nonlinear dynamics;Hamiltonian systems;quantum mechanics;dynamical proprerties;statistical properties;one-dimensional nonlinear systems;time-dependent;dissertations;

Data

Language:	English
Year of publishing:	2015
Typology:	2.08 - Doctoral Dissertation
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	D. Andresas]
UDC:	530.182(043.3)
COBISS:	21209096
Views:	2225
Downloads:	80
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Other data

Secondary language:	Slovenian
Secondary title:	Dinamične in statistične lastnosti časovno odvisnih enodimenzionalnih nelinearnih hamiltonskih sistemov
Secondary abstract:	Preučujemo enodimenzionalne časovno odvisne Hamiltonove sisteme ter njihovo statistično vedenje, ob predpostavki mikrokanoničnega ansambla začetnih pogojev in opišemo razvoj porazdelitve energije v treh značilnih primerih: 1) parametrično brcanje, ki po definiciji pomeni nezvezne skoke kontrolnega parametra sistema, 2) linearna modulacija ter 3) periodična modulacija. V prvem primeru specifično analiziramo spremembo adiabatske invariante (kanonične akcije) sistema ob paramaterični brci: Papamikos in Robnik (2011) sta predlagala domnevo, da se akcija pri povprečni energiji vedno poveča, kar pomeni, za dani statistični ansambel, da se šrednjem Gibbsova entropija poveča (PR lastnost). Ob podrobni rigorozni analizi cele vrste sistemov pokažemo, da domneva večinoma velja, razen če potencial ni dovolj gladek (n.pr. ima nezvezen prvi odvod), ali če je energija preveč blizu stacionarni točki potenciala (separatrisi v faznem prostoru). Formuliramo domnevo šplošni obliki, izvedemo lokalno teoretično analizo ter uvedemo t.i. ABR lastnost. Za primer linearne modulacije študiramo najprej enodimenzionalne hamiltonske sisteme s homogenim potenčnim potencialom in njihove statistične lastnosti ob monotono naraščajoči časovno odvisni funkciji A(t) (predfaktor potenciala). Uporabimo nelinearno WKB podobno metodo Papamikosa in Robnika J. Phys. A: Math. Theor., 44:315102, (2012) in sledimo njunemu delu v članku J. Phys. A: Math. Theor., 45:015206, (2011), ter specifično analiziramo povprečno energijo, varianco in adiabatsko invarianto (akcijo) sistema za velike čase. Analitično pokažemo, da se povprečna energija in varianca povečujeta kot potenci A(t), medtem ko akcija oscilira and na koncu zavzame konstantno vrednost. Na osnovi študija posebnih primerov pokažemo, da je teoretična napoved točna. Za primer periodične modulacije preučujemo enodimenzionalni kvartični oscilator in njegovo statistično vedenje, ko je A(t) (predfaktor potenciala) periodična funkcija časa. Primerjamo rezultate za tri različne modulacije, periodično parametrično brcanje (nezvezni skoki parametra A(t)), odsekoma linearno modulacjo (žagasto), ter gladko (harmonično) modulacijo. Obravnavamo Floquetovo preslikavo in porazdelitev energije ter izvedemo temeljito numerično analizo uporabljajoč simplektične integratorje 8. reda ter predstavimo fazne portrete za vsak primer, razvoj povprečne energije ter porazdelitev končne energije. V primerih, ko vidimo veliko kaotično območje povezano do neskončnosti, najdemo ubežne orbite, ki pobegnejo v neskončnost, kar pomeni, da je lahko rast povprečne energije neomejena, in sicer je tipično eksponentna v času. Glavni rezultati so objavljeni v dveh člankih: Andresas, Batistić in Robnik Phys. Rev. E, 89:062927, (2014) in Andresas in Robnik J. Phys. A: Math. Theor., 47:355102, (2014).
Secondary keywords:	nelinearna dinamika;Hamiltonski sistemi;kvantna mehanika;dinamične lastnosti;statistične lastnosti;enodimenzionalni nelinearni sistemi;časovna odvisnost;disertacije;
URN:	URN:SI:UM:
Type (COBISS):	Doctoral dissertation
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za fiziko
Pages:	VII, 127 str.
ID:	8701053