Sampling procedures for low temperature dynamics on complex energy landscapes [Elektronische Ressource] / vorgelegt von George Alexandru Nemnes

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Sampling procedures for low temperaturedynamics on complex energy landscapesvon der Fakult¨at fur¨ Naturwissenschaftender Technischen Universit¨at Chemnitzgenehmigte Dissertation zur Erlangung des akademischen Gradesdoctor rerum naturaliumvorgelegt von Dipl.-Phys. George Alexandru Nemnesgeboren am 11. Februar 1980 in Bukaresteingereicht am 19. M¨arz 2008Gutachter:Prof. Dr. Karl Heinz HoffmannProf. Dr. Michael SchreiberProf. Dr. Paolo SibaniTag der Verteidigung: 21. Mai 2008http://archiv.tu-chemnitz.de/pub/2008/00682Bibliographic descriptionGeorge Alexandru NemnesSampling procedures for low temperature dynamics on complexenergy landscapesTechnische Universit¨at Chemnitz, Fakult¨at fur¨ Naturwissenschaften,Dissertation, 200883 pages, 34 figures, 6 tables, 46 citationsAbstractThe present work deals with relaxation dynamics on complex energy land-scapes. The state space of a complex system possesses, as a hallmark, themultitude of local minima separated by higher states, called barrier states.This feature gives rise to a host of non-equilibrium phenomena. From case tocase, for different complex systems, ranging from atomic clusters, spin glassesand proteins to neural networks or financial markets, the key quantities likeenergy and temperature may have different meanings, though their function-ality is the same.

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• Physik

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• Publié par: technische_universitat_chemnitz
• Publié le: 01 janvier 2008
• Nombre de lectures: 22
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Sampling procedures for low temperature dynamics on complex energy landscapes

vonderFakult¨atf¨urNaturwissenschaften derTechnischenUniversita¨tChemnitz genehmigte Dissertation zur Erlangung des akademischen Grades

doctor rerum naturalium

vorgelegt von Dipl.-Phys. George Alexandru Nemnes

geboren am 11. Februar 1980 in Bukarest

eingereichtam19.M¨arz2008 Gutachter: Prof. Dr. Karl Heinz Hoffmann Prof. Dr. Michael Schreiber Prof. Dr. Paolo Sibani

Tag der Verteidigung: 21. Mai 2008

http://archiv.tu-chemnitz.de/pub/2008/0068

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Bibliographic description George Alexandru Nemnes Sampling procedures for low temperature dynamics on complex energy landscapes TechnischeUniversit¨tChemnitz,Fakulta¨tfu¨rNaturwissenschaften, a Dissertation, 2008 83 pages, 34 figures, 6 tables, 46 citations

Abstract The present work deals with relaxation dynamics on complex energy land- scapes. The state space of a complex system possesses, as a hallmark, the multitude of local minima separated by higher states, called barrier states. This feature gives rise to a host of non-equilibrium phenomena. From case to case, for different complex systems, ranging from atomic clusters, spin glasses and proteins to neural networks or financial markets, the key quantities like energyandtemperaturemay have different meanings, though their function- ality is the same. The numerical handling of relaxational dynamics in such complex systems, even for relatively small sizes, poses a tough challenge if the entire state space is to be considered. Here, state space sampling pro- cedures are introduced that provide an accurate enough description for the low temperature dynamics, using small subsets from the original state space. As test cases, short range Ising spin systems were considered. The samples - depending on the way they are constructed - provide either lower bounds for the largest relaxation timescales in a quasi-ergodic component of the state space or the isothermal relaxation of the mean energy, like in the proposed DRS method. Upon the latter procedure, a parallel heuristic is built which gives the possibility of handling large samples. The collected structural data provides information of the state space topology in systems with different le- vels of frustration, like disordered ferromagnets and spin glasses. It provides insights into the focusing/anti-focusing types of landscapes, which give rise to different ground state accessibilities. For the large samples, the domain forma- tion and growth has been analysed and compared with existing experimental and numerical data in literature. The algorithms proposed here become more and more accurate as the temperature is decreased and therefore they can pro- vide an alternative to the classical Monte Carlo approach for this temperature range.

Keywords complex system, energy landscape, sampling, spin glass, disordered ferromag- net, parallel computing.

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Contents

1 Introduction 7 2 Model 11 2.1 Spin glasses and disordered ferromagnets . . . . . . . . . . . . . 11 2.2 Experimental facts . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Energy landscapes . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.1 The Master equation . . . . . . . . . . . . . . . . . . . . 17 2.4.2 Application to small lattices . . . . . . . . . . . . . . . . 20 3 Relaxation in quasi-ergodic components 23 3.1 State space reduction procedures . . . . . . . . . . . . . . . . . 23 3.2 State space sampling procedures . . . . . . . . . . . . . . . . . . 26 3.2.1 Decreasing the energy of a barrier state . . . . . . . . . . 27 3.2.2 Lumping states in local equilibrium . . . . . . . . . . . . 29 3.2.3 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Implementation and results . . . . . . . . . . . . . . . . . . . . 32 3.3.1 State space reduction procedures . . . . . . . . . . . . . 33 3.3.2 State space sampling procedures . . . . . . . . . . . . . . 34 4 Dynamically relevant sequences 41 4.1 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Application to disordered ferromagnets and spin glasses . . . . . 43 4.3 Comparison with Monte Carlo Metropolis method . . . . . . . . 45 4.4 Focusing vs. anti-focusing landscapes . . . . . . . . . . . . . . . 47 5 Plaquette parallelism 51 5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2 The parallel algorithm . . . . . . . . . . . . . . . . . . . . . . . 52 5.3 Method accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . 54

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5.4 Parallel performance . . . . . . . . . . . . . 5.5 Application to different boundary conditions 5.6 Analyzing the domain growth . . . . . . . . 5.7 Two replica overlap . . . . . . . . . . . . . .

Conclusions

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Chapter 1

Introduction

The study of complex systems has become an interdisciplinary field, break- ing down the barriers between physics, chemistry, biology on one hand, but also between these disciplines and other so-called soft sciences like psychology, sociology, economics and anthropology [1]. As examples of complex systems together with the processes of interest one may count: relaxation of ultrafast cooled atomic clusters [2–4], low temperature dynamics of spin glasses, the protein folding problem [5], learning and adaptation in neural networks [6, 7], optimization in combinatorial problems [8], the evolution of economical mar- kets [9] or human societies. All these quite different systems (enumerated above) have the fundamental features of a complex system, i.e. there are many interconnected elements interacting with each-other in a competitive manner and often subjected to a certain external environment. The mutual interacti- vity of the parts is here essential to describe the system acting as a whole and not as a sum of its parts. The time evolution of a complex system like the ones mentioned in the above examples is of great interest. Without going into the specificity of each of the systems, it is widely believed that simple universal laws may be formulated that would further enhance and structure the specific differentiation in the rather diverse fields of research. In the following one considers as prototypical complex systems short range Ising spin systems, namely disordered ferromagnets and spin glasses. Such systems have proved to posses a non-ergodic behaviour at reasonable large time scales. This means that the system is trapped for a long time in a certain region of its state (configuration) space, a fact which can be explained through the existence of high energy barriers separating that particular region from the rest of the state space. Furthermore, one knows that the two key ingredients,disorderandfrustrationin case of a spin glass system, induce a

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CHAPTER 1. INTRODUCTION

rough potential energy surface (PES) with many local minima surrounded by higher, so-called barrier-states. This leads to the non-equilibrium phenomena observed experimentally, such as aging and reinitialization effects [10]. For the infinite range systems the thermodynamical properties [11] and many dynamical implications are known analytically. By contrast theshort range systems are more realistic and until the present date they have been mainly subject to intensive computational studies. The main tool to describe dynamics is the Monte Carlo (MC) approach, usually with the Metropolis one-spin-flip acceptance probability [12], which provides the time evolution for the quantities of interest. Still, the classical Monte Carlo Metropolis (MCM) approach becomes inefficient at very low temperatures, when the acceptance probability becomes very small and the system is trapped very long times in a confined region of state space. Several improvements exist in the literature in form ofdynamical(i.e temperature dependent) MC algorithms [13]. The aim of this thesis is to bring a contribution to the issue of solving efficiently the relaxation problem in short range Ising spin systems. More precisely, the developed methods offer a reasonably good description of the real (reference) low temperature dynamics. The basic idea is to use statistical means in order to construct samples of the landscape that can be further used to get information about dynamics. The building of samples is based either on random exploration, as inChapter 3, where lower bounds for the transi- tion timescales are obtained or based on identifying relevant structures in the state space, as inChapter 4. Although the dynamical behavior as a Markov chain is a sequential problem, the proposed algorithms allow parallelization (seeChapter 3, 5). This is of great importance since it is less likely that one develops a computing systemxtimes faster, than a parallel environment with x advantage is that the sample construction innodes. AnotherChapter 4is temperature independent. This reduces considerably the computational effort, since the main time-consuming task does not have to be repeated for each tem- perature, as in the case of previously developeddynamicalMetropolis Monte Carlo approaches. The methods are applicable to other complex systems as well, which posses similar features in the state space. In the following a more detailed structure of this thesis is presented. Chapter 2starts with an introduction in the Ising spin systems - spin glasses and disordered ferromagnets - considered here as prototypical complex systems. The characteristic features of the state space are then pointed out. The Metropolis dynamics is presented in the last section of this chapter in form of a master equation approach, which is considered to be the reference dynamics.

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Chapter 3 Bothintroduces state space reduction- and sampling procedures. approaches are intended to reproduce with good accuracy the largest timescales of the relaxation process in the low temperature limit. The first are based on a deterministic algorithm to find a reduced graph from the original (portion of) state space. They also show in-how-far the saddle points only can offer a good dynamical description at different temperatures. The sampling procedures use statistical means in order to construct samples in the energy landscape that provide lower bounds to the largest timescales. Chapter 4presents a statistical method that identifies dynamical relevant state space structures that are able to describe the dynamics of the system when it is quenched from high- to low temperature. The consistency between the proposed method and the reference Metropolis dynamics is tested and the temperature range of validity is established. In the end, comparing the dis- ordered ferromagnetic- and spin glass systems, different state space structures are revealed, leading to different kinetic accessibilities of the ground states. The short range character of the systems is exploited inChapter 5, where a parallel approach is proposed. The dynamically relevant sequences were obtained exactly in the previous chapter and here a parallelizable heuristic is shown to give good results. Large samples can be handled and the domain formation and growth is analysed.

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CHAPTER

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INTRODUCTION

Chapter 2

Model

2.1 Spin glasses and disordered ferromagnets One of the most well known prototypes for complex systems are the spin glasses (SG). These have been in the early years binary metallic alloys such asM n x Cu 1−x (x <5%), where a small fraction of atoms with net magnetic moment (e.g.M n, F e) was spread into the non-magnetic matrix of the other metal (e.g.Cu, Aumagnetic moment (spin) arises from an unpaired net ). The electron in thed couplings between spinsorbital of the transitional metal. The can be described through what is known as RKKY interaction (Ruderman- Kittel-Kasuya-Yosida). This theory has been originally developed for the indi- rect exchange couplings between nuclear spins through a conduction electron (Ruderman-Kittel) [26]. Later, the theory was expanded to describe interact- ingd-electron spins (Kasuya-Yosida) [27, 28]. Characteristic for the RKKY interaction is the dependence of its strength and sign on the distance between the two spins. Due to the random but fixed positions of spins, usually termed asquenched disorder, different pairs of spins may exertferromagneticoranti- ferromagneticinteractions. The presence of both types of interactions gives rise to another feature, calledfrustration the ground state of the system in, i.e. not all of the interactions have the two corresponding spins aligned parallel, in case of ferromagnetic interactions, and anti-parallel, for the anti-ferromagnetic interactions. The two key features,quenched disorderandfrustration(see fig. 2.1), are paramount in the dynamic behavior of spin glasses. Nowadays there is a multitude of new compounds in the current research. Systems like theRb 2 Cu 1−x Co x F 4 family can be tuned to become a 2d random- exchange anti-ferromagnet (x >0.40), spin glass (0.18< x <0.40) or ferro- magnet (x <0. particular interest is how the level of frustration Of18) [14].

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