Analysis of Avalanches in the Landscape of Disordered Systems Pubblico
Rahman, Mahajabin (Spring 2023)
Abstract
The construct of an energy landscape makes it possible to use the same physics machinery to study completely unrelated scenarios, like fitness landscapes in evolution and algorithmic complexity in constraint satisfaction problems. I use this construct to study disordered (particularly, glassy) systems. When a thermal quench is used to access the inherent structure of their landscape, the consequence is aging – a relaxation process that exhibits interesting behaviors like memory effects, that can ideally be harnessed for engineering purposes, but the slow time-scales make this process difficult and tedious to study.
My dissertation starts and ends with the archetypal models of complex systems - the mean field Ising model called the Sherrington Kirkpatrick spin glass, and a sparse lattice model called the Edwards-Anderson spin glass. The first part of my thesis employs these models to study the statistical signatures of this relaxation process, which reflects the hierarchy of time scales of the energy landscape being explored. My research is anchored on a phenomenological description known as record dynamics (RD), which argues that large fluctuations (also called quakes or avalanches) -- the rare events, rather than the averaged events -- can exclusively be used to describe the relaxation process. I show that an RD-borne minimal model can reproduce even anomalous effects quintessential to aging, offering a simple alternative to mean field theoretic tools.
The second part of my thesis explores how critical avalanches are produced. Inspired by the idea that saturation bounds of marginally stable spins are indicators of whether avalanches will be critical, I use different modes of hysteretic driving to study the effects of dislodging the stability distribution to different extents. This helps drive the Edwards Anderson spin model, a difficult problem to solve (specifically NP hard), into a percolation transition, and elucidates the consequent changing correlation structure. In addition, I explore attempts to further understand the relationship between marginal stability and information transmission, and leave some open ended questions.
Table of Contents
1 General Introduction 1
1.1 The trouble with bridging disordered systems . . . . . . . . . . . . . 1
1.1.1 SpatialComplexity........................ 2
1.1.2 TemporalComplexity ...................... 3
1.2 Practical motivation for studying far from equilibrium disordered systems 5
1.3 Orderparameters............................. 6
1.4 IsingModels................................ 7
1.5 Energy landscapes ............................ 9
1.5.1 MeanFieldTheory........................ 12
1.6 Avalanches ................................ 13
1.6.1 RecordStatistics ......................... 14
1.6.2 Hysteresis ............................. 16
1.7 If a spin glass avalanches, does a traveling salesman see it? . . . . . 18
1.8 Outline................................... 19
2 Valleys in aging and coarsening processes 21
2.1 Summary ................................. 21
2.2 Introduction................................ 22
2.3 Computational Methods ......................... 28
2.3.1 Simulation of quenches...................... 28
2.3.2 Detecting records and introducing the RD order parameter . 29
2.4 Numerical results ............................. 31
2.4.1 Edwards Anderson Spin Glass.................. 32
2.4.2 Sherrington-Kirkpatrick Spin Glass . . . . . . . . . . . . . . . 35
2.5 Conclusion................................. 40
3 Hierarchical Landscapes and Memory 42
3.1 Summary ................................. 42
3.2 Introduction................................ 43
3.3 TheClusterModel ............................ 46
3.3.1 How the cluster model represents RD . . . . . . . . . . . . . . 47
3.4 The waiting time method......................... 48
3.5 Simulation Results ............................ 49
3.5.1 Rejuvenation and Memory.................... 49
3.5.2 Replication of results from soft spheres . . . . . . . . . . . . . 52
3.6 Predictive power of the cluster model.................. 54
3.7 Conclusion................................. 55
4 Creation of critical avalanches 57
4.1 Summary ................................. 57
4.2 Introduction................................ 57
4.2.1 Marginally stable spins and mutual frustration: A Case Study with the SK model ........................ 61
4.2.2 ProblemFormulation ...................... 63
4.3 Models and summary of methods .................... 65
4.4 Comparison of Avalanche Distributions................. 66
4.4.1 Limitations on the branching process . . . . . . . . . . . . . . 70
4.5 Percolation ................................ 72
4.5.1 Background ............................ 72
4.5.2 Computational Methods ..................... 75
4.6 Conclusion................................. 76
5 Connections to optimization 78
5.1 Introduction................................ 78
5.1.1 Background ............................ 78
5.2 Problem formulation ........................... 80
5.3 Methods.................................. 80
5.3.1 HO with constant dH=c .................... 80
5.4 Results................................... 81
5.5 Conclusion and outlook.......................... 82
6 Summary 84
6.1 Meaning of results ............................ 84
6.2 Broad scientific context.......................... 87
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