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A simple approach to the dynamics of Ising spin systems
KTH, School of Computer Science and Communication (CSC), Computational Biology, CB.ORCID iD: 0000-0002-1252-2899
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We study a novel variational approach to solve the dynamics of Ising-like discrete spin systems. The equations are derived under mean-field approximations based on the cluster variational method. Comparison with the actual Glauber dynamics of models defined on Erdos-R\'enyi random graphs show that some of these approximations are extremely accurate, both at equilibrium and in the off-equilibrium regime, providing the same result of the Monte Carlo simulation in a much shorter time. The models studied are the ferromagnetic kinetic Ising model (both with symmetric and partially asymmetric interactions), the random field Ising model and the Viana-Bray model. Only for the latter model we find some small discrepancies in the very low temperature phase, probably due to the existence of a large number of metastable states.

Keyword [en]
variational approximations, spin dynamics, Glauber dynamics, cluster variation method, path probability method
National Category
Physical Sciences
Research subject
URN: urn:nbn:se:kth:diva-192098OAI: diva2:958022

QC 20160905

Available from: 2016-09-05 Created: 2016-09-05 Last updated: 2016-09-05Bibliographically approved
In thesis
1. Equilibrium and Dynamics on Complex Networkds
Open this publication in new window or tab >>Equilibrium and Dynamics on Complex Networkds
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Complex networks are an important class of models used to describe the behaviour of a very broad category of systems which appear in different fields of science ranging from physics, biology and statistics to computer science and other disciplines. This set of models includes spin systems on a graph, neural networks, decision networks, spreading disease, financial trade, social networks and all systems which can be represented as interacting agents on some sort of graph architecture.

In this thesis, by using the theoretical framework of statistical mechanics, the equilibrium and the dynamical behaviour of such systems is studied.

For the equilibrium case, after presenting the region graph free energy approximation, the Survey Propagation method, previously used to investi- gate the low temperature phase of complex systems on tree-like topologies, is extended to the case of loopy graph architectures.

For time-dependent behaviour, both discrete-time and continuous-time dynamics are considered. It is shown how to extend the cavity method ap- proach from a tool used to study equilibrium properties of complex systems to the discrete-time dynamical scenario. A closure scheme of the dynamic message-passing equation based on a Markovian approximations is presented. This allows to estimate non-equilibrium marginals of spin models on a graph with reversible dynamics. As an alternative to this approach, an extension of region graph variational free energy approximations to the non-equilibrium case is also presented. Non-equilibrium functionals that, when minimized with constraints, lead to approximate equations for out-of-equilibrium marginals of general spin models are introduced and discussed.

For the continuous-time dynamics a novel approach that extends the cav- ity method also to this case is discussed. The main result of this part is a Cavity Master Equation which, together with an approximate version of the Master Equation, constitutes a closure scheme to estimate non-equilibrium marginals of continuous-time spin models. The investigation of dynamics of spin systems is concluded by applying a quasi-equilibrium approach to a sim- ple case. A way to test self-consistently the assumptions of the method as well as its limits is discussed.

In the final part of the thesis, analogies and differences between the graph- ical model approaches discussed in the manuscript and causal analysis in statistics are presented.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016. 189 p.
TRITA-CSC-A, ISSN 1653-5723 ; 2016:17
Statistical mechanics, complex networks, spin systems, non equilibrium dynamics, generalized belief propagation, message passing, cavity method, variational approaches
National Category
Other Physics Topics Physical Sciences
Research subject
urn:nbn:se:kth:diva-191991 (URN)978-91-7729-058-2 (ISBN)
Public defence
2016-09-09, Kollegiesalen, Brinellvägen 8, plan 4, Stockholm, 10:00 (English)

QC 20160904

Available from: 2016-09-04 Created: 2016-09-03 Last updated: 2016-09-05Bibliographically approved

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