A random walk in statistical physics
Pontus Svenson2001, DOKTORSAVHANDLINGAR-CHALMERS TEKNISKA …
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Abstract
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The thesis presents an introduction to statistical physics, focusing on random walks and their implications in spin glass systems. It examines the computational methods used to analyze complex systems, exploring topics such as graph coloring, ferromagnetic behavior in random graphs, and damage spreading in small-world models. Through the appended research papers, the work aims to deepen understanding of these phenomena and offers insights into the computational challenges posed by large configuration spaces.
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Physical review letters, 2005
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Dynamical phase transitions in disordered systems: the study of a random walk model
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A new type of collective excitations, due exclusively to the topology of a complex random network that can be characterized by a fractal dimension $D_F$, is investigated. We show analytically that these excitations generate phase transitions due to the non-periodic topology of the $D_F>1$ complex network. An Ising system, with long range interactions over such a network, is studied in detail to support the claim. The analytic treatment is possible because the evaluation of the partition function can be decomposed into closed factor loops, in spite of the architectural complexity. This way we compute the magnetization distribution, magnetization loops, and the two point correlation function; and relate them to the network topology. In summary, the removal of the infrared divergences leads to an unconventional phase transition, where spin correlations are robust against thermal fluctuations.
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Physical Review E, 2011
Randomness is known to affect the dynamical behaviour of many systems to a large extent. In this paper we investigate how the nature of randomness affects the dynamics in a zero temperature quench of Ising model on two types of random networks. In both the networks, which are embedded in a one dimensional space, the first neighbour connections exist and the average degree is four per node. In the random model A, the second neighbour connections are rewired with a probability p while in the random model B, additional connections between neighbours at Euclidean distance l (l > 1) are introduced with a probability P (l) ∝ l −α . We find that for both models, the dynamics leads to freezing such that the system gets locked in a disordered state. The point at which the disorder of the nonequilibrium steady state is maximum is located. Behaviour of dynamical quantities like residual energy, order parameter and persistence are discussed and compared. Overall, the behaviour of physical quantities are similar although subtle differences are observed due to the difference in the nature of randomness.
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We determine the geometrical properties of the most probable paths at ÿnite temperatures T , between two points separated by a distance r, in one-dimensional lattices with positive energies of interaction i associated with bond i. The most probable path-length tmp in a homogeneous medium ( i = , for all i) is found to undergo a phase transition, from an optimal-like form (tmp ∼ r) at low temperatures to a random walk form (tmp ∼ r 2 ) near the critical temperature Tc = =ln 2. At T ¿ Tc the most probable path-length diverges, discontinuously, for all ÿnite endpoint separations greater than a particular value r * (T ). In disordered lattices, with i homogeneously distributed between − =2 and + =2, the random walk phase is absent, but a phase transition to diverging tmp still takes place. Di erent disorder conÿgurations have di erent transition points. A way to characterize the whole ensemble of disorder, for a given distribution, is suggested.
Quantum Chaos in Random Ising Networks
arXiv (Cornell University), 2024
Non equilibrium phase transitions of the Ising model on networks
A zero temperature quench of the Ising model is known to lead to a frozen steady state on random and small world networks. We study such quenches on random scale free networks (RSF) and compare the scenario with that in the Barab\'{a}si-Albert network (BA) and the Watts Strogatz (WS) addition type network. While frozen states are present in all the cases, the RSF shows an order-disorder phase transition of mean field nature as in the WS model as well as the existence of two absorbing phases separated by an active phase. The WS network also shows an active-absorbing (A-A) phase transition occurring at the known order-disorder transition point. The comparison of the RSF and the BA network results show interesting difference in finite size dependence.
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Physical Review E, 2006
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Since computers are able to simulate the equilibrium properties of model systems, they may also prove useful for solving the hard optimization problems that arise in the engineering of complex systems.
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We study the performance of Markov chains for the $q$-state ferromagnetic Potts model on random regular graphs. It is conjectured that their performance is dictated by metastability phenomena, i.e., the presence of "phases" (clusters) in the sample space where Markov chains with local update rules, such as the Glauber dynamics, are bound to take exponential time to escape, and therefore cause slow mixing. The phases that are believed to drive these metastability phenomena in the case of the Potts model emerge as local, rather than global, maxima of the so-called Bethe functional, and previous approaches of analysing these phases based on optimisation arguments fall short of the task. Our first contribution is to detail the emergence of the metastable phases for the $q$-state Potts model on the $d$-regular random graph for all integers $q,d\geq 3$, and establish that for an interval of temperatures, delineated by the uniqueness and the Kesten-Stigum thresholds on the $d$-re...
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