Spin Models

After World War II, quite a few mathematicians were attracted to the modeling of phase transitions as this area of physics was seeing considerable mathematical difficulties. This paper studies their contributions to the physics of phase transitions, and in particular those of the by far most productive and successful of them, the Polish-American mathematician Mark Kac (1914-1984). The focus is on the resources, values, and traditions that the mathematicians brought with them and how these differed from those of contemporary physicists as well as the mathematicians' relations with the physicists in terms of collaboration and reception of results. 1. Introduction After World War II, quite a few mathematicians, including Mark Kac, John von Neumann, and Nobert Wiener, worked on the physical problem of phase transitions, i.e. changes in the state of matter caused by gradual changes of physical parameters such as the condensation of a gas to a liquid and the loss of magnetization of a ferromagnet above a certain temperature (called the Curie temperature). The significance of these mathematicians was not so much that they brought mathematical rigor to the theoretical description of the phenomena, 1 but that they applied their mathematical

In this paper we introduce a variant of the honeycomb lattice in which we create defects by randomly exchanging adjacent bonds, producing a random tiling with a distribution of polygon edges. We study the percolation properties on these lattices as a function of the number of exchanged bonds using a novel computational method. We find the site and bond percolation thresholds are consistent with other three-coordinated lattices with the same standard deviation in the degree distribution of the dual; here we can produce a continuum of lattices with a range of standard deviations in the distribution. These lattices should be useful for modeling other properties of random systems as well as percolation.

Study of the cluster-or community structure of complex networks makes an important contribution to the understanding of networks at a functional level. Despite the many efforts, no definition of community has been agreed on and important aspects such as the statistical significance and theoretical limits of community detection are not well understood. We show how the problem of community detection can be mapped onto finding the ground state of an infinite range spin glass. The ground state energy then corresponds directly to the quality of the partition. The network modularity Q previously defined by Girvan and Newman [1] turns out to be a special case of this spin glass energy. Through this spin glass analogy, we are able to give expectation values for the modularity of random graphs that can be used in the assessment of the statistical significance of real network clusterings. Further, it allows for assessing the theoretical limits of community detection.

Two different models are presented that allow for efficiently performing routing of a quantum state. Both cases involve an XX spin chain working asspin chain working as a data bus and additional spins that play the role of sender and receivers, one of which is selected to be the target of the quantum state transmission protocol via a coherent quantum coupling mechanism making use of local and/or global magnetic fields. Quantum routing is achieved in the first of the models considered by weakly coupling the sender and the receiver to the data bus. On the other hand, in the second model, local magnetic fields acting on additional spins located between the sender and receiver and the data bus allow us to perform high-fidelity routing.

Some of the recent developments concerning the propagation of quantum correlations across spin channels are reviewed. In particular, we focus on the improvement of the transport efficiency obtained by the manipulation of few energy parameters (either end-bond strengths or local magnetic fields) near the sending and receiving sites. We give a physically insightful description of various such schemes and discuss the transfer of both entanglement and of quantum discord.

Some of the recent developments concerning the propagation of quantum correlations across spin channels are reviewed. In particular, we focus on the improvement of the transport efficiency obtained by the manipulation of few energy parameters (either end-bond strengths or local magnetic fields) near the sending and receiving sites. We give a physically insightful description of various such schemes and discuss the transfer of both entanglement and of quantum discord.

We describe the hardwired implementation of algorithms for Monte Carlo simulations of a large class of spin models. We have implemented these algorithms as VHDL codes and we have mapped them onto a dedicated processor based on a large FPGA device. The measured performance on one such processor is comparable to O(100) carefully programmed high-end PCs: it turns out to be even better for some selected spin models. We describe here codes that we are currently executing on the IANUS massively parallel FPGA-based system.

Abstract:- In this paper we deal with the problem of state estimation based on interval-valued stochastic descriptions of noise. To perform the estimation, we derive a probabilistic description of unobserved states conditioned on the observations, called a statistical inference rule. Moreover, we link the derivation of the rules to the notion of estimation risk. In the context of confident estimation, risk is defined as a variational expression dependent of the statistical inference rule. More specifically, we draw upon a compound risk description, comprising two additive terms. The a-level represents the first ingredient of risk and is found to be a function of the lower expectation of the statistical inference rule, while the second term represents the upper expectation of the rule’s fuzziness taken with respect to the observations. Both terms are connected through a variational parameter. Finally, the compound risk reduces into a function of the statistical inference rule and the...

The Watts and Strogatz model of small-world network systems has been successfully applied to explain features in several data sets. We have carried out a set of extensive simulation studies of stochastically generated small-world systems. We report on the data structures, memory and run-time complexity of our simulation codes and discuss how these ideas may be applied to similar simulations studies. Our packed k-index mechanism encodes physical spatial information and supports interpolation between pure list and multi-dimensional array models. The k-index mechanism allows most procedures in simulations of this ilk to be written in a dimensionally independent fashion.

The Watts and Strogatz model of small-world network systems has been successfully applied to explain features in several data sets. We have carried out a set of extensive simulation studies of stochastically generated small-world systems. We report on the data structures, memory and run-time complexity of our simulation codes and discuss how these ideas may be applied to similar simulations studies. Our packed k-index mechanism encodes physical spatial information and supports interpolation between pure list and multi-dimensional array models. The k-index mechanism allows most procedures in simulations of this ilk to be written in a dimensionally independent fashion.

Based on research involving Monte Carlo simulations, this paper discusses the role of models in physics, in particular, the modelling of magnetic material by means of various lattice geometries.  It explains both single spin flip and cluster spin flip dynamics algorithms (namely, Metropolis, Glauber, Swendsen-Wang and Wolf), and considers the analog of time in computational models.