Replica symmetry breaking in the `small world' spin glass

We apply the cavity method to a spin glass model on a `small world' lattice, a random bond graph super-imposed upon a 1-dimensional ferromagnetic ring. We show the correspondence with a replicated transfer matrix approach, up to the level of one step replica symmetry breaking (1RSB). Using the scheme developed by M\'ezard & Parisi for the Bethe lattice, we evaluate observables for a model with fixed connectivity and $\pm J$ long range bonds. Our results agree with numerical simulations significantly better than the replica symmetric (RS) theory.Comment: 21 pages, 3 figure

Dynamic rewiring in small world networks

We investigate equilibrium properties of small world networks, in which both connectivity and spin variables are dynamic, using replicated transfer matrices within the replica symmetric approximation. Population dynamics techniques allow us to examine order parameters of our system at total equilibrium, probing both spin- and graph-statistics. Of these, interestingly, the degree distribution is found to acquire a Poisson-like form (both within and outside the ordered phase). Comparison with Glauber simulations confirms our results satisfactorily.Comment: 21 pages, 5 figure

A solvable model of the genesis of amino-acid sequences via coupled dynamics of folding and slow genetic variation

We study the coupled dynamics of primary and secondary structure formation (i.e. slow genetic sequence selection and fast folding) in the context of a solvable microscopic model that includes both short-range steric forces and and long-range polarity-driven forces. Our solution is based on the diagonalization of replicated transfer matrices, and leads in the thermodynamic limit to explicit predictions regarding phase transitions and phase diagrams at genetic equilibrium. The predicted phenomenology allows for natural physical interpretations, and finds satisfactory support in numerical simulations.Comment: 51 pages, 13 figures, submitted to J. Phys.

Diagonalization of replicated transfer matrices for disordered Ising spin systems

We present an alternative procedure for solving the eigenvalue problem of replicated transfer matrices describing disordered spin systems with (random) 1D nearest neighbor bonds and/or random fields, possibly in combination with (random) long range bonds. Our method is based on transforming the original eigenvalue problem for a $2^n\times 2^n$ matrix (where $n\to 0$) into an eigenvalue problem for integral operators. We first develop our formalism for the Ising chain with random bonds and fields, where we recover known results. We then apply our methods to models of spins which interact simultaneously via a one-dimensional ring and via more complex long-range connectivity structures, e.g. $1+\infty$ dimensional neural networks and `small world' magnets. Numerical simulations confirm our predictions satisfactorily.Comment: 24 pages, LaTex, IOP macro

The Effect of Solvent Vapor Annealing on Drug-Loaded Electrospun Polymer Fibers

Electrospinning has emerged as a powerful strategy to develop controlled release drug delivery systems but the effects of post-fabrication solvent vapor annealing on drug-loaded electrospun fibers have not been explored to date. In this work, electrospun poly(ԑ-caprolactone) (PCL) fibers loaded with the hydrophobic small-molecule spironolactone (SPL) were explored. Immediately after fabrication, the fibers are smooth and cylindrical. However, during storage the PCL crystallinity in the fibers is observed to increase, demonstrating a lack of stability. When freshly-prepared fibers are annealed with acetone vapor, the amorphous PCL chains recrystallize, resulting in the fiber surfaces becoming wrinkled and yielding shish-kebab like structures. This effect does not arise after the fibers have been aged. SPL is found to be amorphously dispersed in the PCL matrix both immediately after electrospinning and after annealing. In vitro dissolution studies revealed that while the fresh fibers show a rapid burst of SPL release, after annealing more extended release profiles are observed. Both the rate and extent of release can be varied through changing the annealing time. Further, the annealed formulations are shown to be stable upon storage

Spin models on random graphs with controlled topologies beyond degree constraints

We study Ising spin models on finitely connected random interaction graphs which are drawn from an ensemble in which not only the degree distribution $p(k)$ can be chosen arbitrarily, but which allows for further fine-tuning of the topology via preferential attachment of edges on the basis of an arbitrary function Q(k,k') of the degrees of the vertices involved. We solve these models using finite connectivity equilibrium replica theory, within the replica symmetric ansatz. In our ensemble of graphs, phase diagrams of the spin system are found to depend no longer only on the chosen degree distribution, but also on the choice made for Q(k,k'). The increased ability to control interaction topology in solvable models beyond prescribing only the degree distribution of the interaction graph enables a more accurate modeling of real-world interacting particle systems by spin systems on suitably defined random graphs.Comment: 21 pages, 4 figures, submitted to J Phys

Slowly evolving random graphs II: Adaptive geometry in finite-connectivity Hopfield models

We present an analytically solvable random graph model in which the connections between the nodes can evolve in time, adiabatically slowly compared to the dynamics of the nodes. We apply the formalism to finite connectivity attractor neural network (Hopfield) models and we show that due to the minimisation of the frustration effects the retrieval region of the phase diagram can be significantly enlarged. Moreover, the fraction of misaligned spins is reduced by this effect, and is smaller than in the infinite connectivity regime. The main cause of this difference is found to be the non-zero fraction of sites with vanishing local field when the connectivity is finite.Comment: 17 pages, 8 figure

Replicated Transfer Matrix Analysis of Ising Spin Models on `Small World' Lattices

We calculate equilibrium solutions for Ising spin models on `small world' lattices, which are constructed by super-imposing random and sparse Poissonian graphs with finite average connectivity c onto a one-dimensional ring. The nearest neighbour bonds along the ring are ferromagnetic, whereas those corresponding to the Poisonnian graph are allowed to be random. Our models thus generally contain quenched connectivity and bond disorder. Within the replica formalism, calculating the disorder-averaged free energy requires the diagonalization of replicated transfer matrices. In addition to developing the general replica symmetric theory, we derive phase diagrams and calculate effective field distributions for two specific cases: that of uniform sparse long-range bonds (i.e. `small world' magnets), and that of (+J/-J) random sparse long-range bonds (i.e. `small world' spin-glasses).Comment: 22 pages, LaTeX, IOP macros, eps figure

On metastable configurations of small-world networks

We calculate the number of metastable configurations of Ising small-world networks which are constructed upon superimposing sparse Poisson random graphs onto a one-dimensional chain. Our solution is based on replicated transfer-matrix techniques. We examine the denegeracy of the ground state and we find a jump in the entropy of metastable configurations exactly at the crossover between the small-world and the Poisson random graph structures. We also examine the difference in entropy between metastable and all possible configurations, for both ferromagnetic and bond-disordered long-range couplings.Comment: 9 pages, 4 eps figure

Diluted antiferromagnet in a ferromagnetic enviroment

The question of robustness of a network under random ``attacks'' is treated in the framework of critical phenomena. The persistence of spontaneous magnetization of a ferromagnetic system to the random inclusion of antiferromagnetic interactions is investigated. After examing the static properties of the quenched version (in respect to the random antiferromagnetic interactions) of the model, the persistence of the magnetization is analysed also in the annealed approximation, and the difference in the results are discussed