Nematic-nematic demixing in polydisperse thermotropic liquid crystals

We consider the effects of polydispersity on isotropic-nematic phase equilibria in thermotropic liquid crystals, using a Maier-Saupe theory with factorized interactions. A sufficient spread (approx. 50%) in the interaction strengths of the particles leads to phase separation into two or more nematic phases, which can in addition coexist with an isotropic phase. The isotropic-nematic coexistence region widens dramatically as polydispersity is increased, leading to re-entrant isotropic-nematic phase separation in some regions of the phase diagram. We show that similar phenomena will occur also for non-factorized interactions as long as the interaction strength between any two particle species is lower than the mean of the intra-species interactions.Comment: 6 pages, revtex4, 4 figures include

Weakly polydisperse systems: Perturbative phase diagrams that include the critical region

The phase behaviour of a weakly polydisperse system, such as a colloid with a small spread of particle sizes, can be related perturbatively to that of its monodisperse counterpart. I show how this approach can be generalized to remain well-behaved near critical points, avoiding the divergences of existing methods and giving access to some of the key qualitative features of polydisperse phase equilibria. The analysis explains also why in purely size polydisperse systems the critical point is, unusually, located very near the maximum of the cloud and shadow curves.Comment: 4.1 pages. Revised version, as published: expanded discussion of Fisher renormalization for systems with non-classifical critical exponents; coefficients "a" and "b" re-defined to simplify statement of critical point shifts and cloud/shadow curve slope

Gaussian Process Regression with Mismatched Models

Learning curves for Gaussian process regression are well understood when the `student' model happens to match the `teacher' (true data generation process). I derive approximations to the learning curves for the more generic case of mismatched models, and find very rich behaviour: For large input space dimensionality, where the results become exact, there are universal (student-independent) plateaux in the learning curve, with transitions in between that can exhibit arbitrarily many over-fitting maxima. In lower dimensions, plateaux also appear, and the asymptotic decay of the learning curve becomes strongly student-dependent. All predictions are confirmed by simulations.Comment: 7 pages, style file nips01e.sty include

General Solutions for Multispin Two-Time Correlation and Response Functions in the Glauber-Ising Chain

The kinetic Glauber-Ising spin chain is one of the very few exactly solvable models of non-equilibrium statistical mechanics. Nevertheless, existing solutions do not yield tractable expressions for two-time correlation and response functions of observables involving products of more than one or two spins. We use a new approach to solve explicitly the full hierarchy of differential equations for the correlation and response functions. From this general solution follow closed expressions for arbitrary multispin two-time correlation and response functions, for the case where the system is quenched from equilibrium at T_i > 0 to some arbitrary T >= 0. By way of application, we give the results for two and four-spin two-time correlation and response functions. From the standard mapping, these also imply new exact results for two-time particle correlation and response functions in one-dimensional diffusion limited annihilation.Comment: 35 Pages, 4 Figure

Dynamical selection of Nash equilibria using Experience Weighted Attraction Learning: emergence of heterogeneous mixed equilibria

We study the distribution of strategies in a large game that models how agents choose among different double auction markets. We classify the possible mean field Nash equilibria, which include potentially segregated states where an agent population can split into subpopulations adopting different strategies. As the game is aggregative, the actual equilibrium strategy distributions remain undetermined, however. We therefore compare with the results of Experience-Weighted Attraction (EWA) learning, which at long times leads to Nash equilibria in the appropriate limits of large intensity of choice, low noise (long agent memory) and perfect imputation of missing scores (fictitious play). The learning dynamics breaks the indeterminacy of the Nash equilibria. Non-trivially, depending on how the relevant limits are taken, more than one type of equilibrium can be selected. These include the standard homogeneous mixed and heterogeneous pure states, but also \emph{heterogeneous mixed} states where different agents play different strategies that are not all pure. The analysis of the EWA learning involves Fokker-Planck modeling combined with large deviation methods. The theoretical results are confirmed by multi-agent simulations.Comment: 35 pages, 16 figure

Inference for dynamics of continuous variables: the Extended Plefka Expansion with hidden nodes

We consider the problem of a subnetwork of observed nodes embedded into a larger bulk of unknown (i.e. hidden) nodes, where the aim is to infer these hidden states given information about the subnetwork dynamics. The biochemical networks underlying many cellular and metabolic processes are important realizations of such a scenario as typically one is interested in reconstructing the time evolution of unobserved chemical concentrations starting from the experimentally more accessible ones. We present an application to this problem of a novel dynamical mean field approximation, the Extended Plefka Expansion, which is based on a path integral description of the stochastic dynamics. As a paradigmatic model we study the stochastic linear dynamics of continuous degrees of freedom interacting via random Gaussian couplings. The resulting joint distribution is known to be Gaussian and this allows us to fully characterize the posterior statistics of the hidden nodes. In particular the equal-time hidden-to-hidden variance -- conditioned on observations -- gives the expected error at each node when the hidden time courses are predicted based on the observations. We assess the accuracy of the Extended Plefka Expansion in predicting these single node variances as well as error correlations over time, focussing on the role of the system size and the number of observed nodes.Comment: 30 pages, 6 figures, 1 Appendi

Error counting in a quantum error-correcting code and the ground-state energy of a spin glass

Upper and lower bounds are given for the number of equivalence classes of error patterns in the toric code for quantum memory. The results are used to derive a lower bound on the ground-state energy of the +/-J Ising spin glass model on the square lattice with symmetric and asymmetric bond distributions. This is a highly non-trivial example in which insights from quantum information lead directly to an explicit result on a physical quantity in the statistical mechanics of disordered systems.Comment: 15 pages, 7 figures, JPSJ style, latex style file include

Rectification of asymmetric surface vibrations with dry friction: an exactly solvable model

We consider a stochastic model for the directed motion of a solid object due to the rectification of asymmetric surface vibrations with Poissonian shot-noise statistics. The friction between the object and the surface is given by a piecewise-linear friction force. This models the combined effect of dynamic friction and singular dry friction. We derive an exact solution of the stationary Kolmogorov-Feller (KF) equation in the case of two-sided exponentially distributed amplitudes. The stationary density of the velocity exhibits singular features such as a discontinuity and a delta-peak singularity at zero velocity, and also contains contributions from non-integrable solutions of the KF equation. The mean velocity in our model generally varies non-monotonically as the strength of the dry friction is increased, indicating that transport improves for increased dissipation.Comment: 9 pages, 5 figure

Glassy dynamics of kinetically constrained models

We review the use of kinetically constrained models (KCMs) for the study of dynamics in glassy systems. The characteristic feature of KCMs is that they have trivial, often non-interacting, equilibrium behaviour but interesting slow dynamics due to restrictions on the allowed transitions between configurations. The basic question which KCMs ask is therefore how much glassy physics can be understood without an underlying ``equilibrium glass transition''. After a brief review of glassy phenomenology, we describe the main model classes, which include spin-facilitated (Ising) models, constrained lattice gases, models inspired by cellular structures such as soap froths, models obtained via mappings from interacting systems without constraints, and finally related models such as urn, oscillator, tiling and needle models. We then describe the broad range of techniques that have been applied to KCMs, including exact solutions, adiabatic approximations, projection and mode-coupling techniques, diagrammatic approaches and mappings to quantum systems or effective models. Finally, we give a survey of the known results for the dynamics of KCMs both in and out of equilibrium, including topics such as relaxation time divergences and dynamical transitions, nonlinear relaxation, aging and effective temperatures, cooperativity and dynamical heterogeneities, and finally non-equilibrium stationary states generated by external driving. We conclude with a discussion of open questions and possibilities for future work.Comment: 137 pages. Additions to section on dynamical heterogeneities (5.5, new pages 110 and 112), otherwise minor corrections, additions and reference updates. Version to be published in Advances in Physic