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

Isotropic-nematic phase equilibria of polydisperse hard rods: The effect of fat tails in the length distribution

We study the phase behaviour of hard rods with length polydispersity, treated within a simplified version of the Onsager model. We give a detailed description of the unusual phase behaviour of the system when the rod length distribution has a "fat" (e.g. log-normal) tail up to some finite cutoff. The relatively large number of long rods in the system strongly influences the phase behaviour: the isotropic cloud curve, which defines the where a nematic phase first occurs as density is increased, exhibits a kink; at this point the properties of the coexisting nematic shadow phase change discontinuously. A narrow three-phase isotropic-nematic-nematic coexistence region exists near the kink in the cloud curve, even though the length distribution is unimodal. A theoretical derivation of the isotropic cloud curve and nematic shadow curve, in the limit of large cutoff, is also given. The two curves are shown to collapse onto each other in the limit. The coexisting isotropic and nematic phases are essentially identical, the only difference being that the nematic contains a larger number of the longest rods; the longer rods are also the only ones that show any significant nematic ordering. Numerical results for finite but large cutoff support the theoretical predictions for the asymptotic scaling of all quantities with the cutoff length.Comment: 21 pages, 13 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

Liquid-gas coexistence and critical point shifts in size-disperse fluids

Specialized Monte Carlo simulations and the moment free energy (MFE) method are employed to study liquid-gas phase equilibria in size-disperse fluids. The investigation is made subject to the constraint of fixed polydispersity, i.e. the form of the `parent' density distribution $\rho^0(\sigma)$ of the particle diameters $\sigma$, is prescribed. This is the experimentally realistic scenario for e.g. colloidal dispersions. The simulations are used to obtain the cloud and shadow curve properties of a Lennard-Jones fluid having diameters distributed according to a Schulz form with a large (40%) degree of polydispersity. Good qualitative accord is found with the results from a MFE method study of a corresponding van der Waals model that incorporates size-dispersity both in the hard core reference and the attractive parts of the free energy. The results show that polydispersity engenders considerable broadening of the coexistence region between the cloud curves. The principal effect of fractionation in this region is a common overall scaling of the particle sizes and typical inter-particle distances, and we discuss why this effect is rather specific to systems with Schulz diameter distributions. Next, by studying a family of such systems with distributions of various widths, we estimate the dependence of the critical point parameters on $\delta$. In contrast to a previous theoretical prediction, size-dispersity is found to raise the critical temperature above its monodisperse value. Unusually for a polydisperse system, the critical point is found to lie at or very close to the extremum of the coexistence region in all cases. We outline an argument showing that such behaviour will occur whenever size polydispersity affects only the range, rather than the strength of the inter-particle interactions.Comment: 14 pages, 12 figure

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

Trap models with slowly decorrelating observables

We study the correlation and response dynamics of trap models of glassy dynamics, considering observables that only partially decorrelate with every jump. This is inspired by recent work on a microscopic realization of such models, which found strikingly simple linear out-of-equilibrium fluctuation-dissipation relations in the limit of slow decorrelation. For the Barrat-Mezard model with its entropic barriers we obtain exact results at zero temperature $T$ for arbitrary decorrelation factor $\kappa$. These are then extended to nonzero $T$, where the qualitative scaling behaviour and all scaling exponents can still be found analytically. Unexpectedly, the choice of transition rates (Glauber versus Metropolis) affects not just prefactors but also some exponents. In the limit of slow decorrelation even complete scaling functions are accessible in closed form. The results show that slowly decorrelating observables detect persistently slow out-of-equilibrium dynamics, as opposed to intermittent behaviour punctuated by excursions into fast, effectively equilibrated states.Comment: 29 pages, IOP styl