Biomedical subjects

Peter Sollich

Publications and source records attributed to Peter Sollich.

14 recordsLinked to original sources

PubMed2006

Simulation estimates of cloud points of polydisperse fluids.

We describe two distinct approaches to obtaining the cloud-point densities and coexistence properties of polydisperse fluid mixtures by Monte Carlo simulation within the grand-canonical ensemble. The first method determines the chemical potential distribution mu(sigma) (with the polydisperse attribute) under the constraint that the ensemble average of the particle density distribution rho(sigma) match a prescribed parent form. Within the region of phase coexistence (delineated by the cloud curve) this leads to a distribution of the fluctuating overall particle density n, p(n), that necessarily has unequal peak weights in order to satisfy a generalized lever rule. A theoretical analysis shows that as a consequence, finite-size corrections to estimates of coexistence properties are power laws in the system size. The second method assigns mu(sigma) such that an equal-peak-weight criterion is satisfied for p(n) for all points within the coexistence region. However, since equal volumes of the coexisting phases cannot satisfy the lever rule for the prescribed parent, their relative contributions must be weighted appropriately when determining mu(sigma). We show how to ascertain the requisite weight factor operationally. A theoretical analysis of the second method suggests that it leads to finite-size corrections to estimates of coexistence properties which are exponentially small in the system size. The scaling predictions for both methods are tested via Monte Carlo simulations of a polydisperse lattice-gas model near its cloud curve, the results showing excellent quantitative agreement with the theory.

Journal Article↗

PubMed2006

Activated aging dynamics and negative fluctuation-dissipation ratios.

In glassy materials, aging proceeds at large times via thermal activation. We show that this can lead to negative dynamical response functions and novel and well-defined violations of the fluctuation-dissipation theorem, in particular, negative fluctuation-dissipation ratios. Our analysis is based on detailed theoretical and numerical results for the activated aging regime of simple kinetically constrained models. The results are relevant to a variety of physical situations, such as aging in glass formers, thermally activated domain growth, and granular compaction.

Journal Article↗

PubMed2005

Finite-size scaling and particle-size cutoff effects in phase-separating polydisperse fluids.

We study the liquid-vapor phase behavior of a polydisperse fluid using grand canonical simulations and moment free energy calculations. The strongly nonlinear variation of the fractional volume of liquid across the coexistence region prevents naive extrapolation from detecting the cloud point. We describe a finite-size scaling method which, nevertheless, permits accurate determination of cloud points from simulations of a single system size. By varying a particle-size cutoff, we find that the cloud point density is highly sensitive to the presence of rare large particles; this could affect the reproducibility of experimentally measured phase behavior in colloids and polymers.

Journal Article↗

PubMed2005

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 [Z. Naturforsch. A 13A, 564 (1958)] with factorized interactions. A sufficient spread (approximately 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 reentrant isotropic-nematic phase separation in some regions of the phase diagram. We show that similar phenomena will occur also for nonfactorized interactions as long as the interaction strength between any two particle species is lower than the mean of the intraspecies interactions.

Journal Article↗

PubMed2005

Observable dependent quasi-equilibrium in slow dynamics.

We present examples demonstrating that quasi-equilibrium fluctuation-dissipation behavior at short time differences is not a generic feature of systems with slow nonequilibrium dynamics. We analyze in detail the nonequilibrium fluctuation-dissipation ratio X(t, t(w)) associated with a defect-pair observable in the Glauber-Ising spin chain. It turns out that X not = 1 throughout the short-time regime and in particular X(t(w), t(w)) = 3/4 for t(w) --> infinity. The analysis is extended to observables detecting defects at a finite distance from each other, where similar violations of quasi-equilibrium behavior are found. We discuss our results in the context of metastable states, which suggests that a violation of short-time quasi-equilibrium behavior could occur in general glassy systems for appropriately chosen observables.

Journal Article↗

PubMed2005

Effects of colloid polydispersity on the phase behavior of colloid-polymer mixtures.

We study theoretically the equilibrium phase behavior of a mixture of polydisperse hard-sphere colloids and monodisperse polymers, modeled using the Asakura-Oosawa model [S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255 (1954)] within the free volume approximation of H. N. W. Lekkerkerker, W. C. K. Poon, P. N. Pusey, A. Stroobants, and P. B. Warren [Europhys. Lett. 20, 559 (1992)]. We compute full phase diagrams in the plane of colloid and polymer volume fractions, using the moment free energy method. The intricate features of phase separation in pure polydisperse colloids combine with the appearance of polymer-induced gas-liquid coexistence to give a rich variety of phase diagram topologies as the polymer-colloid size ratio xi and the colloid polydispersity delta are varied. Quantitatively, we find that polydispersity disfavors fluid-solid against gas-liquid separation, causing a substantial lowering of the threshold value xi(c) above which stable two-phase gas-liquid coexistence appears. Phase splits involving two or more solids can occur already at low colloid concentration, where they may be kinetically accessible. We also analyze the strength of colloidal size fractionation. When a solid phase separates from a fluid, its polydispersity is reduced most strongly if the phase separation takes place at low colloid concentration and high polymer concentration, in agreement with experimental observations. For fractionation in gas-liquid coexistence we likewise find good agreement with experiment, as well as with perturbative theories for near-monodisperse systems.

Journal Article↗

PubMed2004

Fractionation effects in phase equilibria of polydisperse hard-sphere colloids.

The equilibrium phase behavior of hard spheres with size polydispersity is studied theoretically. We solve numerically the exact phase equilibrium equations that result from accurate free energy expressions for the fluid and solid phases, while accounting fully for size fractionation between coexisting phases. Fluids up to the largest polydispersities that we can study (around 14%) can phase separate by splitting off a solid with a much narrower size distribution. This shows that experimentally observed terminal polydispersities above which phase separation no longer occurs must be due to nonequilibrium effects. We find no evidence of reentrant melting; instead, sufficiently compressed solids phase separate into two or more solid phases. Under appropriate conditions, coexistence of multiple solids with a fluid phase is also predicted. The solids have smaller polydispersities than the parent phase as expected, while the reverse is true for the fluid phase, which contains predominantly smaller particles but also residual amounts of the larger ones. The properties of the coexisting phases are studied in detail; mean diameter, polydispersity, and volume fraction of the phases all reveal marked fractionation. We also propose a method for constructing quantities that optimally distinguish between the coexisting phases, using principal component analysis in the space of density distributions. We conclude by comparing our predictions to Monte Carlo simulations at imposed chemical potential distribution, and find excellent agreement.

Journal Article↗

PubMed2004

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 (delta approximately 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 interparticle 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 behavior will occur whenever polydispersity affects only the range, rather than the strength of the interparticle interactions.

Journal Article↗

PubMed2003

Equilibrium phase behavior of polydisperse hard spheres.

We calculate the phase behavior of hard spheres with size polydispersity, using accurate free energies for the fluid and solid phases. Cloud and shadow curves are found exactly by the moment free energy method, but we also compute the complete phase diagram, taking full account of fractionation. In contrast to earlier, simplified treatments we find no point of equal concentration between fluid and solid or reentrant melting at higher densities. Rather, the fluid cloud curve continues to the largest polydispersity that we study (14%); from the equilibrium phase behavior a terminal polydispersity can thus be defined only for the solid, where we find it to be around 7%. At sufficiently large polydispersity, fractionation into several solid phases can occur, consistent with previous approximate calculations; we find, in addition, that coexistence of several solids with a fluid phase is also possible.

Journal Article↗

PubMed2003

Fluctuation-dissipation relations in the nonequilibrium critical dynamics of Ising models.

We investigate the relation between two-time multispin correlation and response functions in the nonequilibrium critical dynamics of Ising models in d=1 and d=2 spatial dimensions. In these nonequilibrium situations, the fluctuation-dissipation theorem (FDT) is not satisfied. We find FDT "violations" qualitatively similar to those reported in various glassy materials, but quantitatively dependent on the chosen observable, in contrast to the results obtained in infinite-range glass models. Nevertheless, all FDT violations can be understood by considering separately the contributions from large wave vectors, which are at quasiequilibrium and obey the FDT, and from small wave vectors where a generalized FDT holds with a nontrivial fluctuation-dissipation ratio X infinity. In d=1, we get X(infinity)=1/2 for spin observables, which measure the orientation of domains, while X(infinity)=0 for observables that are sensitive to the domain-wall motion. Numerical simulations in d=2 reveal a unique X infinity approximately equal 0.34 for all observables. Measurement protocols for X infinity are discussed in detail. Our results suggest that the definition of an effective temperature T(eff)=T/X(infinity) for large length scales is generically possible in nonequilibrium critical dynamics.

Journal Article↗

PubMed2003

Isotropic-nematic phase equilibria in the Onsager theory of hard rods with length polydispersity.

We analyze the effect of a continuous spread of particle lengths on the phase behavior of rodlike particles, using the Onsager theory of hard rods. Our aim is to establish whether "unusual" effects such as isotropic-nematic-nematic (I-N-N) phase separation can occur even for length distributions with a single peak. We focus on the onset of I-N coexistence. For a log-normal distribution, we find that a finite upper cutoff on rod lengths is required to make this problem well posed. The cloud curve, which tracks the density at the onset of I-N coexistence as a function of the width of the length distribution, exhibits a kink; this demonstrates that the phase diagram must contain a three-phase I-N-N region. Theoretical analysis shows that in the limit of large cutoff, the cloud point density actually converges to zero, so that phase separation results at any nonzero density; this conclusion applies to all length distributions with fatter-than-exponentail tails. Finally, we consider the case of a Schulz distribution, with its exponential tail. Surprisingly, even here the long rods (and hence the cutoff) can dominate the phase behavior, and a kink in the cloud curve and I-N-N coexistence again result. Theory establishes that there is a nonzero threshold for the width of the length distribution above which these long-rod effects occur, and shows that the cloud and shadow curves approach nonzero limits for a large cutoff, both in good agreement with the numerical results.

Journal Article↗

PubMed2002

Observable dependence of fluctuation-dissipation relations and effective temperatures.

We study the nonequilibrium fluctuation-dissipation theorem (FDT) in the glass phase of Bouchaud's trap model. We incorporate an arbitrary observable m and obtain its correlation and response functions in closed form. A limiting nonequilibrium FDT plot is approached at long times for most choices of m. In contrast to standard mean field models, however, the shape of the plot depends nontrivially on the observable, and its slope varies continuously even though there is a single scaling of relaxation times with age. Nonequilibrium FDT plots can therefore not be used to define a meaningful effective temperature T(eff) in this model. Consequences for the wider applicability of an FDT-derived T(eff) are discussed.

Journal Article↗

PubMed2002

Learning curves for Gaussian process regression: approximations and bounds.

We consider the problem of calculating learning curves (i.e., average generalization performance) of gaussian processes used for regression. On the basis of a simple expression for the generalization error, in terms of the eigenvalue decomposition of the covariance function, we derive a number of approximation schemes. We identify where these become exact and compare with existing bounds on learning curves; the new approximations, which can be used for any input space dimension, generally get substantially closer to the truth. We also study possible improvements to our approximations. Finally, we use a simple exactly solvable learning scenario to show that there are limits of principle on the quality of approximations and bounds expressible solely in terms of the eigenvalue spectrum of the covariance function.

Journal Article↗

PubMedDate not supplied

Bayesian approach to feature selection and parameter tuning for support vector machine classifiers.

A Bayesian point of view of SVM classifiers allows the definition of a quantity analogous to the evidence in probabilistic models. By maximizing this one can systematically tune hyperparameters and, via automatic relevance determination (ARD), select relevant input features. Evidence gradients are expressed as averages over the associated posterior and can be approximated using Hybrid Monte Carlo (HMC) sampling. We describe how a Nyström approximation of the Gram matrix can be used to speed up sampling times significantly while maintaining almost unchanged classification accuracy. In experiments on classification problems with a significant number of irrelevant features this approach to ARD can give a significant improvement in classification performance over more traditional, non-ARD, SVM systems. The final tuned hyperparameter values provide a useful criterion for pruning irrelevant features, and we define a measure of relevance with which to determine systematically how many features should be removed. This use of ARD for hard feature selection can improve classification accuracy in non-ARD SVMs. In the majority of cases, however, we find that in data sets constructed by human domain experts the performance of non-ARD SVMs is largely insensitive to the presence of some less relevant features. Eliminating such features via ARD then does not improve classification accuracy, but leads to impressive reductions in the number of features required, by up to 75%.

Algorithms↗