arXiv subjects

Peter Sollich

Publications and source records attributed to Peter Sollich.

At least 145 records · Page 8Linked to original sources

arXiv2008

Duality between random trap and barrier models

We discuss the physical consequences of a duality between two models with quenched disorder, in which particles propagate in one dimension among random traps or across random barriers. We derive an exact relation between their diffusion fronts at fixed disorder, and deduce from this that their disorder-averaged diffusion fronts are exactly equal. We use effective dynamics schemes to isolate the different physical processes by which particles propagate in the models and discuss how the duality arises from a correspondence between the rates for these different processes.

cond-mat.stat-mech↗

arXiv2008

Phase Transition in a Random Minima Model: Mean Field Theory and Exact Solution on the Bethe Lattice

We consider the number and distribution of minima in random landscapes defined on non-Euclidean lattices. Using an ensemble where random landscapes are reweighted by a fugacity factor $z$ for each minimum they contain, we construct first a `two-box' mean field theory. This exhibits an ordering phase transition at $z\c=2$ above which one box contains an extensive number of minima. The onset of order is governed by an unusual order parameter exponent $β=1$, motivating us to study the same model on the Bethe lattice. Here we find from an exact solution that for any connectivity $μ+1>2$ there is an ordering transition with a conventional mean field order parameter exponent $β=1/2$, but with the region where this behaviour is observable shrinking in size as $1/μ$ in the mean field limit of large $μ$. We show that the behaviour in the transition region can also be understood directly within a mean field approach, by making the assignment of minima `soft'. Finally we demonstrate, in the simplest mean field case, how the analysis can be generalized to include both maxima and minima. In this case an additional first order phase transition appears, to a landscape in which essentially all sites are either minima or maxima.

cond-mat.dis-nn↗

arXiv2008

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.

cond-mat.soft↗

arXiv2008

Fluctuation-dissipation relations in critical coarsening: crossover from unmagnetized to magnetized initial states

We study the non-equilibrium dynamics of the spherical ferromagnet quenched to its critical temperature, as a function of the magnetization of the initial state. The two limits of unmagnetized and fully magnetized initial conditions can be understood as corresponding to times that are respectively much shorter and much longer than a magnetization timescale, as in a recent field theoretical analysis of the $n$-vector model. We calculate exactly the crossover functions interpolating between these two limits, for the magnetization correlator and response and the resulting fluctuation-dissipation ratio (FDR). For $d>4$ our results match those obtained recently from a Gaussian field theory. For $d<4$, non-Gaussian fuctuations arising from the spherical constraint need to be accounted for. We extend our framework from the fully magnetized case to achieve this, providing an exact solution for the relevant integral kernel. The resulting crossover behaviour is very rich, with the asymptotic FDR $X^\infty$ depending non-monotonically on the scaled age of the system. This is traced back to non-monotonicities of the two-time correlator, themselves the consequence of large magnetization fluctuations on the crossover timescale. We correct a trivial error in our earlier calculation for fully magnetized initial states; the corrected FDR is {\em consistent} with renormalization group expansions to first order in $4-d$ for the longitudinal fluctuations of the O(n) model in the limit $n\to\infty$.

cond-mat.dis-nn↗

arXiv2007

A polydisperse lattice-gas model

We describe a lattice-gas model suitable for studying the generic effects of polydispersity on liquid-vapor phase equilibria. Using Monte Carlo simulation methods tailored for the accurate determination of phase behaviour under conditions of fixed polydispersity, we trace the cloud and shadow curves for a particular Schulz distribution of the polydisperse attribute. Although polydispersity enters the model solely in terms of the strengths of the interparticle interactions, this is sufficient to induce the broad separation of cloud and shadow curves seen both in more realistic models and experiments.

cond-mat.stat-mech↗

arXiv2007

Non-equilibrium dynamics of spin facilitated glass models

We consider the dynamics of spin facilitated models of glasses in the non-equilibrium aging regime following a sudden quench from high to low temperatures. We briefly review known results obtained for the broad class of kinetically constrained models, and then present new results for the behaviour of the one-spin facilitated Fredrickson-Andersen and East models in various spatial dimensions. The time evolution of one-time quantities, such as the energy density, and the detailed properties of two-time correlation and response functions are studied using a combination of theoretical approaches, including exact mappings of master operators and reductions to integrable quantum spin chains, field theory and renormalization group, and independent interval and timescale separation methods. The resulting analytical predictions are confirmed by means of detailed numerical simulations. The models we consider are characterized by trivial static properties, with no finite temperature singularities, but they nevertheless display a surprising variety of dynamic behaviour during aging, which can be directly related to the existence and growth in time of dynamic lengthscales. Well-behaved fluctuation-dissipation ratios can be defined for these models, and we study their properties in detail. We confirm in particular the existence of negative fluctuation-dissipation ratios for a large number of observables. Our results suggest that well-defined violations of fluctuation-dissipation relations, of a purely dynamic origin and unrelated to the thermodynamic concept of effective temperatures, could in general be present in non-equilibrium glassy materials.

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arXiv2007

Aging in One-Dimensional Coagulation-Diffusion Processes and the Fredrickson-Andersen Model

We analyse the aging dynamics of the one-dimensional Fredrickson-Andersen (FA) model in the nonequilibrium regime following a low temperature quench. Relaxation then effectively proceeds via diffusion limited pair coagulation (DLPC) of mobility excitations. By employing a familiar stochastic similarity transformation, we map exact results from the free fermion case of diffusion limited pair annihilation to DLPC. Crucially, we are able to adapt the mapping technique to averages involving multiple time quantities. This relies on knowledge of the explicit form of the evolution operators involved. Exact results are obtained for two-time correlation and response functions in the free fermion DLPC process. The corresponding long-time scaling forms apply to a wider class of DLPC processes, including the FA model. We are thus able to exactly characterise the violations of the fluctuation-dissipation theorem (FDT) in the aging regime of the FA model. We find nontrivial scaling forms for the fluctuation-dissipation ratio (FDR) X = X(tw/t), but with a negative asymptotic value X = -3*pi/(6*pi - 16) = -3.307. While this prevents a thermodynamic interpretation in terms of an effective temperature, it is a direct consequence of probing FDT with observables that couple to activated dynamics. The existence of negative FDRs should therefore be a widespread feature in non mean-field systems.

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arXiv2006

Multicomponent adhesive hard sphere models and short-ranged attractive interactions in colloidal or micellar solutions

We investigate the dependence of the stickiness parameters $t_{ij}=1/(12τ_{ij})$ -- where the $τ_{ij}$ are the conventional Baxter parameters -- on the solute diameters $σ_{i}$ and $σ_{j}$ in multicomponent sticky hard sphere (SHS) models for fluid mixtures of mesoscopic neutral particles. A variety of simple but realistic interaction potentials, utilized in the literature to model \textit{short-ranged attractions} present in real solutions of colloids or reverse micelles, is reviewed. We consider: i) van der Waals attractions, ii) hard-sphere-depletion forces, iii) polymer-coated colloids, iv) solvation effects (in particular hydrophobic bonding and attractions between reverse micelles of water-in-oil microemulsions). We map each of these potentials onto an equivalent SHS model, by requiring the equality of the second virial coefficients. The main finding is that, for most of the potentials considered, the size-dependence of $t_{ij}(T,σ_{i},σ_{j})$ can be approximated by essentially the same expression, i.e. a simple polynomial in the variable $σ_{i}σ_{j}/σ_{ij}^{2}$, with coefficients depending on the temperature $T$, or -- for depletion interactions -- on the packing fraction $η_{0}$ of the depletant particles.

cond-mat.soft↗

arXiv2006

Wetting transitions in polydisperse fluids

The properties of the coexisting bulk gas and liquid phases of a polydisperse fluid depend not only on the prevailing temperature, but also on the overall parent density. As a result, a polydisperse fluid near a wall will exhibit density-driven wetting transitions inside the coexistence region. We propose a likely topology for the wetting phase diagram, which we test using Monte Carlo simulations of a model polydisperse fluid at an attractive wall, tracing the wetting line inside the cloud curve and identifying the relationship to prewetting.

cond-mat.stat-mech↗

arXiv2006

Phase behavior of weakly polydisperse sticky hard spheres: Perturbation theory for the Percus-Yevick solution

We study the effects of size polydispersity on the gas-liquid phase behaviour of mixtures of sticky hard spheres. To achieve this, the system of coupled quadratic equations for the contact values of the partial cavity functions of the Percus-Yevick solution is solved within a perturbation expansion in the polydispersity, i.e. the normalized width of the size distribution. This allows us to make predictions for various thermodynamic quantities which can be tested against numerical simulations and experiments. In particular, we determine the leading-order effects of size polydispersity on the cloud curve delimiting the region of two-phase coexistence and on the associated shadow curve; we also study the extent of size fractionation between the coexisting phases. Different choices for the size-dependence of the adhesion strengths are examined carefully; the Asakura-Oosawa model of a mixture of polydisperse colloids and small polymers is studied as a specific example.

cond-mat.soft↗

arXiv2006

Phase behaviour and particle-size cutoff effects in polydisperse fluids

We report a joint simulation and theoretical study of the liquid-vapor phase behaviour of a fluid in which polydispersity in the particle size couples to the strength of the interparticle interactions. Attention is focussed on the case in which the particles diameters are distributed according to a fixed Schulz form with degree of polydispersity $δ=14%$. The coexistence properties of this model are studied using grand canonical ensemble Monte Carlo simulations and moment free energy calculations. We obtain the cloud and shadow curves as well as the daughter phase density distributions and fractional volumes along selected isothermal dilution lines. In contrast to the case of size-{\em independent} interaction strengths (N.B. Wilding, M. Fasolo and P. Sollich, J. Chem. Phys. {\bf 121}, 6887 (2004)), the cloud and shadow curves are found to be well separated, with the critical point lying significantly below the cloud curve maximum. For densities below the critical value, we observe that the phase behaviour is highly sensitive to the choice of upper cutoff on the particle size distribution. We elucidate the origins of this effect in terms of extremely pronounced fractionation effects and discuss the likely appearance of new phases in the limit of very large values of the cutoff.

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arXiv2006

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.

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arXiv2006

Accurate simulation estimates of cloud points of polydisperse fluids

We describe two distinct approaches to obtaining 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 $μ(σ)$ (with $σ$ the polydisperse attribute) under the constraint that the ensemble average of the particle density distribution $ρ(σ)$ matches 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 $μ(σ)$ 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 $μ(σ)$. 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 {\em exponentially small} in the system size. The scaling predictions for both methods are tested via Monte Carlo simulations of a novel polydisperse lattice gas model near its cloud curve, the results showing excellent quantitative agreement with the theory.

cond-mat.stat-mech↗

arXiv2006

Mappings between reaction-diffusion and kinetically constrained systems: A+A <-> A and the FA model have upper critical dimension d_c=2

We present an exact mapping between two simple spin models: the Fredrickson-Andersen (FA) model and a model of annihilating random walks with spontaneous creation from the vacuum, A+A <-> 0. We discuss the geometric structure of the mapping and its consequences for symmetries of the models. Hence we are able to show that the upper critical dimension of the FA model is two, and that critical exponents are known exactly in all dimensions. These conclusions also generalise to a mapping between A+A <-> 0 and the reaction-diffusion system in which the reactions are branching and coagulation, A+A <-> A. We discuss the relation of our analysis to earlier work, and explain why the models considered do not fall into the directed percolation universality class.

cond-mat.stat-mech↗

arXiv2005

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 $κ$. 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.

cond-mat.stat-mech↗

arXiv2005

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

We study the liquid-vapor phase behaviour 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 to detect the cloud point. We describe a finite-size scaling method which nevertheless permits accurate determination of cloud points and spinodals 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.

cond-mat.stat-mech↗

arXiv2005

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 non-equilibrium dynamics. We analyze in detail the non-equilibrium fluctuation-dissipation ratio X(t,tw) associated with a defect-pair observable in the Glauber-Ising spin chain. It turns out that $X \neq 1$ throughout the short-time regime and in particular X(tw,tw) = 3/4 for $tw \to \infty$. The analysis is extended to observables detecting defects at a finite distance from each other, where similar violations of quasi-equilibrium behaviour 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.

cond-mat.stat-mech↗

arXiv2005

Dynamic Heterogeneity in the Glauber-Ising chain

In a recent paper [P. Mayer et al., Phys. Rev. Lett. 93, 115701 (2004)] it was shown, by means of experiments, theory and simulations, that coarsening systems display dynamic heterogeneity analogous to that of glass formers. Here, we present a detailed analysis of dynamic heterogeneities in the Glauber-Ising chain. We discuss how dynamic heterogeneity in Ising systems must be measured through connected multi-point correlation functions. We show that in the coarsening regime of the Ising chain these multi-point functions reveal the growth of spatial correlations in the dynamics, beyond what can be inferred from standard two-point correlations. They have non-trivial scaling properties, which we interpret in terms of the diffusion-annihilation dynamics of domain walls. In the equilibrium dynamics of the Ising chain, on the other hand, connected multi-point functions vanish exactly and dynamic heterogeneity is not observed. Our results highlight the similarities between coarsening systems and glass formers.

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