arXiv subjects

Peter Sollich

Publications and source records attributed to Peter Sollich.

At least 73 records · Page 4Linked to original sources

arXiv2019

Phase separation of mixtures after a second quench: composition heterogeneities

We investigate binary mixtures undergoing phase separation after a second (deeper) temperature quench into two- and three-phase coexistence regions. The analysis is based on a lattice theory previously developed for gas-liquid separation in generic mixtures. Our previous results, which considered an arbitrary number of species and a single quench, showed that, due to slow changes in composition, dense colloidal mixtures can phase-separate in two stages. Moreover, the denser phase contains long-lived composition heterogeneities that originate as the interfaces of shrunk domains. Here we predict several new effects that arise after a second quench, mostly associated with the extent to which crowding can slow down 'fractionation', i.e. equilibration of compositions. They include long-lived regular arrangements of secondary domains; wetting of fractionated interfaces by oppositely fractionated layers; 'surface'-directed spinodal 'waves' propagating from primary interfaces; a 'dead zone' where no phase separation occurs; and, in the case of three-phase coexistence, filamentous morphologies arising out of secondary domains.

cond-mat.soft↗

arXiv2019

Slow coarsening in jammed athermal soft particle suspensions

We simulate a densely jammed, athermal assembly of repulsive soft particles immersed in a solvent. Starting from an initial condition corresponding to a quench from a high temperature, we find non-trivial slow dynamics driven by a gradual release of stored elastic energy, with the root mean squared particle speed decaying as a power law in time with a fractional exponent. This decay is accompanied by the presence within the assembly of spatially localised and temporally intermittent `hot-spots' of non-affine deformation, connected by long-ranged swirls in the velocity field, reminiscent of the local plastic events and long-ranged elastic propagation that have been intensively studied in sheared amorphous materials. The pattern of hot-spots progressively coarsens, with the hot-spot size and separation slowly growing over time, and the associated velocity correlation length increasing as a sublinear power law. Each individual spot however exists only transiently, within an overall picture of strongly intermittent dynamics.

cond-mat.soft↗

arXiv2019

Exploring the link between crystal defects and non-affine displacement fluctuations

We generalize, and then use, a recently introduced formalism to study thermal fluctuations of atomic displacements in several two and three dimensional crystals. We study both close packed as well as open crystals with multi atom bases. Atomic displacement fluctuations in a solid, once coarse-grained over some neighborhood may be decomposed into two mutually orthogonal components. In any dimension $d$ there are always $d^2$ {\em affine} displacements representing local strains and rotations of the ideal reference configuration. In addition, there exists a number of {\em non-affine} localized displacement modes that cannot be represented as strains or rotations. The number of these modes depends on $d$ and the size of the coarse graining region. All thermodynamic averages and correlation functions concerning the affine and non-affine displacements may be computed within a harmonic theory. We show that for compact crystals, such as the square and triangular in $d=2$ and the simple, body-centered and face-centered cubic crystals in $d=3$, a single set of $d-$fold degenerate modes always dominate the non-affine sub-space and are separated from the rest by a large gap. These modes may be identified with specific precursor configurations that lead to lattice defects. Deformation mechanisms such as lattice slips and stacking faults in close packed crystals can also be understood within this framework. The qualitative features of these conclusions are expected to be independent of the details of the atomic interactions.

cond-mat.stat-mech↗

arXiv2019

Market fragmentation and market consolidation: Multiple steady states in systems of adaptive traders choosing where to trade

Technological progress is leading to proliferation and diversification of trading venues, thus increasing the relevance of the long-standing question of market fragmentation versus consolidation. To address this issue quantitatively, we analyse systems of adaptive traders that choose where to trade based on their previous experience. We demonstrate that only based on aggregate parameters about trading venues, such as the demand to supply ratio, we can assess whether a population of traders will prefer fragmentation or specialization towards a single venue. We investigate what conditions lead to market fragmentation for populations with a long memory and analyse the stability and other properties of both fragmented and consolidated steady states. Finally we investigate the dynamics of populations with finite memory; when this memory is long the true long-time steady states are consolidated but fragmented states are strongly metastable, dominating the behaviour out to long times.

q-fin.TR↗

arXiv2019

Glassy dynamics on networks: local spectra and return probabilities

The slow relaxation and aging of glassy systems can be modelled as a Markov process on a simplified rough energy landscape: energy minima where the system tends to get trapped are taken as nodes of a random network, and the dynamics are governed by the transition rates among these. In this work we consider the case of purely activated dynamics, where the transition rates only depend on the depth of the departing trap. The random connectivity and the disorder in the trap depths make it impossible to solve the model analytically, so we base our analysis on the spectrum of eigenvalues $λ$ of the master operator. We compute the local density of states $ρ(λ|τ)$ for traps with a fixed lifetime $τ$ by means of the cavity method. This exhibits a power law behaviour $ρ(λ|τ)\simτ|λ|^T$ in the regime of small relaxation rates $|λ|$, which we rationalize using a simple analytical approximation. In the time domain, we find that the probabilities of return to a starting node have a power law-tail that is determined by the distribution of excursion times $F(t)\sim t^{-(T+1)}$. We show that these results arise only by the combination of finite configuration space connectivity and glassy disorder, and interpret them in a simple physical picture dominated by jumps to deep neighbouring traps.

cond-mat.dis-nn↗

arXiv2018

Plastic deformation of a permanently bonded network: stress relaxation by pleats

We show that a flat two dimensional network of connected vertices, when stretched, may deform plastically by producing `pleats'; system spanning linear structures with width comparable to the lattice spacing, where the network overlaps on itself. To understand the pleating process, we introduce an external field that couples to local {\em non-affine} displacements, i.e. those displacements of neighbouring vertices that cannot be represented as a local affine strain. We obtain both zero and finite temperature phase diagrams in the strain -- field plane. Pleats occur here as a result of an equilibrium first-order transition from the homogeneous network to a heterogeneous phase where stress is localised within pleats and eliminated elsewhere. We show that in the thermodynamic limit the un-pleated state is always metastable at vanishing field for infinitesimal strain. Plastic deformation of the initially homogeneous network is akin to the decay of a metastable phase via a dynamical transition. We make predictions concerning local stress distributions and thermal effects associated with pleats which may be observable in suitable experimental systems.

cond-mat.soft↗

arXiv2018

Critical phase behavior in multi-component fluid mixtures: Complete scaling analysis

We analyze the critical gas-liquid phase behavior of arbitrary fluid mixtures in their coexistence region. We focus on the setting relevant for polydisperse colloids, where the overall density and composition of the system are being controlled, in addition to temperature. Our analysis uses the complete scaling formalism and thus includes pressure mixing effects in the mapping from thermodynamic fields to the effective fields of 3D Ising criticality. Because of fractionation, where mixture components are distributed unevenly across coexisting phases, the critical behavior is remarkably rich. We give scaling laws for a number of important loci in the phase diagram. These include the cloud and shadow curves, which characterise the onset of phase coexistence, a more general set of curves defined by fixing the fractional volumes of the coexisting phases to arbitrary values, and conventional coexistence curves of the densities of coexisting phases for fixed overall density. We identify suitable observables (distinct from the Yang-Yang anomalies discussed in the literature) for detecting pressure mixing effects. Our analytical predictions are checked against numerics using a set of mapping parameters fitted to simulation data for a polydisperse Lennard-Jones fluid, allowing us to highlight crossovers where pressure mixing becomes relevant close to the critical point.

cond-mat.soft↗

arXiv2018

Spectral properties of the trap model on sparse networks

One of the simplest models for the slow relaxation and aging of glasses is the trap model by Bouchaud and others, which represents a system as a point in configuration-space hopping between local energy minima. The time evolution depends on the transition rates and the network of allowed jumps between the minima. We consider the case of sparse configuration-space connectivity given by a random graph, and study the spectral properties of the resulting master operator. We develop a general approach using the cavity method that gives access to the density of states in large systems, as well as localisation properties of the eigenvectors, which are important for the dynamics. We illustrate how, for a system with sparse connectivity and finite temperature, the density of states and the average inverse participation ratio have attributes that arise from a non-trivial combination of the corresponding mean field (fully connected) and random walk (infinite temperature) limits. In particular, we find a range of eigenvalues for which the density of states is of mean-field form but localisation properties are not, and speculate that the corresponding eigenvectors may be concentrated on extensively many clusters of network sites.

cond-mat.dis-nn↗

arXiv2018

Memory functions reveal structural properties of gene regulatory networks

Gene regulatory networks (GRNs) control cellular function and decision making during tissue development and homeostasis. Mathematical tools based on dynamical systems theory are often used to model these networks, but the size and complexity of these models mean that their behaviour is not always intuitive and the underlying mechanisms can be difficult to decipher. For this reason, methods that simplify and aid exploration of complex networks are necessary. To this end we develop a broadly applicable form of the Zwanzig-Mori projection. By first converting a thermodynamic state ensemble model of gene regulation into mass action reactions we derive a general method that produces a set of time evolution equations for a subset of components of a network. The influence of the rest of the network, the bulk, is captured by memory functions that describe how the subnetwork reacts to its own past state via components in the bulk. These memory functions provide probes of near-steady state dynamics, revealing information not easily accessible otherwise. We illustrate the method on a simple cross-repressive transcriptional motif to show that memory functions not only simplify the analysis of the subnetwork but also have a natural interpretation. We then apply the approach to a GRN from the vertebrate neural tube, a well characterised developmental transcriptional network composed of four interacting transcription factors. The memory functions reveal the function of specific links within the neural tube network and identify features of the regulatory structure that specifically increase the robustness of the network to initial conditions. Taken together, the study provides evidence that Zwanzig-Mori projections offer powerful and effective tools for simplifying and exploring the behaviour of GRNs.

q-bio.MN↗

arXiv2018

Do thermodynamically stable rigid solids exist?

Customarily, crystalline solids are defined to be {\em rigid} since they resist changes of shape determined by their boundaries. However, rigid solids cannot exist in the thermodynamic limit where boundaries become irrelevant. Particles in the solid may rearrange to adjust to shape changes eliminating stress without destroying crystalline order. Rigidity is therefore valid only in the {\em metastable} state that emerges because these particle rearrangements in response to a deformation, or strain, are associated with slow collective processes. Here, we show that a thermodynamic collective variable may be used to quantify particle rearrangements that occur as a solid is deformed at zero strain rate. Advanced Monte Carlo simulation techniques are then employed to obtain the equilibrium free energy as a function of this variable. Our results lead to a new view on rigidity: While at zero strain a rigid crystal coexists with one that responds to infinitesimal strain by rearranging particles and expelling stress, at finite strain the rigid crystal is metastable, associated with a free energy barrier that decreases with increasing strain. The rigid phase becomes thermodynamically stable by switching on an external field, which penalises particle rearrangements. This produces a line of first-order phase transitions in the field - strain plane that intersects the origin. Failure of a solid once strained beyond its elastic limit is associated with kinetic decay processes of the metastable rigid crystal deformed with a finite strain rate. These processes can be understood in quantitative detail using our computed phase diagram as reference.

cond-mat.mtrl-sci↗

arXiv2017

Phase transitions in Restricted Boltzmann Machines with generic priors

We study Generalised Restricted Boltzmann Machines with generic priors for units and weights, interpolating between Boolean and Gaussian variables. We present a complete analysis of the replica symmetric phase diagram of these systems, which can be regarded as Generalised Hopfield models. We underline the role of the retrieval phase for both inference and learning processes and we show that retrieval is robust for a large class of weight and unit priors, beyond the standard Hopfield scenario. Furthermore we show how the paramagnetic phase boundary is directly related to the optimal size of the training set necessary for good generalisation in a teacher-student scenario of unsupervised learning.

cond-mat.dis-nn↗

arXiv2017

Spontaneous Segregation of Agents Across Double Auction Markets

In this paper we investigate the possibility of spontaneous segregation into groups of traders that have to choose among several markets. Even in the simplest case of two markets and Zero Intelligence traders, we are able to observe segregation effects below a critical value Tc of the temperature T; the latter regulates how strongly traders bias their decisions towards choices with large accumulated scores. It is notable that segregation occurs even though the traders are statistically homogeneous. Traders can in principle change their loyalty to a market, but the relevant persistence times become long below Tc.

econ.GN↗

arXiv2017

Emergence of Cooperative Long-term Market Loyalty in Double Auction Markets

Loyal buyer-seller relationships can arise by design, e.g. when a seller tailors a product to a specific market niche to accomplish the best possible returns, and buyers respond to the dedicated efforts the seller makes to meet their needs. We ask whether it is possible, instead, for loyalty to arise spontaneously, and in particular as a consequence of repeated interaction and co-adaptation among the agents in a market. We devise a stylized model of double auction markets and adaptive traders that incorporates these features. Traders choose where to trade (which market) and how to trade (to buy or to sell) based on their previous experience. We find that when the typical scale of market returns (or, at fixed scale of returns, the intensity of choice) become higher than some threshold, the preferred state of the system is segregated: both buyers and sellers are segmented into subgroups that are persistently loyal to one market over another. We characterize the segregated state analytically in the limit of large markets: it is stabilized by some agents acting cooperatively to enable trade, and provides higher rewards than its unsegregated counterpart both for individual traders and the population as a whole.

econ.GN↗

arXiv2017

Phase separation dynamics of polydisperse colloids: a mean-field lattice-gas theory

New insights into phase separation in colloidal suspensions are provided via a new dynamical theory based on the Polydisperse Lattice-Gas model. The model gives a simplified description of polydisperse colloids, incorporating a hard-core repulsion combined with polydispersity in the strength of the attraction between neighbouring particles. Our mean-field equations describe the local concentration evolution for each of an arbitrary number of species, and for an arbitrary overall composition of the system. We focus on the predictions for the dynamics of colloidal gas-liquid phase separation after a quench into the coexistence region. The critical point and the relevant spinodal curves are determined analytically, with the latter depending only on three moments of the overall composition. The results for the early-time spinodal dynamics show qualitative changes as one crosses a 'quenched' spinodal that excludes fractionation and so allows only density fluctuations at fixed composition. This effect occurs for dense systems, in agreement with a conjecture by Warren that, at high density, fractionation should be generically slow because it requires inter-diffusion of particles. We verify this conclusion by showing that the observed qualitative changes disappear when direct particle-particle swaps are allowed in the dynamics. Finally, the rich behaviour beyond the spinodal regime is examined, where we find that the evaporation of gas bubbles with strongly fractionated interfaces causes long-lived composition heterogeneities in the liquid phase; we introduce a two-dimensional density histogram method that allows such effects to be easily visualized for an arbitrary number of particle species.

cond-mat.soft↗

arXiv2017

Phase Diagram of Restricted Boltzmann Machines and Generalised Hopfield Networks with Arbitrary Priors

Restricted Boltzmann Machines are described by the Gibbs measure of a bipartite spin glass, which in turn corresponds to the one of a generalised Hopfield network. This equivalence allows us to characterise the state of these systems in terms of retrieval capabilities, both at low and high load. We study the paramagnetic-spin glass and the spin glass-retrieval phase transitions, as the pattern (i.e. weight) distribution and spin (i.e. unit) priors vary smoothly from Gaussian real variables to Boolean discrete variables. Our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard Hopfield model with Boolean patterns. The retrieval region is larger when the pattern entries and retrieval units get more peaked and, conversely, when the hidden units acquire a broader prior and therefore have a stronger response to high fields. Moreover, at low load retrieval always exists below some critical temperature, for every pattern distribution ranging from the Boolean to the Gaussian case.

cond-mat.dis-nn↗

arXiv2017

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.

cs.GT↗

arXiv2017

Accurate Interatomic Force Fields via Machine Learning with Covariant Kernels

We present a novel scheme to accurately predict atomic forces as vector quantities, rather than sets of scalar components, by Gaussian Process (GP) Regression. This is based on matrix-valued kernel functions, on which we impose the requirements that the predicted force rotates with the target configuration and is independent of any rotations applied to the configuration database entries. We show that such covariant GP kernels can be obtained by integration over the elements of the rotation group SO(d) for the relevant dimensionality d. Remarkably, in specific cases the integration can be carried out analytically and yields a conservative force field that can be recast into a pair interaction form. Finally, we show that restricting the integration to a summation over the elements of a finite point group relevant to the target system is sufficient to recover an accurate GP. The accuracy of our kernels in predicting quantum-mechanical forces in real materials is investigated by tests on pure and defective Ni, Fe and Si crystalline systems.

physics.chem-ph↗

arXiv2017

Statistical physics approaches to subnetwork dynamics in biochemical systems

We apply a Gaussian variational approximation to model reduction in large biochemical networks of unary and binary reactions. We focus on a small subset of variables (subnetwork) of interest, e.g. because they are accessible experimentally, embedded in a larger network (bulk). The key goal is to write dynamical equations reduced to the subnetwork but still retaining the effects of the bulk. As a result, the subnetwork-reduced dynamics contains a memory term and an extrinsic noise term with non-trivial temporal correlations. We first derive expressions for this memory and noise in the linearized (Gaussian) dynamics and then use a perturbative power expansion to obtain first order nonlinear corrections. For the case of vanishing intrinsic noise, our description is explicitly shown to be equivalent to projection methods up to quadratic terms, but it is applicable also in the presence of stochastic fluctuations in the original dynamics. An example from the Epidermal Growth Factor Receptor (EGFR) signalling pathway is provided to probe the increased prediction accuracy and computational efficiency of our method.

physics.chem-ph↗