Off-lattice Systems Research Articles

Perpendicular and parallel phase separation in two-species driven diffusive lattice gases.

We study three different lattice models in which two species of diffusing particles are driven in opposite directions by an electric field. We focus on dynamical phase transitions that involve phase separation into domains that may be parallel or perpendicular to a driving field. In all cases, the perpendicular state appears for weak driving, consistent with previous work. For strong driving, we introduce two models that support the parallel state. In one model, this state occurs because of the inclusion of dynamical rules that enhance lateral diffusion during collisions; in the other, it is a result of a nearest-neighbor attractive or repulsive interaction between particles of the same or opposite species. We discuss the connections between these results and the behavior found in off-lattice systems, including laning and freezing by heating.

Application of the dynamic mean field theory to fluid transport in slit pores.

We explore the applicability of the lattice model and dynamic mean field theory as a computationally efficient tool to study transport across heterogeneous porous media, such as mixed matrix membranes. As a starting point and to establish some basic definitions of properties analogous to those in the off-lattice systems, we consider transport across simple models of porous materials represented by a slit pore in a chemical potential gradient. Using this simple model, we investigate the distribution of density and flux under steady state conditions, define the permeability across the system, and explore how this property depends on the length of the pore and the solid-fluid interactions. Among other effects, we observe that the flux in the system goes through a maximum as the solid-fluid interaction is varied from weak to strong. This effect is dominated by the behavior of the fluid near the walls and is also confirmed by off-lattice molecular dynamics simulations. We further extend this study to explore transport across heterogeneous slit pore channels composed of two solids with different values of solid-fluid interaction strengths. We demonstrate that the lattice models and dynamic mean field theory provide a useful framework to pose questions on the accuracy and applicability of the classical theories of transport across heterogeneous porous systems.

Critical behavior in active lattice models of motility-induced phase separation

Lattice models allow for a computationally efficient investigation of motility-induced phase separation (MIPS) compared to off-lattice systems. Simulations are less demanding, and thus, bigger systems can be accessed with higher accuracy and better statistics. In equilibrium, lattice and off-lattice models with comparable interactions belong to the same universality class. Whether concepts of universality also hold for active particles is still a controversial and open question. Here, we examine two recently proposed active lattice systems that undergo MIPS and investigate numerically their critical behavior. In particular, we examine the claim that these systems and MIPS in general belong to the Ising universality class. We also take a more detailed look on the influence and role of rotational diffusion and active velocity in these systems.Graphic

Information density, structure and entropy in equilibrium and non-equilibrium systems

During a spontaneous change, a macroscopic physical system will evolve towards a macro-state with more realizations. This observation is at the basis of the statistical mechanical version of the second law of thermodynamics, and it provides an interpretation of entropy in terms of probabilities. However, we cannot rely on the statistical-mechanical expressions for entropy in systems that are far from equilibrium. In this paper, we compare various extensions of the definition of entropy, which have been proposed for non-equilibrium systems. It has recently been proposed that measures of information density may serve to quantify entropy in both equilibrium and non-equilibrium systems. We propose a new ‘bit-wise’ method to measure the information density for off-lattice systems. This method does not rely on coarse-graining of the particle coordinates. We then compare different estimates of the system entropy, based on information density and on the structural properties of the system, and check if the various entropies are mutually consistent and, importantly, whether they can detect non-trivial ordering phenomena. We find that, except for simple (one-dimensional) cases, the different methods yield answers that are at best qualitatively similar, and often not even that, although in several cases, the different entropy estimates do detect ordering phenomena qualitatively. Our entropy estimates based on bit-wise data compression contain no adjustable scaling factor, and show large quantitative differences with the thermodynamic entropy obtained from equilibrium simulations. Hence, our results suggest that, at present, there is not yet a single, structure-based entropy definition that has general validity for equilibrium and non-equilibrium systems.

Unsupervised machine learning for detection of phase transitions in off-lattice systems. II. Applications.

We outline how principal component analysis can be applied to particle configuration data to detect a variety of phase transitions in off-lattice systems, both in and out of equilibrium. Specifically, we discuss its application to study (1) the nonequilibrium random organization (RandOrg) model that exhibits a phase transition from quiescent to steady-state behavior as a function of density, (2) orientationally and positionally driven equilibrium phase transitions for hard ellipses, and (3) a compositionally driven demixing transition in the non-additive binary Widom-Rowlinson mixture.

Unsupervised machine learning for detection of phase transitions in off-lattice systems. I. Foundations.

We demonstrate the utility of an unsupervised machine learning tool for the detection of phase transitions in off-lattice systems. We focus on the application of principal component analysis (PCA) to detect the freezing transitions of two-dimensional hard-disk and three-dimensional hard-sphere systems as well as liquid-gas phase separation in a patchy colloid model. As we demonstrate, PCA autonomously discovers order-parameter-like quantities that report on phase transitions, mitigating the need for a priori construction or identification of a suitable order parameter-thus streamlining the routine analysis of phase behavior. In a companion paper, we further develop the method established here to explore the detection of phase transitions in various model systems controlled by compositional demixing, liquid crystalline ordering, and non-equilibrium active forces.

Scaling properties of a parallel implementation of the multicanonical algorithm

The multicanonical method has been proven powerful for statistical investigations of lattice and off-lattice systems throughout the last two decades. We discuss an intuitive but very efficient parallel implementation of this algorithm and analyze its scaling properties for discrete energy systems, namely the Ising model and the 8-state Potts model. The parallelization relies on independent equilibrium simulations in each iteration with identical weights, merging their statistics in order to obtain estimates for the successive weights. With good care, this allows faster investigations of large systems, because it distributes the time-consuming weight-iteration procedure and allows parallel production runs. We show that the parallel implementation scales very well for the simple Ising model, while the performance of the 8-state Potts model, which exhibits a first-order phase transition, is limited due to emerging barriers and the resulting large integrated autocorrelation times. The quality of estimates in parallel production runs remains of the same order at the same statistical cost.

Pair force distributions in simple fluids

Analytic expressions are derived for the frequency distribution, P(f), of pair forces, f, and those of their α-Cartesian component, f(α), or P(f(α)), for some typical model simple fluids, expressed in terms of the radial distribution function and known constants. For strongly repulsive inverse power (IP), exponential and Yukawa purely repulsive potentials, P(f) diverges at the origin approximately as ∼f(-1), but with different limiting analytic forms. P(f(α)) is also shown to diverge as ∼f(-1) as f → 0 for the IP fluid. For the Lennard-Jones potential fluid, P(f) is finite for all f ≥ 0 but has two singularities for negative f, corresponding to the zero force limit (i.e., f → 0(-)) and the point of inflection in the potential. The corresponding component force distribution is singular as f(α) → 0 from both positive and negative force sides. The large force limit of P(f), which originates from the close neighbor interactions, is nearly exponential for the IP and LJ fluids, as is also found for granular materials. A more complete picture of force distributions in off-lattice particulate systems as a function of force law and state point (particularly the extent of "thermalization" of the particles) is provided.

Calculation of Partition Functions for the Self-Assembly of Patchy Particles

Partition functions encode all the thermodynamics of a system, but for most systems of practical importance, they cannot be calculated exactly. In this work we present a new hierarchical method for calculating partition functions to arbitrary precision. We discuss the algorithmic details of our implementation, including elements of shape-matching and entropy calculation for on-lattice and off-lattice systems. We highlight computational trade-offs between speed and accuracy, showing how this varies with temperature, and demonstrate its utility in studying nanoscale self-assembly for a system of model patchy particles.

Monte Carlo Methods for Estimating Interfacial Free Energies and Line Tensions

Excess contributions to the free energy due to interfaces occur for many problems encountered in the statistical physics of condensed matter when coexistence between different phases is possible (e.g. wetting phenomena, nucleation, crystal growth, etc.). This article reviews two methods to estimate both interfacial free energies and line tensions by Monte Carlo simulations of simple models, (e.g. the Ising model, a symmetrical binary Lennard-Jones fluid exhibiting a miscibility gap, and a simple Lennard-Jones fluid). One method is based on thermodynamic integration. This method is useful to study flat and inclined interfaces for Ising lattices, allowing also the estimation of line tensions of three-phase contact lines, when the interfaces meet walls (where “surface fields” may act). A generalization to off-lattice systems is described as well. The second method is based on the sampling of the order parameter distribution of the system throughout the two-phase coexistence region of the model. Both the interface free energies of flat interfaces and of (spherical or cylindrical) droplets (or bubbles) can be estimated, including also systems with walls, where sphere-cap shaped wall-attached droplets occur. The curvature-dependence of the interfacial free energy is discussed, and estimates for the line tensions are compared to results from the thermodynamic integration method. Basic limitations of all these methods are critically discussed, and an outlook on other approaches is given.

Efficiencies of dynamic Monte Carlo algorithms for off-lattice particle systems with a single impurity

The efficiency of dynamic Monte Carlo algorithms for off-lattice systems composed of particles is studied for the case of a single impurity particle. The theoretical efficiencies of the rejection-free method and of the Monte Carlo with Absorbing Markov Chains method are given. Simulation results are presented to confirm the theoretical efficiencies.

Phase transition in spin systems with various types of fluctuations.

Various types ordering processes in systems with large fluctuation are overviewed. Generally, the so-called order–disorder phase transition takes place in competition between the interaction causing the system be ordered and the entropy causing a random disturbance. Nature of the phase transition strongly depends on the type of fluctuation which is determined by the structure of the order parameter of the system. As to the critical property of phase transitions, the concept “universality of the critical phenomena” is well established. However, we still find variety of features of ordering processes. In this article, we study effects of various mechanisms which bring large fluctuation in the system, e.g., continuous symmetry of the spin in low dimensions, contradictions among interactions (frustration), randomness of the lattice, quantum fluctuations, and a long range interaction in off-lattice systems.

Controlling the Percolation Threshold of Conductor-Insulator Composites by Changing the Granular Size of Insulators

When one attempts to modulate and control the characteristics of composite materials, limit of the modulation may be dictated by the mathematical threshold of the percolation transition. We report our computer simulation, in which percolation behavior can strongly be controlled by introducing size differences in the insulator particles rather than the conductor particles. This modulation effectively lowers the transition point to 0.52, from about 0.59 achieved with the conventional 2D site percolation model. Although a similar effect has already been reported for off-lattice systems and experiments, this is the first observation and analysis made in a basic 2D lattice model. Also, we show that this effect of threshold reduction is also related to the shape difference of the insulator particles. Such an observation in a basic model is believed to be fundamental to exploiting this phenomenon in real applications.

Can the Pruned-Enriched Method be Used for the Simulation of Fluids?

The flat histogram version of pruned and enriched Rosenbluth method (FLATPERM) is an effective Monte Carlo method for calculating densities of states of polymers on a lattice. In this paper we generalize this method to calculate the densities of states of off-lattice systems. To demonstrate the feasibility of the approach, we perform sample calculations for the Lennard-Jones fluids. The densities of states of Lennard-Jones fluids simulated by Pruned-enriched method, i.e., the generalization of FLATPERM, agree with the densities simulated by Wang-Landau method in the range of high potential energy. However the direct extension of FLATPERM fails at low energy and a useful extension still needs to be found.

Analytic continuation average spectrum method for quantum liquids

We revisit the problem of determining the real-frequency density response in quantum fluids via analytical continuation of imaginary-time quantum Monte Carlo data. We demonstrate that the average spectrum method (ASM) is capable of revealing resolved modes in the dynamic structure factor of both ortho-deuterium and liquid para-hydrogen, in agreement with experiments and quantum mode-coupling theories, while the maximum entropy approach yields only a smooth unimodal spectrum. Outstanding issues are discussed. Our work provides the first application of the ASM method in realistic off-lattice systems.

Lattice Model of Diffusion-Limited Bimolecular Chemical Reactions in Confined Environments

We study the effect of confinement on diffusion-limited bimolecular reactions within a lattice model where a small number of reactants diffuse among a much larger number of inert particles. When the number of inert particles is held constant, the rate of the reaction is slow for small reaction volumes due to limited mobility from crowding and for large reaction volumes due to the reduced concentration of the reactants. The reaction rate proceeds fastest at an intermediate confinement corresponding to a volume fraction near 50%. We generalize the model to off-lattice systems with hydrodynamic coupling and predict that the optimal reaction rate for monodisperse colloidal systems occurs when the volume fraction is approximately 19%. Finally, we discuss the implications of our model for bimolecular reactions inside cells and the dynamics of confined polymers.

A Highly Accurate and Analytic Equation of State for a Hard Sphere Fluid in Random Porous Media

An analytical equation of state (EOS) for a hard sphere fluid confined in random porous media is derived by extending the scaled particle theory to such complex systems with quenched disorders. A simple empirical correction allows us to obtain a highly accurate EOS with errors within the simulation ones. These are the first analytical results for non trivial off-lattice quench-annealed systems.

Off-lattice Dynamic Monte Carlo simulations of aggregation in one dimension

We present an approach to simulate aggregation and gelation based on the extension of the Dynamic Monte Carlo method for chemical reactions to off-lattice inhomogeneous systems. Diffusion is explicitly introduced into the Dynamic Monte Carlo method through the use of first collision time distributions. The resulting method is as fast as diffusion limited cluster aggregation simulations but is more precise, producing results in good agreement with expensive Brownian dynamics simulations. We present results for one-dimensional aggregation using both hard-sphere and continuous potentials. Extensions to more dimensions are discussed.

A sigma-model approach to glassy dynamics

In this contribution we review recent progress in understanding fluctuations in the aging process of macroscopic systems, and we propose further tests of these ideas. We discuss how the emergence of a symmetry in aging systems, global timereparametrization invariance, could be responsible for the observed ‘universal’ behavior of local and mesoscopic non-equilibrium fluctuations. We discuss (i) the two-time scaling and functional form of the distribution of local correlations and responses; (ii) the scaling of multi-time correlations and susceptibilities; (iii) how the above can be derived from a random surface effective action; (iv) the behavior of a diverging two-time dependent correlation length; (v) how these ideas apply to off-lattice particle systems.

Anderson localization in Euclidean random matrices

We study the spectra and localization properties of Euclidean random matrices defined on a random graph. We introduce a powerful method to find the density of states and the localization threshold in topologically disordered (off-lattice) systems. We solve numerically an equation (exact on the random graph) for the probability distribution function of the diagonal elements of the resolvent matrix, with a population dynamics algorithm (PDA). We show that the localization threshold can be estimated by studying the stability of a population of real resolvents under the PDA. An application is given in the context of the instantaneous normal modes of a liquid.