One-dimensional Lattice Gas Model Research Articles

Optimum transport in systems with time-dependent drive and short-ranged interactions.

We consider a one-dimensional lattice gas model of hardcore particles with nearest-neighbor interaction in presence of a time-periodic external potential. We investigate how attractive or repulsive interaction affects particle transport and determine the conditions for optimum transport, i.e., the conditions for which the maximum dc particle current is achieved in the system. We find that the attractive interaction in fact hinders the transport, while the repulsive interaction generally enhances it. The net dc current is a result of the competition between the current induced by the periodic external drive and the diffusive current present in the system. When the diffusive current is negligible, particle transport in the limit of low particle density is optimized for the strongest possible repulsion. But when the particle density is large, very strong repulsion makes particle movement difficult in an overcrowded environment and, in that case, the optimal transport is obtained for somewhat weaker repulsive interaction. Our numerical simulations show reasonable agreement with our mean-field calculations. When the diffusive current is significantly large, the particle transport is still facilitated by repulsive interaction, but the conditions for optimality change. Our numerical simulations show that the optimal transport occurs at the strongest repulsive interaction for large particle density and at a weaker repulsion for small particle density.

One Dimensional Exclusion Process with Dynein Inspired Hops: Simulation and Mean Field Analysis

We introduce a one-dimensional non-equilibrium lattice gas model representing the processive motion of dynein molecular motors over the microtubule. We study both dynamical and stationary state properties for the model consisting of hardcore particles hopping on the lattice with variable step sizes. We find that the stationary state gap-distribution exhibits striking peaks around gap sizes that are multiples of the maximum step size, for both open and periodic boundary conditions. We verified this feature using a mean-field calculation. For open boundary conditions, we observe intriguing damped oscillator-like distribution of particles over the lattice with a periodicity equal to the maximum step size. To characterize transient dynamics, we measure the mean square displacement that shows weak superdiffusive growth with exponent $$\gamma \approx 1.34$$ for periodic boundary and ballistic growth ( $$\gamma \approx 2$$ ) for open boundary conditions at early times. We also study the effect of Langmuir dynamics on the density profile.

Non-equilibrium island size distribution in one dimension

In this paper, a novel analytical expression for the size distribution of one-dimensional structures at thermodynamic non-equilibrium is derived. The expression is obtained using one-dimensional lattice gas model. Ostwald ripening and decay of short one-dimensional islands is taken into account. The theoretical results are compared with kinetic Monte Carlo simulations and the experimental size distribution of one-dimensional wires. A correct method of extracting the binding energy from the experimental size distribution in one dimension is proposed.

Thermodynamic Formalism for Transient Potential Functions

We study the thermodynamic formalism of locally compact Markov shifts with transient potential functions. In particular, we show that the Ruelle operator admits positive continuous eigenfunctions and positive Radon eigenmeasures in forms of Martin kernels. These eigenmeasures can be characterized in terms of the direction of escape to infinity of their orbits, when viewed inside a suitable Martin-like compactification of the underlying shift space. We relate these results to first-order phase transitions in one-dimensional lattice gas models with infinite set of states. This work complements earlier works by Sarig (Ergod Theory Dyn Syst 19(06):1565–1593, 1999; Isr J Math 121(1):285–311, 2001) who focused on the recurrent scenario.

Диаграммы основных состояний для адсорбции на нанокластерах при учете взаимодействия неближайших соседей

В связи c бурным развитием нанотехнологий и микроэлектроники много внимания уделяется изучению различных адсорбционных процессов. Как правило, адсорбция наблюдается на поверхности кристалла или жидкости, но иногда имеет место и одномерная адсорбция. В частности, особый интерес вызывает вопрос о распределении фаз при адсорбции на атомных цепочках. В статье рассматривается адсорбция атомов одного сорта на цепочке конечного размера при температуре абсолютного нуля в рамках обобщенной модели Изинга с учетом взаимодействия первых и вторых соседей и внешнего поля. В качестве граничных условий выбраны «оборванные концы», что позволяет моделировать процессы на реальной цепочке со свободными концами. Построены диаграммы основных состояний — диаграммы стабильности фаз в координатной плоскости энергетических параметров при температуре абсолютного нуля: по вертикали откладывается энергия взаимодействия ближайших или вторых соседей; по горизонтали — внешнее поле. Расчет диаграмм основных состояний позволяет определить возможные адсорбционные превращения и новые фазы при достаточно низких температурах, а также построить изотерму одномерной адсорбции. Некоторые из рассмотренных фаз уже обнаружены экспериментально.DOI 10.14258/izvasu(2018)4-08

Critical dynamics of classical systems under slow quench

We study the slow quench dynamics of a one-dimensional nonequilibrium lattice gas model which exhibits a phase transition in the stationary state between a fluid phase with homogeneously distributed particles and a jammed phase with a macroscopic hole cluster. Our main result is that in the critical region (i.e., at the critical point and in its vicinity) where the dynamics are assumed to be frozen in the standard Kibble-Zurek argument, the defect density exhibits an algebraic decay in the inverse annealing rate with an exponent that can be understood using critical coarsening dynamics. However, in a part of the critical region in the fluid phase, the standard Kibble-Zurek scaling holds. We also find that when the slow quench occurs deep into the jammed phase, the defect density behavior is explained by the rapid quench dynamics in this phase.

Computation of Large Deviation Statistics via Iterative Measurement-and-Feedback Procedure

We propose a computational method for large deviation statistics of time-averaged quantities in general Markov processes. In our proposed method, we repeat a response measurement against external forces, where the forces are determined by the previous measurement as feedback. Consequently, we obtain a set of stationary states corresponding to an exponential family of distributions, each of which shows rare events in the original system as the typical behavior. As a demonstration of our method, we study large deviation statistics of one-dimensional lattice gas models.

The Kinetic Monte Carlo Simulations of the Self-Diffusivity in Zeolites

The diffusion of guest molecules in zeolites is investigated in the framework of a onedimensionallattice-gas model with two non-equivalent sites. The concentration dependenciesof the tracer and center-of-mass diffusion coefficients are calculated for some representative valuesof the lateral interaction between the guest molecules. Using simple ideas about moleculesdiffusion we propose analytical expressions for the diffusion coefficients. We compare the theoreticaldependencies with the numerical data obtained by the kinetic Monte Carlo simulations.A very good coincidence of the data obtained by the two quite independent different methodscorroborates strongly the validity of the proposed approach.

A one-dimensional lattice-gas model for simulating diffusion in channel pores with side pockets: The analytical approach and kinetic Monte Carlo technique

The diffusion of particles adsorbed in a one-dimensional channel with side pockets is investigated in the framework of a lattice-gas model. The concentration dependencies of the tracer, center-of-mass and Fickian diffusion coefficients are calculated for some representative values of the lateral interactions between molecules. Using simple ideas about the molecule diffusion we propose the analytical expressions for the diffusion coefficients. The theoretical dependencies are compared with the numerical data obtained by kinetic Monte Carlo simulations. The data obtained by the two quite different methods well coincide in the whole concentration and wide interaction regions.

Self-similarity of phase-space networks of frustrated spin models and lattice gas models

We studied the self-similar properties of the phase spaces of two frustrated spin models and two lattice gas models. The two frustrated spin models were (1) the antiferromagnetic Ising model on a two-dimensional triangular lattice (1a) at the ground states and (1b) above the ground states and (2) the six-vertex model. The two lattice gas models were (3) the one-dimensional lattice gas model and (4) the two-dimensional lattice gas model. Their phase spaces were mapped to networks so that the fractal analysis of complex networks can be applied. These phase spaces, in turn, establish new classes of networks with unique self-similar properties. Models 1a, 2, and 3 with long-range power-law correlations in real space exhibit fractal phase spaces, while models 1b and 4 with short-range exponential correlations in real space exhibit nonfractal phase spaces. This behavior agrees with one of the untested assumptions in Tsallis nonextensive statistics. All the phase spaces have power-law "mass" -radius relations that reflect the local self-similar structures.

Discussion notes on “Spurious character of singularities associated with phase transitions in cylindrical pores”, by K. Binder et al.

Comments are provided for [6]. It would be of some interest if the quasi-one-dimensional nature of phase coexistence in a cylindrical pore could be mapped directly onto an exactly solvable one-dimensional model, to thereby compare the observed cluster statistics with rigorous results for isodesmic chemical equilibria (linear self-assembly). For example, a simple coarsegraining of simulation density profile data into coin-shaped discs should allow for a direct mapping onto the one-dimensional lattice-gas (LG) model, [1]. Early simulation observations of the multi-domain structure of phase-coexistence in cylindrical pores showed a qualitative resemblance to the polydisperse domains of one-dimensional selfassembled cluster distributions [2]. With modern computer resources and a suitable mapping it should be possible to collect entire equilibrated cluster distributions from simulations of standard models used by physical chemists. Of most relevance here are cases where the self-assembled phase wants to wet the pore wall, since then there can be no pore-induced nucleation barrier to interfere with one-dimensional self-assembly. The physical chemistry of one-dimensional self-assembly has previously been elucidated from exact solutions of the cluster distributions of one-dimensional lattice-gas models. These results include the effect of solvent-induced self-assembly on isodesmic chemical equilibria [3], the effect of altering the binary equilibrium constant to depart from isodesmic behaviour [4] and the effects of confinement and/or a finite amount of solute [5]. The last situation is directly relevant to the ‘transition’ between normal isodesmic behaviour and ‘polymerisation.’ Namely, if there is sufficient space and amount of solute then the well-known polydisperse exponential cluster distribution is found. Here, the maximum in the isodesmic cluster distribution will always be isolated monomers. However, if the system runs out of space or sufficient solute to be able to generate the largest cluster lengths then the exponential distribution of cluster sizes will rapidly switch over to the opposite case of polymerisation, where the most probable cluster is the longest one possible. Due to the one-dimensional geometry, this ‘transition’ cannot be a true phase-transition, but it is still a clearly identifiable qualitative change in behaviour. In particular, a cross-over line in phase space can be a e-mail: jrh.leeds@gmail.com 244 The European Physical Journal Special Topics uniquely defined by solving the one-dimensional LG model for the ‘transition’ states at which isolated monomers are equally probable as the largest possible cluster, [5]. Note that this cross-over is controlled by the ratio of the free-energy cost of a coarsegrained disc free-surface to the temperature energy scale kBT . This appears to be the only reason why linear polymerisation is far more dominant in chemical self-assembly than observed in physical self-assembly!

Equilibrium island-size distribution in one dimension

We derive an analytical expression for the size distribution of monoatomic wires in the framework of a one-dimensional lattice gas model at thermodynamic equilibrium. The theoretical results are compared with the size distribution of one-dimensional Ag wires obtained via nucleation at the step edges of the Pt(997) surface.

One-dimensional lattice gas models with infinitely many absorbing states

We determined, by numerical simulations, the critical behavior one-dimensional systems with infinitely many absorbing states. These models are modified versions of the conserved lattice gas model in which two adjacent active particles instead of just one can jump to their neighboring sites. The models studied here are examples of systems with many absorbing states and order parameter coupled to a nondiffusive conserved field that show nontrivial critical behavior.

Dynamics of molybdenum nano structure formation on the TiO2(110) surface: A kinetic Monte Carlo approach

The rutile TiO 2 (1 1 0) surface is a highly anisotropic surface exhibiting “channels” delimited by oxygen rows. In previous experimental and theoretical DFT works we could identify the molybdenum adsorption sites. They are located inside the channels. Moreover, experimental studies have shown that during subsequent annealing after deposition, special molybdenum nano structures can be formed, especially two monolayer high pyramidal chains of atoms. In order to better understand the dynamics of nano structure formation, we present a kinetic Monte Carlo study on diffusion and adsorption of molybdenum atoms on a TiO 2 (1 1 0) surface. A quasi one-dimensional lattice gas model has been used which describes the possible adsorption sites of a Mo atom in a single channel of the surface. The atomic positions of a 1.5 monolayer thick Mo film formed of pyramidal chains define the lattice sites of the model. Thereby the formation of three-dimensional clusters could be studied. Here we have studied the cluster formation as a function of parameters that can be controlled in a growth experiment by physical vapor deposition: deposition and annealing temperature, flux and total amount of deposited Mo. Good qualitative agreement with recent experiments is obtained.

Microscopic approach to the collective diffusion in the interacting lattice gas

A variational approach to collective diffusion in the interacting lattice gas, based on kinetics of microscopic states of the system, is presented. The approach accounts for equilibrium correlations and is capable of predicting the coverage dependence of the diffusion coefficient D ( θ ) in an analytic form. It provides a viable alternative to approaches based on a hierarchy of kinetic equations for correlation functions and in contrast to them is free of uncertainties often introduced by various truncation schemes. Applications to one-dimensional lattice gas models with increasing degree of complexity are presented. A two-dimensional gas with strong interparticle repulsive interactions is chosen to illustrate the application to a system with structural ordering. In each case, analytic predictions agree very well with the results of the Monte Carlo simulations. In particular, rapid changes/discontinuities of D ( θ ) , discovered in numerical simulations, are confirmed and their origin understood.

Exact solution of a one-dimensional model of strained epitaxy on a periodically modulated substrate

We consider a one-dimensional lattice gas model of strained epitaxy with the elastic strain accounted for through a finite number of cluster interactions comprising contiguous atomic chains. Interactions of this type arise in the models of strained epitaxy based on the Frenkel-Kontorova model. Furthermore, the deposited atoms interact with the substrate via an arbitrary periodic potential of period L. This model is solved exactly with the use of an appropriately adopted technique developed recently in the theory of protein folding. The advantage of the proposed approach over the standard transfer-matrix method is that it reduces the problem to finding the largest eigenvalue of a matrix of size L instead of 2(L-1), which is vital in the case of nanostructures where L may measure in hundreds of interatomic distances. Our major conclusion is that the substrate modulation always facilitates the size calibration of self-assembled nanoparticles in one- and two-dimensional systems.

Equilibrium and kinetics: water confined in carbon nanotubes as one-dimensional lattice gas.

We present a simple one-dimensional lattice gas model, which describes very well the equilibrium and kinetic behaviors of water confined in a thin carbon nanotube found in an atomistic molecular dynamics simulation [G. Hummer, J. C. Rasaiah, and J. P. Noworyta, Nature (London), 414, 188 (2001)]. The model parameters correspond to various physical interactions and can be calculated or estimated by using statistical mechanics. Then, the roles of all interactions in the water filling, emptying, and transporting processes are clearly understood. Our results indicate that the interaction from the water molecules outside the nanotube plays a key role in these processes and the interaction can be simply treated as an average effect of the bulk water.

Infinite-Parameter Potential Symmetries and a New ExactSolution for the Particle-Cluster Dynamic Equation

The master equation of a one-dimensional lattice-gas model with orderpreservation where the occupation probabilities of sites correspondingto Bose statistics as a consequence of the prescribed dynamics isstudied with the potential symmetry method. The infinite-parameterpotential symmetry and a new exact solution are obtained. The resultillustrates that there remains the possibility of the above nonlinearequation to a linear partial differential equation by a non-invertiblemapping.

Analytical solution of a one-dimensional lattice gas model with an infinite number of multiatom interactions.

We consider a one-dimensional lattice gas model in which the atoms interact via an infinite number of cluster interactions within contiguous atomic chains plus the next-nearest-neighbor pairwise interaction. All interactions are of arbitrary strength. An analytical expression for the size distribution of atomic chain lengths is obtained in the framework of the canonical ensemble formalism. Application of the exact solution to the problems of self-assembly and self-organization is briefly discussed.

Exact solution of the particle–cluster dynamic equation

The symmetry reduction of the master equation of a one-dimensional lattice-gas model with order preservation where the occupation probabilities of sites corresponding to Bose statistics as a consequence of the prescribed dynamics is studied. Some exact solutions are found.