Local Mean-field Approximation Research Articles

A comparative study between Monte Carlo entropic sampling method and local mean field investigations of thermal properties of spin-crossover nanoparticles based on Ising-like model

The thermally induced transitions between low-spin (LS) and high-spin (HS) configurations of spin-crossover (SCO) nanoparticles are simulated, focusing on the effects of localized surface and bulk interactions on the average magnetization of 2D square lattices. The thermal behaviors and hysteresis cycles are investigated within the framework of the Ising model Hamiltonian and are conducted following two approaches: local mean field approximation (LMFA) and Monte Carlo entropic sampling (MCES) techniques. The results obtained by these two methods are compared for the two square lattice sizes, 6 $$ \times $$ 6 and 7 $$ \times $$ 7. Thus, when the bulk-surface interaction term is set to zero, the two approaches lead to identical values of the surface and bulk transition temperatures separated by a long intermediate plateau in both cases. Although hysteresis curves exhibit a similar shape, LMFA shows slightly larger widths $$ \Delta T $$ than MCES. On increasing bulk-surface interaction term, the two methods lead to different shifts in equilibrium temperature values for both bulk and surface components, respectively, to lower and higher values by MCES. In general, it is found that LMFA shifts surface equilibrium temperature differently to lower values and enhances the hysteresis effect, particularly for surface molecules. On the other hand, for the 7 $$ \times $$ 7 square lattice, the equilibrium temperatures are slightly higher by 1.5% and 3.2% for bulk and surface molecules, respectively, with a narrower hysteresis width in the surface. Moreover, with the MCES method, an abrupt transition instead of a hysteresis transition is calculated for surface molecules ($$ \Delta T_{surf}^{MCES}=0 $$ K).

Cascade-enhanced transport efficiency of biochemical systems.

Recent developments in nonequilibrium thermodynamics, known as thermodynamic uncertainty relations, limit the system's accuracy by the amount of free-energy consumption. A transport efficiency, which can be used to characterize the capacity to control the fluctuation by means of energy cost, is a direct result of the thermodynamic uncertainty relation. According to our previous research, biochemical systems consume much lower energy cost by noise-induced oscillations to keep almost equal efficiency to maintain precise processes than that by normal oscillations. Here, we demonstrate that the performance of noise-induced oscillations propagating can be further improved through a cascade reaction mechanism. It has been discovered that it is possible to considerably enhance the transport efficiency of the biochemical reactions attained at the terminal cell, allowing the cell to use the cascade reaction mechanism to operate more precisely and efficiently. Moreover, an optimal reaction coupling strength has been predicted to maximize the transport efficiency of the terminal cell, uncovering a concrete design strategy for biochemical systems. By using the local mean field approximation, we have presented an analytical framework by extending the stochastic normal form equation to the system perturbed by external signals, providing an explanation of the optimal coupling strength.

Hexagonal-Shaped Spin Crossover Nanoparticles Studied by Ising-Like Model Solved by Local Mean Field Approximation

The properties of spin crossover (SCO) nanoparticles were studied for five 2D hexagonal lattice structures of increasing sizes embedded in a matrix, thus affecting the thermal properties of the SCO region. These effects were modeled using the Ising-like model in the framework of local mean field approximation (LMFA). The systematic combined effect of the different types of couplings, consisting of (i) bulk short- and long-range interactions and (ii) edge and corner interactions at the surface mediated by the matrix environment, were investigated by using parameter values typical of SCO complexes. Gradual two and three hysteretic transition curves from the LS to HS states were obtained. The results were interpreted in terms of the competition between the structure-dependent order and disorder temperatures (TO.D.) of internal coupling origin and the ligand field-dependent equilibrium temperatures (Teq) of external origin.

Three states and three steps simulated within Ising like model solved by local mean field approximation in 3D spin crossover nanoparticles

Coordination iron (II) compounds are studied to simulate switching properties between low spin (LS, S = 0) and high-spin (HS, S = 2) states in spin-crossover materials. These two states are diamagnetic (LS) and paramagnetic (HS) in nature, and the switching between these two states is achieved through external excitations which may be of thermal or of pressure origin. In this contribution, a local mean-field approach is proposed to study SCO nano/micro-particles, for which distinctions among the contributions of molecules localized at the edge, corner, surface or the bulk, as well as for the external coupling that concerns only surface particles have been introduced. In this first attempt, the model is solved using a rough approximation which simplifies its treatment, leading to finding out three steps switching and three states, simulated under temperature effect while two steps transitions are obtained under pressure effect.

Local mean field approximation applied to a 3D spin crossover nanoparticles configuration: free energy analysis of the relative stability of the stationary states

The local mean field approximation is applied to an inhomogeneous 3D spin crossover (SCO) nanoparticle configuration with a special focus on its systemic effect on molecules in the bulk, at the corner, at the edge and at surface. The matrix effect at the surface is introduced through a specific interaction term, L. The partition function for each region allows the determination of the total free-energy F from which the stability of each configuration is analyzed through thermodynamic considerations.

2D Spin Crossover Nanoparticles described by the Ising-like model solved in Local Mean-Field Approximation

Some six-coordinate iron (II) coordination compounds exhibit thermal-, optical-, electrical-, magnetic- and pressure-induced switching between the diamagnetic low-spin (LS, S=0) and the paramagnetic high-spin (HS; S=2) states [1]. This may lead to potential application of these complexes in molecular devices such as temperature and pressure sensors [2]. An Ising-like model has been proposed to explain the occurrence of the thermal hysteresis behaviour [3,4] of this switchable solids. In this contribution, the local mean field approximation is applied to solve the Hamiltonian modelling interactions pertaining to 2D nanoparticles embedded in a magnetically-inactive matrix.

Magnetic-state control in a 3d dimer on a metallic substrate

The Alexander-Anderson model is used to investigate the magnetic states of a dimer composed of 3d-element atoms on a metallic substrate in the local mean-field approximation enabling the examination of noncollinear magnetic states. The magnetic moments localized near the dimer atoms are calculated self-consistently depending on the angle θ between them under the action of an external magnetic field. The dependence between the angle θ of the ground state and the number N of d electrons around the dimer atom is determined. The transition between the antiferromagnetic and ferromagnetic exchange interactions of the dimer magnetic moments is implemented with increasing N. The dimer ground state near the transition is noncollinear, cannot be described by the Heisenberg model, and is very sensitive to the influence of an external magnetic field. The interaction between 3d-dimer atoms and the substrate, gas adsorption, and external fields can vary the number of d electrons, making it possible to control magnetic states on an atomic scale.

Coarse-graining Calcium Dynamics on Stochastic Reaction-diffusion Lattice Model

We develop a coarse grained (CG) approach for efficiently simulating calcium dynamics in the endoplasmic reticulum membrane based on a fine stochastic lattice gas model. By grouping neighboring microscopic sites together into CG cells and deriving CG reaction rates using local mean field approximation, we perform CG kinetic Monte Carlo (kMC) simulations and find the results of CG-kMC simulations are in excellent agreement with that of the microscopic ones. Strikingly, there is an appropriate range of coarse proportion m, corresponding to the minimal deviation of the phase transition point compared to the microscopic one. For fixed m, the critical point increases monotonously as the system size increases, especially, there exists scaling law between the deviations of the phase transition point and the system size. Moreover, the CG approach provides significantly faster Monte Carlo simulations which are easy to implement and are directly related to the microscopics, so that one can study the system size effects at the cost of reasonable computational time.

Coarse-grained Simulations of Chemical Oscillation in Lattice Brusselator System

The oscillation behavior of a two-dimension lattice-gas Brusselator model was investigated. We have adopted a coarse-grained kinetic Monte Carlo (CG-KMC) procedure, where m×m microscopic lattice sites are grouped together to form a CG cell, upon which CG processes take place with well-defined CG rates. Such a CG approach almost fails if the CG rates are obtained by a simple local mean field (s-LMF) approximation, due to the ignorance of correlation among adjcent cells resulting from the trimolecular reaction in this nonlinear system. By proper incorporating such boundary effects, thus introduce the so-called b-LMF CG approach. Extensive numerical simulations demonstrate that the b-LMF method can reproduce the oscillation behavior of the system quite well, given that the diffusion constant is not too small. In addition, the deviation from the KMC results reaches a nearly zero minimum level at an intermediate cell size, which lies in between the effective diffusion length and the minimal size required to sustain a well-defined temporal oscillation.

Spin Models for Ferroelastics: towards a Spin Glass Description of Strain Glass

We review the description of ferroelastic transitions in terms of spin models. We show how one can systematically obtain a pseudo-spin Hamiltonian from the Landau energy describing the first order transition between Austenite/Martensite phases. It is shown that a Local Mean-field approximation predicts the same microstructure as the continuous Landau model in terms of strain variables. This method can be applied to a wide range of two and three dimensional transitions. We then demonstrate how quenched disorder in such pseudo-spin models yields the existence of a glass phase, characterized by the Edwards-Anderson order parameter. Our approach uses Mean-field approximation and Monte-Carlo simulations (using Zero Field Cooling/Field Cooling experiments) to study the influence of the long-range interactions. Although our model captures the salient features of a ferroelastic material in the presence of disorder, the influence of the disorder on the high symmetry austenite phase is not quite consistent with expected behavior. We examine different means of introducing disorder that can improve upon the results.

Microstructure from ferroelastic transitions using strain pseudospin clock models in two and three dimensions: A local mean-field analysis

We show how microstructure can arise in first-order ferroelastic structural transitions, in two and three spatial dimensions, through a local meanfield approximation of their pseudospin hamiltonians, that include anisotropic elastic interactions. Such transitions have symmetry-selected physical strains as their $N_{OP}$-component order parameters, with Landau free energies that have a single zero-strain 'austenite' minimum at high temperatures, and spontaneous-strain 'martensite' minima of $N_V$ structural variants at low temperatures. In a reduced description, the strains at Landau minima induce temperature-dependent, clock-like $\mathbb{Z}_{N_V +1}$ hamiltonians, with $N_{OP}$-component strain-pseudospin vectors ${\vec S}$ pointing to $N_V + 1$ discrete values (including zero). We study elastic texturing in five such first-order structural transitions through a local meanfield approximation of their pseudospin hamiltonians, that include the powerlaw interactions. As a prototype, we consider the two-variant square/rectangle transition, with a one-component, pseudospin taking $N_V +1 =3$ values of $S= 0, \pm 1$, as in a generalized Blume-Capel model. We then consider transitions with two-component ($N_{OP} = 2$) pseudospins: the equilateral to centred-rectangle ($N_V =3$); the square to oblique polygon ($N_V =4$); the triangle to oblique ($N_V =6$) transitions; and finally the 3D cubic to tetragonal transition ($ N_V =3$). The local meanfield solutions in 2D and 3D yield oriented domain-walls patterns as from continuous-variable strain dynamics, showing the discrete-variable models capture the essential ferroelastic texturings. Other related hamiltonians illustrate that structural-transitions in materials science can be the source of interesting spin models in statistical mechanics.

Effect of the reservoir size on gas adsorption in inhomogeneous porous media

We study the influence of the relative size of the reservoir on the adsorption isotherms of afluid in disordered or inhomogeneous mesoporous solids. We consider both an atomisticmodel of a fluid in a simple, yet structured pore, whose adsorption isotherms are computedby molecular simulation, and a coarse-grained model for adsorption in a disorderedmesoporous material, studied by a density functional approach in a local mean-fieldapproximation. In both cases, the fluid inside the porous solid exchanges matterwith a reservoir of gas that is at the same temperature and chemical potentialand whose relative size can be varied, and the control parameter is the totalnumber of molecules present in the porous sample and in the reservoir. Varyingthe relative sizes of the reservoir and the sample within experimental range maychange the shape of the hysteretic isotherms, leading to a ‘re-entrant’ behaviorcompared to the grand-canonical isotherm when the latter displays a jump in density.We relate these phenomena to the organization of the metastable states thatare accessible for the adsorbed fluid at a given chemical potential or density.

Multibody Interactions in Coarse-Graining Schemes for Extended Systems

In this paper we address the role of multibody interactions for the coarse-grained approximation of stochastic lattice systems. Such interaction potentials are often not included in coarse-graining schemes, as they can be computationally expensive. The multibody interactions are obtained from the error expansion of the reference measure which is, in many cases, chosen as a Gibbs measure corresponding to a local mean-field approximation. We identify the parameter $\epsilon$ that characterizes the level of approximation and its relation to the underlying interaction potential. The error analysis suggests strategies to overcome the computational costs due to evaluations of multibody interactions by additional approximation steps with controlled errors. We present numerical examples demonstrating that the inclusion of multibody interactions shows substantial improvement in dynamical simulations, e.g., of rare events and metastability in phase transitions regimes.

Transition to burst synchronization in coupled neuron networks

Transition to burst synchronization (BS), where all the neurons start and end bursting simultaneously, has been studied on a diffusively coupled network of Hindmash-Rose bursting neurons. When the coupling strength epsilon is increased from zero, spatiotemporal chaos can be first tamed into one type of BS states with fold-homoclinic bursting, which then undergoes spike-adding and transits into another type of BS states with fold-Hopf bursting. The latter transition takes place via dynamic cluster separation, during which all the neurons with degree k>k(c) change to show fold-Hopf bursting, with epsilonk(c) nearly a constant. A reasonable mechanism behind such phenomena is given by using local mean field approximation, and the role of network topology is also discussed. The case when the neurons are coupled via chemical synapses is also briefly discussed.

Relationship between strain localization and catastrophic rupture

In order to explore a prior warning to catastrophic rupture of heterogeneous media, like rocks, the present study investigates the relationship between surface strain localization and catastrophic rupture. Instrumented observations on the evolution of surface strain field and the catastrophic rupture of a rock under uniaxial compression were carried out. It is found that the evolution of surface strain field displays two phases: at the early stage, the strain field keeps nearly uniform with weak fluctuations increasing slowly; but at the stage prior to catastrophic rupture, a certain accelerating localization develops and a localized zone emerges. Based on the measurements, an analysis was performed with local mean-field approximation. More importantly, it is found that the scale of localized zone is closely related to the catastrophic rupture strain and the rupture strain can be calculated in accord with the local-mean-field model satisfactorily. This provides a possible clue to the forecast of catastrophic rupture.

Phase behavior of an asymmetric vector lattice model for oil-water-amphiphile mixtures

A vector model describing oil-water-amphiphile mixtures, originally proposed by Ciach, Ho/ye and Stell, is investigated on the simple cubic lattice. Our previous approach, involving an exact integration over the amphiphile orientational degrees of freedom, is reformulated to consider the more general case of asymmetric amphiphiles. The resulting spin-1 effective Hamiltonian, containing many-site temperature-dependent interactions, is then treated in a local mean field approximation and the phase behavior at finite temperature is extensively analyzed. In order to stress asymmetry effects, a detailed comparison with the symmetric case is performed.

Statistical thermodynamics of lattice models in zeolites: Implications of local versus global mean field interactions

The statistical thermodynamics of adsorption in caged zeolites is developed by treating the zeolite as an ensemble of M identical cages or subsystems. Within each cage adsorption is assumed to occur onto a lattice of n identical sites. Expressions for the average occupancy per cage are obtained by minimizing the Helmholtz free energy in the canonical ensemble subject to the constraints of constant M and constant number of adsorbates N. Adsorbate–adsorbate interactions in the Brag–Williams or mean field approximation are treated in two ways. The local mean field approximation (LMFA) is based on the local cage occupancy and the global mean field approximation (GMFA) is based on the average coverage of the ensemble. The GMFA is shown to be equivalent in formulation to treating the zeolite as a collection of interacting single site subsystems. In contrast, the treatment in the LMFA retains the description of the zeolite as an ensemble of identical cages, whose thermodynamic properties are conveniently derived in the grand canonical ensemble. For a z coordinated lattice within the zeolite cage, with εaa as the adsorbate–adsorbate interaction parameter, the comparisons for different values of εaa*=εaaz/2kT, and number of sites per cage, n, illustrate that for −1<εaa*<0 and n⩾10, the adsorption isotherms and heats of adsorption predicted with the two approaches are similar. In general, the deviation between the LMFA and GMFA is greater for smaller n and less sensitive to n for εaa*>0. We compare the isotherms predicted with the LMFA with previous GMFA predictions [K. G. Ayappa, C. R. Kamala, and T. A. Abinandanan, J. Chem. Phys. 110, 8714 (1999)] (which incorporates both the site volume reduction and a coverage-dependent εaa) for xenon and methane in zeolite NaA. In all cases the predicted isotherms are very similar, with the exception of a small steplike feature present in the LMFA for xenon at higher coverages.

Partial integration and local mean field approach for a vector lattice model of microemulsions: unbalanced case

We consider a vector lattice model for mixtures of water, oil and amphiphile, and it extends to the (unbalanced) case of different water and oil volume fractions, an approach previously developed for the balanced case. After an exact summation of the orientational degrees of freedom we get an effective hamiltonian with multi-site couplings, which is then studied in a local mean-field approximation. The phase diagram for several temperatures and strong amphiphilic interactions is mapped out both in terms of the chemical potentials and of the volume fractions of the components. We find several structured phases (lamellar, micellar and cubic) as well as homogeneous phases, and coexistence phenomena.

Structured Phases of a Three-Dimensional Vector Lattice Model for Oil-Water-Surfactant Mixtures

Abstract A three-dimensional vector lattice model for a mixture of water, oil and amphiphile is considered on the simple cubic lattice. For a balanced system with equal water and oil concentrations the ground state exhibits, when the surfactant is strong, an ordered bicontinuous structure in addition to pure oil/water and pure surfactant states. The model is transformed, integrating out exactly the surfactant orientational degrees of freedom, in an effective spin-1 model with two-, three, five-and seven-body interactions with temperature dependent interaction strengths. This effective model is then investigated using the local mean-field approximation. The phase diagram of the model is derived and discussed, with particular attention to the structured ordered phase. In the disordered phase a microemulsion region is also found.

Dipolar systems — transition to glassy ordering on dilution

Diluted dipolar Ising systems on a regular lattice are studied numerically within the local mean field approximation used for quenched systems. We compensate for the demagnetization field by adding a magnetization dependent term to the local field and study different solutions for different initial conditions. This added term enables us to distinguish the glassy state from the ferromagnetic state which otherwise would break up into domains. For high densities of dipoles we obtain a well defined ferromagnetic phase, where the solution stays the same, independently of the initial conditions. For diluted systems on the other side, the ferromagnetic phase disappears and is replaced by the glassy state. Different solutions are obtained with almost the same free energy. We find that the concentration x c, separating these two phases, agrees relatively well with experimental findings. The found transition temperatures, however are somewhat higher than the experimental ones for ferromagnetic samples. In the glassy regime this difference increases. These deviations are reduced slightly by using the one loop expansion, where a reaction term has been subtracted from the effective field.