A stochastic lattice gas for Burgers' equation: A practical study

We continue our investigation of stochastic lattice gases as a (highly parallel) means of simulating given PDEs, in this case Burgers' equation in one dimension. The lattice dynamics consists of stochastic unidirectional particle displacement, and our attention is turned toward the reliability of the model, i.e., its ability to reproduce the unique physical solution of Burgers' equation. Lattice gas results are discussed and compared against finite-difference calculations and exact solutions in examples which include shocks and rarefaction waves.     

Percolation in hard-core lattice gases and a model ferrofluid

A lattice gas on ³ consisting of hard spheres with exclusions extending through third neighbors is proved to undergo a percolation transition. If spins with ferromagnetic couplings are attached to the spheres, spontaneous magnetization is proved to occur. This may provide a model for a ferrofluid, a system which exhibits spontaneous magnetization without crystalline order. Similar results are also obtained for an analogous model on ².     

Damage spreading in a driven lattice gas model

We studied damage spreading in a Driven Lattice Gas (DLG) model as a function of the temperature T$T$, the magnitude of the external driving field E$E$, and the lattice size. The DLG model undergoes an order–disorder second-order phase transition at the critical temperature _Tc(E)$T_c\left(E\right)$, such that the ordered phase is characterized by high-density strips running along the direction of the applied field; while in the disordered phase one has a lattice-gas-like behavior. It is found that the damage always spreads for all the investigated temperatures and reaches a saturation value _Dsat$D_{sat}$ that depends only on T$T$. _Dsat$D_{sat}$ increases for T<_Tc(E=∞)$T<T_c(E=\infty )$, decreases for T>_Tc(E=∞)$T>T_c(E=\infty )$ and is free of finite-size effects. This behavior can be explained as due to the existence of interfaces between the high-density strips and the lattice-gas-like phase whose roughness depends on T$T$. Also, we investigated damage spreading for a range of finite fields as a function of T$T$, finding a behavior similar to that of the case with E=∞$E=\infty$.     

Density profile and flow of miscible fluid with dissimilar constituent masses

A computer simulation model is used to study the density profile and flow of a miscible gaseous fluid mixture consisting of differing constituent masses (m_A=m_B/3) through an open matrix. The density profile is found to decay with the height ∝exp (−m_A(B)h), consistent with the barometric height law. The flux density shows a power-law increase ∝(p_c−p)^μ with μ2.3 at the porosity 1−p above the pore percolation threshold 1−p_c.     

Phase transitions in a driven lattice gas in two planes

We report on a Monte Carlo study of ordering in a nonequilibrium system. The system is a lattice gas that comprises two equal, parallel square lattices with stochastic particle-conserving irreversible dynamics. The particles are driven along a principal direction under the competition of the heat bath and a large, constant external electric field. There is attraction only between particles on nearest-neighbor sites within the same lattice. Particles may jump from one plane to the other; therefore, density fluctuations have an extra mechanism to decay and build up. It helps to obtain the steady-state accurately. Spatial correlations decay with distance according to a power law at high enough temperature, as for the ordinary two-dimensional case. We find two kinds of nonequilibrium phase transitions. The first one has a critical point for half occupation of the lattice, and seems to be related to the anisotropic phase transition reported before for the plane. This transition becomes discontinuous for low enough density. The difference of density between the planes changes discontinuously for any density at a lower temperature. This seems to correspond to a phase transition that does not have a counterpart in equilibrium nor in the two-dimensional nonequilibrium case.     

小体积比两相分离早期过程的三维格子气模型研究

采用Monte Carlo模拟方法研究了小体积比三维格子气模型中的相分离过程, 比较了sc, fcc, bcc 等3种晶格类型和T/T_c=0.45–0.85等 5个相对温度的系统演化行为, 计算了系统的结构因子函数, 发现结构因子在相分离早期不满足标度关系, 生长指数等于1/6, 小于经典Lifshitz-Slyozov理论的值. 关键词： 格子气模型 相分离过程 Monte Carlo模拟 生长律     

Finite Size Effect in Path Integral Monte Carlo Simulations of ^4He Systems

Path integral Monte Carlo （PIMC） simulations are a powerful computational method to study interacting quantum systems at finite temperatures. In this work, PIMC has been applied to study the finite size effect of the simulated systems of ^4He. We determine the energy as a function of temperature at saturated-vapor-pressure （SVP） conditions in the temperature range of T ∈ [1.0 K,4.0 K], and the equation of state （EOS） in the grmmd state For systems consisted of 32, 64 and 128 ^4He atoms, respectively, We find that the energy at SVP is influenced significantly by the size of the simulated system in the temperature range of T ∈ [2.1 K, 3.0 K] and the larger the system is, the better results are obtained in comparison with the experimental values; while the EOS appeared to be unrelated to it.     

Finite Size Effect in Path Integral Monte Carlo Simulations of 4He Systems

Path integral Monte Carlo (PIMC) simulations are a powerful computational method to study interacting quantum systems at finite temperatures. In this work, PIMC has been applied to study the finite size effect of the simulated systems of ⁴He. We determine the energy as a function of temperature at saturated-vapor-pressure (SVP) conditions in the temperature range of T∈[1.0 K, 4.0 K], and the equation of state (EOS) in the ground state for systems consisted of 32, 64 and 128 ⁴He atoms, respectively. We find that the energy at SVP is influenced significantly by the size of the simulated system in the temperature range of T∈[2.1 K, 3.0 K] and the larger the system is, the better results are obtained in comparison with the experimental values; while the EOS appeared to be unrelated to it.     

Monte Carlo studies of percolation phenomena for a simple cubic lattice

The site-percolation problem on a simple cubic lattice is studied by the Monte Carlo method. By combining results for periodic lattices of different sizes through the use of finite-size scaling theory we obtain good estimates forp _c (0.3115±0.0005), (0.41±0.01), (1.6±0.1), and(0.8±0.1). These results are consistent with other studies. The shape of the clusters is also studied. The average surface area for clusters of sizek is found to be close to its maximal value for the low-concentration region as well as for the critical region. The percentage of particles in clusters of different sizesk is found to have an exponential tail for large values ofk forP <p _c. Forp >p _c there is too much scatter in the data to draw firm conclusions about the size distribution.Work supported in part by USAFOSR Grant #73-2430B and by ERDA #E(11-1)-3077.     

Distribution for fermionic discrete lattice gas within the canonical ensemble

The distinct deviations from the Fermi-Dirac statistics ascertained recently at low temperatures for a one-dimensional, spinless fermionic discrete lattice gas with conserved number of noninteracting particles hopping on the non-degenerated, well-separated single-particle energy levels are studied in numerical and theoretical terms. The generalized distribution is derived in the formn(h)={Y _h exp [(_h–)]+1{su–1 valid even in the thermodynamic limit, when the discreteness of the energy levels is kept. This distribution demonstrates good agreement with the data obtained numerically both by the canonical partition function technique and by Monte Carlo simulation.     

A remark on the condensation in the hard-core lattice Bose gas

We point out that Bose-Einstein condensation occurs at sufficiently low temperature in a hard-core ^d-lattice Bose gas for d3 and particle density 1/2, by exploiting its equivalence to a spin-1/2XY model.     

Phase transitions in a driven lattice gas at two temperatures

By suitably combining the uniformly driven lattice gas and the two-temperature kinetic Ising model, we obtain a generalized model that allows us to probe a variety of nonequilibrium phase transitions, including a type not previously observed. This new type of transition involves longitudinally ordered steady states, which are phase-segragated states with interface normalsparallel to the drive. Using computer simulations on a two-dimensional lattice gas, we map out the structure of the phase diagram, and the nature of the transitions, in the three-dimensional space of the drive and the two temperatures. While recovering anticipated results in most cases, we find one surprise, namely, that the transition from disorder to longitudinal order is continuous. Unless it turns out to be very weakly first order, this result is inconsistent with the expectation of field-theoretic renormalization group calculations.     

Static correlations of higher order and diffusion in an interacting lattice gas

For a lattice gas with extended hard core interaction on a square lattice the static correlation functions of higher order, which determine the average jump rate in the diffusion process, are calculated both by the Monte Carlo method and by analytic approximations. It is found that the superposition approximation is very inaccurate for the correlation functions of third and fourth order, but gives better results for the average jump rate. Up to concentrations ofc = 0.3 better consistency with the Monte Carlo data for both quantities is obtained by treating the site occupation numbers as Gaussian random variables and accordingly expressing the correlation functions of higher order by products of averages of two particle correlations. For concentrationsc > 0.3, however, a Bethe-Peierls cluster approximation is superior to the superposition approximation.The results of this paper were presented at the I.L.L, workshop Beyond Radial Distribution, Grenoble, July 15–16, 1985.     

Field-induced percolation in a polarized lattice gas

We introduce a lattice gas model with particles carrying a charge either +1 or –1 and drifting in opposite directions due to the presence of an external field. Our numerical simulations show the formation of polarized clusters elongated along the direction of the field. At low enough temperatures the clusters percolate through the system in a similar way as in the strip phase of the driven lattice gas model. A possible application of the model can be found in microemulsions.     

Nonequilibrium phase transitions in a simple three-state lattice gas

We investigate the dynamics of a three-state stochastic lattice gas consisting of holes and two oppositely charged species of particles, under the influence of an electric field at zero total charge. Interacting only through an excluded-volume constraint, particles exchange with holes and, on a slower time scale, with each other. Using a combination of Monte Carlo simulations and meanfield equations of motion, we study a set of suitably defined order parameters, their histograms and fluctuations, as well as the current through the system. With increasing particle density and drive, the system first orders into a charge-segregated state, and then disorders again near complete filling. The transition is first order at low densities and turns second order at higher ones. The finite-size and aspect-ratio dependence of characteristic quantities is discussed at the mean-field level.     

A lattice gas of prime numbers and the Riemann Hypothesis

In recent years, there has been some interest in applying ideas and methods taken from Physics in order to approach several challenging mathematical problems, particularly the Riemann Hypothesis. Most of these kinds of contributions are suggested by some quantum statistical physics problems or by questions originated in chaos theory. In this article, we show that the real part of the non-trivial zeros of the Riemann zeta function extremizes the grand potential corresponding to a simple model of one-dimensional classical lattice gas, the critical point being located at 1/2 as the Riemann Hypothesis claims.     

Measures of critical exponents in the four-dimensional site percolation

Using finite-size scaling methods we measure the thermal and magnetic exponents of the site percolation in four dimensions, obtaining a value for the anomalous dimension very different from the results found in the literature. We also obtain the leading corrections-to-scaling exponent and, with great accuracy, the critical density.     

Exact analysis of a lattice gas on the 3–12 lattice with two- and three-site interactions

Exact results are obtained for a lattice gas on the 3–12 lattice with two- and three-site interactions. By using a decoration transformation, we map the lattice gas into one on the honeycomb lattice with pure two-site interactions. This procedure permits us to draw exact results for the original 3–12 lattice gas. In particular, we obtain its exact two-phase boundary, and confirm the fact that an experimentally observed anomaly in the critical behavior of the coexistence-curve diameter is present if, and only if, the three-site interactions are present.     

具有表面弛豫的液体自扩散的Monte Carlo模拟

研究多孔介质中液体的自扩散行为能获得介质的微观结构信息,有助于了解介质中液体的传输性质.以具有不同孔隙大小的无限长圆柱体模型中的液体为对象,采用Monte Carlo随机行走方法,模拟存在表面弛豫时液体的自扩散系数和核自旋磁化强度随时间的变化,导出将NMR弛豫参数和随机行走参数联系在一起的表达式.结果表明:在快扩散区,液体的核自旋磁化强度随弛豫时间呈单指数衰减,且自扩散系数在短时情况下独立于表面弛豫率;在慢扩散区,液体的核自旋磁化强度衰减和自扩散系数在短时情况下均与表面弛豫率无关.模拟结果与理论分析相吻合,可用于求解介质的表面积与体积之比及孔径大小等结构信息.