Diffusion on random lattices

We study the motion of a point particle along the bonds of a two-dimensional random lattice, whose sites are randomly occupied with right and left rotators, which scatter the particle according to deterministic scattering rules. We consider both a Poisson (PRL) and a vectorized random lattice (VRL) and fixed as well as flipping scatterers. On both lattices, for fixed scatterers and equal concentrations of right and left rotators the same anomalous diffusion of the particle is obtained as before for the triangular lattice, where the mean square displacement is t, the diffusion process non-Gaussian, and the particle trajectories exhibit scaling behavior as at a percolation threshold. For unequal concentrations the particle is trapped exponentially rapidly. This system can be considered as an extreme case of the Lorentz lattice gases on regular lattices discussed before or as an example of the motion of a particle along cracks or (grain or cellular) boundaries on a two-dimensional surface.     

Diffusion in Lorentz lattice gas cellular automata: The honeycomb and quasi-lattices compared with the square and triangular lattices

We study numerically the nature of the diffusion process on a honeycomb and a quasi-lattice, where a point particle, moving along the bonds of the lattice, scatters from randomly placed scatterers on the lattice sites according to strictly deterministic rules. For the honeycomb lattice fully occupied by fixed rotators two (symmetric) isolated critical points appear to be present, with the same hyperscaling relation as for the square and the triangular lattices. No such points appear to exist for the quasi-lattice. A comprehensive comparison is made with the behavior on the previously studied square and triangular lattices. A great variety of diffusive behavior is found, ranging from propagation, superdiffusion, normal, quasi-normal, and anomalous, to absence of diffusion. The influence of the scattering rules as well as of the lattice structure on the diffusive behavior of a point particle moving on the all lattices studied so far is summarized.     

Diffusion in Lorentz lattice gas automata with backscattering

The probability of first return to the initial intervalx and the diffusion tensorD _x are calculated exactly for a ballistic Lorentz gas on a Bethe lattice or Cayley tree. It consists of a moving particle and a fixed array of scatterers, located at the nodes, and the lengths of the intervals between scatterers are determined by a geometric distribution. The same values forx andD _x apply also to a regular space lattice with a fraction of sites occupied by a scatterer in the limit of a small concentration of scatterers. If backscattering occurs, the results are very different from the Boltzmann approximation. The theory is applied to different types of lattices and different types of scatterers having rotational or mirror symmetries.     

Recurrence properties of Lorentz lattice gas cellular automata

Recurrence properties of a point particle moving on a regular lattice randomly occupied with scatterers are studied for strictly deterministic, nondeterministic, and purely random scattering rules.On leave from Institute of Oceanology, USSR Academy of Sciences, 117218 Moscow, USSR     

Lorentz lattice gases,abnormal diffusion,and polymer statistics

Diffusive behavior in various Lorentz lattice gases, especially wind-tree-like models, is discussed. Comparisons between lattice and continuum models as well as deterministic and probabilistic models are made. In one deterministic model, where the scatterers behave like double-sided mirrors, a new kind of abnormal diffusion is found, viz., the mean square displacement is proportional to the time, but the probability density distribution function is non-Gaussian. The connections of this mirror model with the percolation problem and the statistics of polymer chains on a lattice are also discussed.     

New results for diffusion in Lorentz lattice gas cellular automata

New calculations to over ten million time steps have revealed a more complex diffusive behavior than previously reported of a point particle on a square and triangular lattice randomly occupied by mirror or rotator scatterers. For the square lattice fully occupied by mirrors where extended closed particle orbits occur, anomalous diffusion was still found. However, for a not fully occupied lattice the superdiffusion, first noticed by Owczarek and Prellberg for a particular concentration, obtains for all concentrations. For the square lattice occupied by rotators and the triangular lattice occupied by mirrors or rotators, an absence of diffusion (trapping) was found for all concentrations, except on critical lines, where anomalous diffusion (extended closed orbits) occurs and hyperscaling holds for all closed orbits withuniversal exponentsd _f =7/4 and =15/7. Only one point on these critical lines can be related to a corresponding percolation problem. The questions arise therefore whether the other critical points can be mapped onto a new percolation-like problem and of the dynamical significance of hyperscaling.     

Quantum diffusion in semi-infinite periodic and quasiperiodic systems

This paper studies quantum diffusion in semi-infinite one-dimensional periodic lattice and quasiperiodic Fibonacci lattice. It finds that the quantum diffusion in the semi-infinite periodic lattice shows the same properties as that for the infinite periodic lattice. Different behaviour is found for the semi-infinite Fibonacci lattice. In this case, there are still C（t） - t^-δ and d（t） - t^β. However, it finds that 0 〈δ 〈 1 for smaller time, and δ = 0 for larger time due to the influence of surface localized states. Moreover, β for the semi-infinite Fibonacci lattice is much smaller than that for the infinite Fibonacci lattice. Effects of disorder on the quantum diffusion are also discussed.     

Diffusion in lattices with anisotropic scatterers

We study diffusion in lattices with periodic and random arrangements of anisotropic scatterers. We show, using both analytical techniques based upon our previous work on asymptotic properties of multistate random walks and computer calculation, that the diffusion constant for the random arrangement of scatterers is bounded above and below at an arbitrary density by the diffusion constant for an appropriately chosen periodic arrangement of scatterers at the same density. We also investigate the accuracy of the low-density expansion for the diffusion constant up to second order in the density for a lattice with randomly distributed anisotropic scatterers. Comparison of the analytical results with numerical calculations shows that the accuracy of the density expansion depends crucially on the degree of anisotropy of the scatterers. Finally, we discuss a monotonicity law for the diffusion constant with respect to variation of the transition rates, in analogy with the Rayleigh monotonicity law for the effective resistance of electric networks. As an immediate corollary we obtain that the diffusion constant, averaged over all realizations of the random arrangement of anisotropic scatterers at density, is a monotone function of the density.     

One-Dimensional Lorentz Gas with Rotating Scatterers: Exact Solutions

The simplest solutions (orbits) to the recently introduced Lorentz gas with rotating scatterers are found by considering its one-dimensional one-particle reduction. This model has only one parameter which can be viewed as the amount of energy transfer between the scatterers and the particle during a collision. Exact solutions of the system are found for several values of this parameter. For some of these values, the dynamics is shown to be in many respects similar to the dynamics of the deterministic Lorentz lattice gases.     

Decay of correlations in the regular Lorentz gas

The regular Lorentz gas on triangular lattice is studied numerically and analytically. The velocity correlation function is shown to decay exponentially in the number of collisions with a decay rate which vanishes as the scatterers approach close packing. The crossover to power law decay at close packing is described by a scaling function.     

Diffusion Lattice Boltzmann Scheme on a Orthorhombic Lattice

We present a diffusion lattice Boltzmann (DLB) scheme which is derived from first principles. As opposed to the traditional lattice BGK schemes the DLB is valid for orthorhombic lattices and it has two eigenvalues of the collision operator. It is shown that the diffusion coefficient depends only on one eigenvalue of the collision operator. Hence, the DLB scheme can be optimized with means of the additional eigenvalue of the collision operator and with different lattice spacing along the principal axes. The properties of the DLB scheme concerning consistency, stability, and accuracy are studied with eigenmode analysis. This analysis shows that the DLB scheme is consistent with diffusion for a wide range of diffusion coefficients, it has unconditional stability, and that it has third-order accuracy. Furthermore, it is shown that accuracy is improved by setting the additional eigenvalue to zero and by densifying the lattice spacing along the direction of the density gradient.     

镶嵌正方晶格上Gauss模型的相图

利用等效变换和自旋重标相结合的方法, 研究了镶嵌正方晶格上的Gauss模型. 研究 发现, 该系统可以变换为正方晶格上具有最近邻和次近邻相互作用的Gauss系统, 由此严格求得了镶嵌正方晶格上Gauss模型的临界温度, 得到了该系统的精确相图.     

Diffusion in lattice Lorentz gases : Nearest scatterers approximation

The expressions for diffusion coefficients and for velocity autocorrelation functions of lattice Lorentz gases are derived both in the nearest scatterers and Boltzmann approximations. The results are obtained for linear chain, square, triangular, simple cubic, body centred cubic, face centred cubic and face centred hyper cubic lattices. The diffusion coefficients are compared with those from the effective medium approximation for the square lattice and with computer simulation results for triangular, simple cubic and body centred cubic lattices.     

A Solvable Decorated Ising Lattice Model

A decorated lattice is suggested and the Ising model on it with three kinds of interactions K₁, K₂, and K₃ is studied. Using an equivalent transformation, the square decorated Ising lattice is transformed into a regular square Ising lattice with nearest-neighbor, next-nearest-neighbor, and four-spin interactions, and the critical fixed point is found at K₁=0.5769, K₂=-0.0671, and K₃=0.3428, which determines the critical temperature of the system. It is also found that this system and the regular square Ising lattice, and the eight-vertex model belong to the same universality class.     

Multipartite Entanglement of a Tetrahedron Lattice

Three-dimensional Heisenberg model in the form of a tetrahedron lattice is investigated. The concurrence and multipartite entanglement are calculated through 2-concurrence C and 4-concurrence C4. The concurrence C and multipartite entanglement G4 depend on different coupling strengths Ji and are decreased when the temperature T is increased. For a symmetric tetrahedron lattice, the concurrence C is symmetric about J1 when J~ is negative while the multipartite entanglement G4 is symmetric about J1 when J2 〈 2. For a regular tetrahedron lattice, the concurrence G of ground state is 1/3 for ferromagnetic case while G = 0 for antiferromagnetic ca.se. However, there is no multipartitc entanglement since C4=0 in a regular tetrahedron lattice. The external magnetic field 13 can increase the maximum value of the concurrence GB and induce two or three peaks in Cn. There is a peak in the multipartite entanglement G4 B when G4B is varied as a function of the temperature T. This peak is mainly induced by the magnetic field B.     

Approximations to wave propagation through a lattice of Dirichlet scatterers

A scheme of matched asymptotic expansions is used to obtain approximations to the dispersion relation when waves, governed by the Helmholtz equation, propagate through a two-dimensional lattice of scatterers on each of which a homogeneous Dirichlet boundary condition is imposed. The scatterers must be identical, but can be of any shape as long as each is small relative to the wavelength and the lattice periodicity. The results differ from those obtained using homogenisation in that there is no requirement that the wavelength be much longer than the lattice periodicity, and hence it is possible to describe band gaps.     

Field-dependent conductivity and diffusion in a two-dimensional Lorentz gas

The conductivity and diffusion of a color-charged two-dimensional thermostatted Lorentz gas in a color field is studied by a variety of methods. In this gas, point particles move through a regular triangular array of soft scatterers, where, in the presence of a field, a nonequilibrium stationary state is reached by coupling to a Gaussian thermostat. The zero-field conductivity and diffusion coefficient are computed with equilibrium molecular dynamics dynamics from the Green-Kubo formula and the Einstein relation. Their values are consistent and approach those obtained by Machta and Zwanzig in the limit of hard (disk) scatterers. The field-dependent conductivity is obtained from its constitutive relation, from the coupling constant to the thermostat, and by using the recently derived conjugate pairing rule of Evans, Cohen, and Morriss, from the two maximal Lyapunov exponents of the Lorentz gas in the stationary state. All these methods give consistent results. Finally, elements of the field-dependent diffusion tensor have been computed. At zero field, they are consistent with the zero-field conductivity, but they vanish beyond a critical field strength, suggesting a dynamical phase transition at the critical field; the conductivity appears to remain finite, approaching a constant value for large field strengths.     

Non-Markoffian diffusion in a one-dimensional disordered lattice

Recent treatments of diffusion in a one-dimensional disordered lattice by Machta using a renormalization-group approach, and by Alexander and Orbach using an effective medium approach, lead to a frequency-dependent (or non-Markoffian) diffusion coefficient. Their results are confirmed by a direct calculation of the diffusion coefficient.Research supported by NSF Grant No. CHE 77-16308.     

Noncommutat ive Differential Calculus on Discrete Abelian Groups and Its Applications

A new noncommutative differential calculus on function space of discrete Abelian groups is proposed. The derivatives are introduced with respect to the generators of the groups only. It is applied to discrete symplectic geometry and Hamiltonian systems with H(p, q) = T(p) + V(q) as well as the lattice gauge theory on regular lattice.