An anomalous diffusion in a long-range disordered lattice

We construct an exactly solvable model of a two-and three-dimensional disordered lattice which presents an anomalous diffusive behaviour.     

Diffusive motion in a model for particle hopping

The diffusive motion of adsorbates on crystal planes is studied by means of a lattice gas model with stochastic dynamics, in the disordered phase and at half coverage. The diffusion coefficient and the time-correlation functions measured in field-emission experiments are calculated. These correlation functions are shown to have the proper hydrodynamic power law decay at long times. It is pointed out that if experiments are done at times before the onset of the hydrodynamic regime the value of the diffusion coefficient obtained will be too small. Our results show also that correlations among the adsorbed particles persist for times longer than predicted by a hydrodynamical approximation.     

Quantum transport in dynamically disordered continuum tight binding systems

We consider the continuum limit of three distinct models describing tightly bound electron systems in one dimension. The first model is the usual tight binding hamiltonian for monatomic lattices with nearest-neighbour hopping between sites. The second model describes a two-subband tight binding system involving two different atoms per unit cell. Finally, the third model represents a monatomic system with two energy levels per atomic site and different nearest-neighbour hopping parameters for hopping between equivalent and non-equivalent levels. The continuum limits of these models result in field-theoretic hamiltonians showing similarities with the Dirac hamiltonian. Assuming the different types of site energies to be dynamically disordered with gaussian whitenoise spectra, we calculate exactly the quantum mechanical mean square displacement <x ²(t)>. Due to the use of Novikov's theorem for the evaluation of configuration averages our analysis for the two-band models is restricted to the degenerate case, where the average positions of the two types of atomic levels coincide. Fort we find coherent motion, <x ²(t)>t ², for the one-band model and disorder induced diffusive contributions for the two-band models. However, for the two-level atomic model the diffusive term is dominated by at ²-term describing coherent hopping between equivalent levels. These findings are discussed in relation to previous results for both discrete and continuum models.     

Chiral metal as a ferromagnetic super spin chain

The electrons on the surface of a disordered multi-layer integer quantum Hall system constitute an unusual chiral metal with ballistic motion transverse to the field, and diffusive motion parallel to it. We present a non-perturbative analytic treatment of an appropriate model, consisting of disordered chiral fermions in two dimensions. A supersymmetric generating functional is set up for the correlation functions of this system. The strong disorder limit is mapped into a supersymmetric spin chain, with ferromagnetic exchange coupling, reflecting the electron's chiral motion. The ferromagnetic ground state and the spin wave excitations, corresponding to the diffusion modes of the chiral metal, are found exactly. The parametric density of states correlator in the ergodic limit is computed from a Boltzmann-weighted sum over low-energy spin states. The result is of a universal form and coincides with that for a hermitian random matrix.     

Quantum treatment of Brownian motion and influence of dissipation on diffusion in dynamically disordered systems

The time-dependent hamiltonian formulation of the Langevin equation is used as a starting point for a quantum treatment of the motion of a free Brownian particle. From an exact solution of the time-dependent Schrödinger equation for the density-matrix in one dimension we obtain the mean square displacement, x ²(t), of the Brownian particle, as well as the mean displacement induced by a uniform electric fieldE(t). While quantum effects are significant for time intervals up to the frictional relaxation time, the long time results are identical to those obtained directly from the solution of the Langevin equation. Next, we analyse in a similar way the motion of an electron in a dynamically disordered continuum where the effect of a classical friction force (dissipation) is taken into account. The friction effect is described using the phenomenological time-dependent hamiltonian inferred from the Langevin equation and the potential fluctuations are assumed to have a generalized gaussian white-noise form. The final result for x ² (t) shows a time-dependence similar to that obtained for the case of Brownian motion. In particular, it corresponds to diffusive behavior at long times, in contrast to thet ³-dependence obtained in a recent study for the case where friction is absent.     

QCD sum rules,the spontaneous breakdown of chiral symmetry and short-distance behaviour in lattice gauge theories

We analyze the behaviour that correlation functions ought to have on the lattice in order to reproduce QCD sum rules in the continuum limit. We formulate a set of relations between lattice correlation functions of meson operators at small time separation and the quark condensates responsible for spontaneous breakdown of chiral symmetery. We suggest that the degree to which such relations are satisfied will provide a set of consistency checks on the ability of lattice Monte Carlo simulations to reproduce the correct spontaneous chiral symmetry breaking of the continuum theory.     

Two-dimensional quantum antiferromagnet in a strong magnetic field

The behaviour of the two-dimensional quantum antiferromagnet of arbitrary spin in a strong transversal magnetic field on a square lattice is studied in terms of the equivalent Bose gas problem. The existence of phase transition from the state characterized by “quasi-long-range” magnetic order to the disordered ferromagnetic state is demonstrated. The expressions for correlation functions, thermodynamical and magnetic characteristics are derived.     

An isotropic dynamically consistent gradient elasticity model derived from a 2D lattice

This paper presents a derivation of a second-order isotropic continuum from a 2D lattice. The derived continuum is isotropic and dynamically consistent in the sense that it is unconditionally stable and prohibits the infinite speed of energy propagation. The Lagrangian density of the continuum is obtained from the Lagrange function of the underlying lattice. This density is used to obtain the expressions for standard and higher-order stresses in direct correspondence with the equations of the continuum motion. The derived continuum is characterized by two additional parameters relative to the classical elastic continuum. These are the characteristic lengthscale and a dimensionless continualization parameter, which characterizes indirectly the timescale of the derived continuum. The margins for the latter parameter are found from the stability analysis. It is envisaged that the continualization parameter could be measured employing a high-frequency pulse propagating along the surface of the continuum. Excitation and propagation of such pulse is studied theoretically in this paper.     

Ground state phase diagram of a spin-2 antiferromagnet on the square lattice

We use the vertex state model approach to construct optimum ground states for a large class of quantum spin-2 antiferromagnets on the square lattice. Optimum ground states are exact ground states of the model which minimize all local interaction operators. The ground state contains two continuous parameters and exhibits a second order phase transition from a disordered phase with exponentially decaying correlation functions to a Néel ordered phase. The behaviour is very similar to that of the corresponding ground state of a quantum spin-3/2 model on the hexagonal lattice, which has been investigated in an earlier paper. Received 8 April 1999     

The diffusive motion of silver ions in α-AgI: Results from quasielastic neutron scattering

The diffusive motion of silver ions in σ-AgI at 250°C has been studied by quasielastic cold neutron scattering. Spectra were taken in the range of wavevector transfer 0.5 < Q < 2.2Å^?1 for elastic scattering. The quasielastic line shapes contain a narrow and a broad component. They are compared to model calculations allowing for the superposition of two kinds of motion on two different time scales, a local random motion and a translational motion of the jump-diffusion type. The model closely fits the data. The local random motion takes place on a time scale of the order of 10^?12 s, with amplitudes of the order of 1 Å. It is probably caused by rapid fluctuations of the local potentials due to the diffusive motion of the other cations. The translational motion results in a mean displacement of the silver ion over a distance of the order of a lattice constant (5 Å) with a correlation time of the order of 10^?11s. This correlation time is composed of a residence time and a time-of-flight, which are both of comparable magnitude.     

Light transport and correlation length in a random laser

A random laser is a strongly disordered, laser‐active optical medium. The coherent laser feedback, which has been demonstrated experimentally to be present in these systems beyond doubt, requires the existence of spatially localized photonic quasimodes. However, the origin of these quasimodes has remained controversial. We develop an analytical theory for diffusive random lasers by coupling the transport theory of the disordered medium to the semiclassical laser rate equations, accounting for (coherent) stimulated and (incoherent) spontaneous emission. From the causality of wave propagation in an amplifying, diffusive medium we derive a novel length scale which we identify with the average mode radius of the lasing quasi‐modes. We show that truly localized modes do not exist in the system without photon number conservation. However, we find that causality in the amplifying medium implies the existence of a novel, finite intensity correlation length which we identify with the average mode volume of the lasing quasimodes. We show further that the surface of the laser‐active medium is crucial in order to stabilize a stationary lasing state. We solve the laser transport theory with appropriate surface boundary conditions to obtain the spatial distributions of the light intensity and of the occupation inversion. The dependence of the intensity correlation length on the pump rate agrees with experimental findings.     

Mode-coupling theory for purely diffusive systems

A mode-coupling formalism is developed for multicomponent systems of particles performing diffusive motion in a uniform host medium. The mode-coupling equations are derived from a set of nonlinear fluctuating diffusion equations by expanding the concentration-dependent diffusion constants about their equilibrium values. From the mode-coupling equations the dominant long time behavior of current-current and super-Burnett correlation functions is derived. As specific applications I consider the long time behaviors of these correlation functions for collective and tracer diffusion in a one-component lattice gas with particle-conserving stochastic dynamics. The results agree with those from exactly solvable models and computer simulations.     

Critical ultrasonic attenuation in a disordered binary alloy

The expression for ultrasonic attenuation in a binary alloy (annealed disordered) system has been obtained solving the Green function equation of motion and performing configurational averaging. Its typical dependence on the configurational fluctuation correlation is seen to lead to a non-divergent behaviour in the ultrasonic attenuation near the order-disorder transition point, where the correlations diverge. The results compare well with the experimental observations on β-brass.     

Continuum limit on a 2-dimensional random lattice

We investigate the continuum limit of the Ising model on a 2-dimensional random lattice using Monte Carlo and strong coupling techniques. No evidence is found for deviation from the behaviour of the standard Ising model on a regular lattice; we do not see the modified critical properties found in the bond-diluted Ising model. Our results suggest that the continuum limit is the usual non-interacting Majorana fermion theory.     

Spatio-temporal patterns in a square-lattice Hodgkin-Huxley neural network

The collective behaviour of a square-lattice Hodgkin-Huxley neural network model with white noise is investigated by numerical methods. It is found that for an intermediate value of noise the Hodgkin-Huxley neurons in the square lattice exhibit an ordered circular structure. However, as the noise level increases, the ordered circular structures are distorted, and eventually totally destroyed. Thereby, the constructive role of appropriately pronounced random perturbations in the studied network is revealed. Furthermore, it is shown that as the diffusive coefficient increases, the typical width of the spatial waves also increases accordingly, which results in a decrease of the number of cycles by a given size of the spatial grid. More interestingly, it is observed that the spatio-temporal coherence resonance is enhanced as the diffusive coefficient is increased. Finally, the dependence of the typical width and the average period of the firing rate function on the diffusive coefficient is studied. Results presented in this paper should prove valuable for the understanding of information processing of neural systems in the presence of noise.     

Two-polaron energy diffusion in a 1D lattice of hydrogen bonded peptide units

A generalized Pauli master equation is established for describing the vibrational energy flow in a 1D lattice of hydrogen bounded peptide units. A Lang-Firsov transformation is applied so that the relevant excitations are small polarons corresponding to vibrational excitons dressed by virtual phonons. A special attention is thus paid to characterize the energy transfer mediated by two polarons. At biological temperature, it is shown that the polaron-phonon coupling is sufficiently strong to prevent any coherent motion. The polaron-polaron interaction occurring in such a nonlinear lattice does not affect the long time behavior of the energy flow which results from the diffusion of two independent polarons. This diffusive motion originates from the competition between two contributions related to phonon mediated transitions (incoherent contribution) and to dephasing limited coherent motion (coherent contribution).     

Langevin model of the rotational diffusion of molecules

Globular molecules in dense gases, liquids, and orientationally disordered crystals experience an incessant fluctuating torque of rather weak magnitude. In gases and liquids the fluctuating torque perturbs the otherwise free rotational motion which becomes diffusive. We treat the rotational diffusion within the Langevin model where the molecules are driven by a stochastic torque and hindered by a friction term. The Langevin equation for molecules with one, two, and three angular degrees of freedom is solved numerically and the dipole and Raman correlation functions are extracted. In the one-dimensional case we compare with the exact solution of the Langevin equation, in two and three dimensions with earlier work and with experimental results.     

Hydrodynamic effect on the correlation functions of a gaussian polymer chain

Time-dependent correlation functions of the conformations of a single gaussian polymer chain immersed in a solvent are calculated renormalization-group theoretically. As a byproduct is is confirmed that to order ? (=4-d, d being the spatial dimensionality) the translational diffusion constant obtained fully dynamically (i.e. by the Green-Kubo formalism) is identical to that obtained by the Kirkwood theory (or the Kirkwood-Riseman formalism) even with the self-avoiding interaction.     

Anomalous biennial oscillations in a Fisher equation with a discretized verhulst term

The dynamics of a biological population governed by a modified Fisher equation is studied by means of Monte Carlo simulations. Reproduction of the population occurs at discrete times, while transport caused by diffusion and conduction takes place on shorter time scales. The discrete reproduction, modeled with a set of coupled logistic maps, exhibits phenomena which are not evident in the usual continuum version of the Fisher equation. Several mechanisms for biennial oscillations of the total population are investigated. One of these shows an ordered coupling between random diffusive motion and the chaotic attractor of the logistic map.     

Universality in Ising model gauge-field analogues

The lattice dependence of a class of gauge-invariant Ising models is investigated. Any lattice dependence would indicate that the lattice could not be regarded as irrelevent and that it would be incorrect to define gauge models on a lattice as a basis for investigating the continuum limit. The models investigated lie within the class of multispin Ising models which show a wide variety of lattice-dependent behaviour and so these models should provide a significant test of the importance of the gauge-invariance constraint. Two and three dimensional models are investigated and lattice independence is confirmed. This indicates that imposing gauge symmetries on lattice models can restrict the possible behaviour in such a way that lattice independent continuum limits can be defined.