Power-law Long-range Interactions Research Articles

Synchronization of oscillators with long-range power law interactions

We present analytical calculations and numerical simulations for the synchronization of oscillators interacting via a long-range power law interaction on a one-dimensional lattice. We have identified the critical value of the power law exponent α(c) across which a transition from a synchronized to an unsynchronized state takes place for a sufficiently strong but finite coupling strength in the large system limit. We find α(c)=3/2. Frequency entrainment and phase ordering are discussed as a function of α≥1 . The calculations are performed using an expansion about the aligned phase state (spin-wave approximation) and a coarse graining approach. We also generalize the spin-wave results to the d -dimensional problem.

Long-range effects on superdiffusive algebraic solitons in anharmonic chains

Studies on thermal diffusion of lattice solitons in Fermi-Pasta-Ulam (FPU)-like lattices were recently generalized to the case of dispersive long-range interactions (LRI) of the Kac-Baker form. The position variance of the soliton shows a stronger than linear time-dependence (superdiffusion) as found earlier for lattice solitons on FPU chains with nearest neighbour interactions (NNI). In contrast to the NNI case where the position variance at moderate soliton velocities has a considerable linear time-dependence (normal diffusion), the solitons with LRI are dominated by a superdiffusive mechanism where the position variance mainly depends quadratic and cubic on time. Since the superdiffusion seems to be generic for nontopological solitons, we want to illuminate the role of the soliton shape on the superdiffusive mechanism. Therefore, we concentrate on a FPU-like lattice with a certain class of power-law long-range interactions where the solitons have algebraic tails instead of exponential tails in the case of FPU-type interactions (with or without Kac-Baker LRI). A collective variable (CV) approach in the continuum approximation of the system leads to stochastic integro-differential equations which can be reduced to Langevin-type equations for the CV position and width. We are able to derive an analytical result for the soliton diffusion which agrees well with the simulations of the discrete system. Despite of structurally similar Langevin systems for the two soliton types, the algebraic solitons reach the superdiffusive long-time limit with a characteristic $t^{1.5}$ time-dependence much faster than exponential solitons. The soliton shape determines the diffusion constant in the long-time limit that is approximately a factor of $\pi$ smaller for algebraic solitons.

On the extended Kolmogorov–Nagumo information-entropy theory, the [formula omitted] duality and its possible implications for a non-extensive two-dimensional Ising model

The aim of this paper is to investigate the q → 1 / q duality in an information-entropy theory of all q -generalized entropy functionals (Tsallis, Renyi and Sharma–Mittal measures) in the light of a representation based on generalized exponential and logarithm functions subjected to Kolmogorov's and Nagumo's averaging. We show that it is precisely in this representation that the form invariance of all entropy functionals is maintained under the action of this duality. The generalized partition function also results to be a scalar invariant under the q → 1 / q transformation which can be interpreted as a non-extensive two-dimensional Ising model duality between systems governed by two different power law long-range interactions and temperatures. This does not hold only for Tsallis statistics, but is a characteristic feature of all stationary distributions described by q -exponential Boltzmann factors.

Coupled oscillators with power-law interaction and their fractional dynamics analogues

A one-dimensional chain of coupled oscillators with the long-range power-law interactions is considered. Equations of motion in the infrared limit are mapped onto the continuum equation with the Riesz fractional derivative of order α, when 0 < α < 2. The evolution of soliton-like and breather-like structures is obtained numerically and compared for two types of simulations: using the chain of oscillators and using the continuous medium equation with the fractional derivative.

Nonlinear fractional dynamics on a lattice with long range interactions

A unified approach has been developed to study nonlinear dynamics of a 1D lattice of particles with long-range power-law interaction. A classical case is treated in the framework of the generalization of the well-known Frenkel–Kontorova chain model for the non-nearest interactions. Quantum dynamics is considered following Davydov's approach for molecular excitons. In the continuum limit the problem is reduced to dynamical equations with fractional derivatives resulting from the fractional power of the long-range interaction. Fractional generalizations of the sine-Gordon, nonlinear Schrödinger, and Hilbert–Schrödinger equations have been found. There exists a critical value of the power s of the long-range potential. Below the critical value ( s < 3 , s ≠ 1 , 2) we obtain equations with fractional derivatives while for s ⩾ 3 we have the well-known nonlinear dynamical equations with space derivatives of integer order. Long-range interaction impact on the quantum lattice propagator has been studied. We have shown that the quantum exciton propagator exhibits transition from the well-known Gaussian-like behavior to a power-law decay due to the long-range interaction. A link between 1D quantum lattice dynamics in the imaginary time domain and a random walk model has been discussed.

Reaction zones and quenched charged-particle systems with long-range interactions

We determine the evolving segregated or mixed morphology of charged-particle systems with long-range power-law interactions and overall charge neutrality that have been quenched to a low temperature. Segregated morphology systems are characterized by the size of uniformly charged domains, $L(t),$ the particle separation within the domains, ${l}_{\mathrm{AA}}(t),$ the particle flux density leaving the domains, $J(t),$ the width of reaction zones between domains, $W(t),$ the particle spacing within the reaction zones, ${l}_{\mathrm{AB}}(t),$ and the particle lifetime in the reaction zones, $\ensuremath{\tau}(t).$ Mixed morphology systems are essentially one large reaction zone, with $L\ensuremath{\sim}{l}_{\mathrm{AB}}\ensuremath{\sim}{l}_{\mathrm{AA}}.$ By relating these quantities through the scaling behavior of particle fluxes and microscopic annihilation rates within reaction zones, we determine the characteristic time exponents of these quantities at late times. The morphology of the system, segregated or mixed, is also determined self-consistently. With this unified approach, we consider systems with diffusion and/or long-range interactions, and with either uncorrelated or correlated high-temperature initial conditions. Finally, we discuss systems with particlelike topological defects and electronic systems in various substrate dimensions---including quantum Hall devices with Skyrmions.

Solitons in anharmonic chains with power-law long-range interactions

We study the influence of long-range interactions with distance dependence ${r}^{\ensuremath{-}s}$ of the elastic coupling constant on the properties of pulse solitons in a one-dimensional anharmonic chain. Introducing the approximations of small amplitude and long wavelength, we have arrived at the Boussinesq equation for $s&gt;5$ and the Benjamin-Ono equation for $s=4.$ For $s&gt;5$ the soliton tails are exponential while for $3&lt;s&lt;~5$ they are algebraic. For $s&lt;~3.5$ there is an energy gap between the spectra of plane waves and the soliton states.

Scaling model of annihilation-diffusion kinetics for charged particles with long-range interactions.

We propose a general scaling model for the diffusion-annihilation reaction ${A}_{+}+{A}_{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\emptyset}$ with long-range power-law interactions. The presented scaling arguments lead to the finding of three different regimes, depending on the space dimensionality $d$ and the long-range force power exponent $n$. The obtained kinetic phase diagram agrees well with existing simulation data and approximate theoretical results.

Scaling theory of particle annihilation in systems with a long-range interaction.

A scaling theory is presented which takes into account the influence of long-range power-law interactions on the density decay rate in a system of charged particles which move in a viscous medium and annihilate via the bimolecular reaction ${\mathit{A}}_{+}$+${\mathit{A}}_{\mathrm{\ensuremath{-}}}$\ensuremath{\rightarrow}0. Various regimes of the annihilation process depending on system dimensionality d and long-range-force power exponent n, are discovered and analyzed.

Spherical random-field systems with long-range interactions: general results and application to the Coulomb glass

A classical spherical random-field Hamiltonian with long-range (power-law) interactions is investigated by means of the replica theory. Both ferromagnetic and anti-ferromagnetic interactions are considered. The use of continuous variables instead of Ising variables in the spherical version of the model allows one to calculate the free energy exactly. The existence of an equilibrium phase transition is investigated based on the replica-symmetric solution. The results are applied to the Coulomb-glass model of interacting localized electrons in a disordered solid. This model is shown not to have an equilibrium phase transition for spatial dimensions D 4 the model has a phase transition to an ordered phase; however, it does not have a phase transition to a 'glassy' phase.

Domain walls subject to quenched impurities (invited)

Interfaces between domains become rough as a result of pinning to impurities. The degree of roughening can be characterized by the exponent describing the scaling of fluctations with the size of the domain wall. Current understanding of the roughening process will be reviewed. The simple Imry–Ma scaling arguments are expected to give the correct roughening exponent in systems subject to random-field impurities, but they clearly fail in the case of random bonds. Possible reasons for this failure are examined. A novel class of impurities, with long-range power-law interactions with the interface, are considered. By changing the exponent of the power law, this model interpolates between random bonds and random fields.

Low-temperature renormalization group for the Lifshitz point

The momentum-shell technique is employed to derive the recursion relations for the Lifshitz point above the lower critical dimensionality. For the uniaxial point ($m=1$) the expansion is made in $\ensuremath{\epsilon}=d\ensuremath{-}2.5$ and the critical exponents are calculated. The results are ${\ensuremath{\lambda}}_{\mathrm{t}}={\ensuremath{\nu}}_{4}^{\ensuremath{-}1}=2\ensuremath{\epsilon}$, ${\ensuremath{\eta}}_{2}=\frac{\ensuremath{\epsilon}}{(n\ensuremath{-}2)}$, ${\ensuremath{\eta}}_{4}=\frac{2\ensuremath{\epsilon}}{(n\ensuremath{-}2)}$, ${\ensuremath{\beta}}_{k}=\frac{1}{2}+\frac{3\ensuremath{\epsilon}}{4(n\ensuremath{-}2)}$, where $n$ is the number of components of the spin. The critical behavior in the presence of long-range power-law interactions is also studied.