Interdimensional scaling laws with long range interactions

Scaling laws relating the critical indices in different dimensionalities with the same long range (1/r^x) interactions are derived. Critical exponents are then generated from the short range values and from recent near-mean-field expansions.     

Metastability in a long range one-dimensional Ising model

Decay of the metastable state of the one-dimensional Ising model with an x^-α potential is investigated using instanton (critical droplet) techniques. Calculations indicating that the analytic structure of the free energy is modified by droplet-droplet interactions for 1 < α ? 1.2 are presented.     

Langevin dynamics of a π4-model with long range interactions

The Dominicis-Peliti generating functional (GF) method is used for the investigation of a Langevin dynamics of the π⁴-model: the symmetric double-well on-site potential and the infinite range interparticle interaction. We limit ourselves to the range above the temperature of the second order phase transition. The role of the 1/N-fluctuations (where N is the number of particles) is systematically investigated by using the steepest descent method. It is shown that the functional Legendre transformation directly results in the kinetic equation for the complete correlation function. Although this equation resembles the mode coupling equations used to describe the glass transition, it is qualitatively different. The solutions of this non-linear equation are investigated. It is shown that 1/N-fluctuations do not result in a breaking or ergodicity if the mean-field correlator is ergodic. On the other hand, if the mean-field correlator is nonergodic (e.g. if the time is much less than the inverse Kramers rate) then 1/N-fluctuations restore the ergodicity with characteristic relaxation time proportional to N.     

Nonlinear damping in spin systems: long range spin-spin interactions

The effective field theory of nonlinear damping for strongly interacting spins driven by external fields and weakly coupled to a bath is derived starting from the microscopic dynamics. The use of a block-spin model with long range spin-spin interactions makes it possible to treat the spin dynamics exactly and to eliminate the microscopic degrees of freedom. The results are presented in form of a nonlinear integro-differential equation for the time and space dependent magnetization. This equation is of memory type, applies to spin-spin interactions of any symmetry and is valid for inhomogeneous external fields of arbitrary strength and time dependence. For an isotropic ferromagnet in weak and slow external fields the general integro-differential equation is shown to reduce to a differential equation of motion of Landau-Lifshitz-Bloch type.This work was performed within a program of the Sonderforschungsbereich 185 Nichlineare Dynamik Frankfurt-Darmstadt     

Relaxation of quasi-stationary states in long range interacting systems and a classification of the range of pair interactions

Systems of particles interacting with long range interactions present generically ”quasi-stationary states” (QSS), which are approximately time-independent out of equilibrium states. In this proceedings, we explore the generalization of the formation of such QSS and their relaxation from the much studied case of gravity to a generic pair interaction with the asymptotic form of the potential v(r) ∼ 1/r ^γ with γ > 0 in d dimensions. We compute analytic estimations of the relaxation time calculating the rate of two body collisionality in a virialized system approximated as homogeneous. We show that for γ < (d − 1/2), the collision integral is dominated by the size of the system, while for γ > (d − 1/2), it is dominated by small impact parameters. In addition, the lifetime of QSS increases with the number of particles if γ < d − 1 (i.e. the force is not integrable) and decreases if γ > d − 1. Using numerical simulations we confirm our analytic results. A corollary of our work gives a ”dynamical” classification of interactions: the dynamical properties of the system depend on whether the pair force is integrable or not.     

Spin glasses and other lattice systems with long range interactions

We study classical lattice systems, in particular real spin glasses with Ruderman-Kittel interactions and dipole gases, with interactions of very long (non-summable) range but variable sign. Using the Kac-Siegert representation of such systems and Brascamp-Lieb inequalities we are able to establish detailed properties of the high-temperature phase, such as decay of connected correlations, for these systems.     

Critical relaxation for spin system with long range interactions

The relaxation rates for time independent spin correlation functions are estimated for a long range interaction. The damping of a critical spin variable in the critical region is found to be a function of this range.     

Effect of long range coulomb interactions on the mott transition

We reconsider the Mott transition problem in the presence of long range Coulomb interactions. Using an extended dynamical mean field theory (DMFT) that sums an important class of diagrams absent in ordinary DMFT, we show that in the presence of Coulomb interactions, the zero temperature Mott transition is, as envisioned by Mott, discontinuous in two and three dimensions.     

Lattice Lotka-Volterra model with long range mixing

A spatio-temporal process in the Lattice Lotka Volterra (LLV) model, when realized on low dimensional support, is studied. It is shown that the introduction of a long-range mixing causes a drastic change in the system’s behavior, which transits from small random-like fluctuations to global oscillations when the mixing rate transcends above a critical point. The amplitude of the induced oscillations is well defined by the mixing rate and is insensitive to the initial conditions and the lattice size variations. The observed behavior essentially differs from that predicted by the Mean-Field model which is conservative. The oscillations are of limit-cycle type and appear as a stochastic analog of a Hopf bifurcation.     

From mechanical to biological oscillator networks: The role of long range interactions

The study of one-dimensional particle networks of Classical Mechanics, through Hamiltonian models, has taught us a lot about oscillations of particles coupled to each other by nearest neighbor (short range) interactions. Recently, however, a careful analysis of the role of long range interactions (LRI) has shown that several widely accepted notions concerning chaos and the approach to thermal equilibrium need to be modified, since LRI strongly affects the statistics of certain very interesting, long lasting metastable states. On the other hand, when LRI (in the form of non-local or all-to-all coupling) was introduced in systems of biological oscillators, Kuramoto’s theory of synchronization was developed and soon thereafter researchers studied amplitude and phase oscillations in networks of FitzHugh Nagumo and Hindmarsh Rose (HR) neuron models. In these models certain fascinating phenomena called chimera states were discovered where populations of synchronous and asynchronous oscillators are seen to coexist in the same system. Currently, their synchronization properties are being widely investigated in HR mathematical models as well as realistic neural networks, similar to what one finds in simple living organisms like the C.elegans worm.     

Competition and cooperation on a toy Autobahn model

Traffic on an one-lane freeway is simulated using a continuous space-discrete time probabilistic cellular automata model.The effet of different individual driving patterns is estimated by monitoring the traffic flow, the velocity and acceleration distributions, the aver-age number of accidents, and the distribution of density-waves (traffic jams) as a function of traffic density. The number of accidents, traffic jams, and the fuel consumption are drastically reduced by driving strategies adapting to local traffic conditions. At high traffic densities this leads, however, to a decrease in the global traffic throughout.     

Conductance of ion channels and nanopores with charged walls: a toy model

We consider ion transport through protein ion channels in lipid membranes and water-filled nanopores in silicon films. It is known that, due to the large ratio of dielectric constants of water and the surrounding material, an ion placed inside the channel faces a large electrostatic self-energy barrier. The barrier leads to an exponentially large resistance of the channel. We study reduction of the electrostatic barrier by immobile charges located on the internal walls of the channel. We show that the barrier practically vanishes already at relatively small concentration of wall charges.     

Fermion condensation and non Fermi liquid behavior in a model with long range forces

The phenomenon of the so called Fermion condensation, a phase transition analogous to Bose condensation but for Fermions, postulated in the past to occur in systems with strong momentum dependent forces, is reanalyzed in a model with infinite range interactions. The strongly non Fermi Liquid behavior of this system is demonstrated analytically at T = 0 and at T ≠ 0 in the superconducting and normal phases. The validity of the quasiparticle picture is investigated and seen to hold true for temperatures less than the characteristic temperature T _f of the Fermion condensation.     

Low energy behaviour of a one dimensional Fermi model with short and long range interaction

A one dimensional Fermi model is investigated for a general four parameter interaction, containing both short and long range components. Using the second order renormalization approach it is shown that the type of interaction with all the four participating electrons around the same Fermi point has a drastic effect. It tends to suppress the superconducting type instability and favours a normal metal, antiferromagnetic or charge density wave state.     

Effect of long range interactions on the growth of compact clusters under deposition

In the paper the role of long range interactions on the growth of a volume conserving surface is studied using the Nonlocal Conserved Kardar-Parisi-Zhang (NCKPZ) equation. It is shown that previous theoretical predictions are inconsistent with an exact one-dimensional result. This serves as a motivation for construction of a Self-Consistent Expansion (SCE) that recovers the exact one-dimensional result, and gives the scaling exponents in higher dimensions as well. A possible application of this result to colloidal systems is discussed.