Mean-Field Driven First-Order Phase Transitions in Systems with Long-Range Interactions

We consider a class of spin systems on ℤ ^d with vector valued spins (S _x ) that interact via the pair-potentials J _x,y S _x ⋅S _y . The interactions are generally spread-out in the sense that the J _x,y 's exhibit either exponential or power-law fall-off. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a first-order phase transition whenever the associated mean-field theory signals such a transition. As a consequence, e.g., in dimensions d≥3, we can finally provide examples of the 3-state Potts model with spread-out, exponentially decaying interactions, which undergoes a first-order phase transition as the temperature varies. Similar transitions are established in dimensions d = 1,2 for power-law decaying interactions and in high dimensions for next-nearest neighbor couplings. In addition, we also investigate the limit of infinitely spread-out interactions. Specifically, we show that once the mean-field theory is in a unique “state,” then in any sequence of translation-invariant Gibbs states various observables converge to their mean-field values and the states themselves converge to a product measure.     

Tricritical and Critical Exponents in Microcanonical Ensemble of Systems with Long-Range Interactions

We explore the tricritical points and the critical lines of both Blume Emery Griffiths and Ising model within long-range interactions in the microcanonical ensemble.For K = Kmtp,the tricritical exponents take the valuesβ = 1/4,1 =γ~-≠γ~+ = 1/2 and 0 =α~-≠α~+ =-1/2,which disagree with classical(mean ffeld) values.When K Kmtp,the phase transition becomes second order and the critical exponents have classical values except close to the canonical tricritical parameters(Kctp),where the values of the critical expoents become β = 1/2,1 = γ~-≠γ~+= 2and 0 =α~-≠α~+ = 1.     

Microcanonical Analysis on a System with Long-Range Interactions

We study a long-range interacting spin chain placed in a staggered magnetic field using microcanonical approach and obtain the global phase diagram. We find that this model exhibits both first order phase transition and second order phase transition separated by a tricritical point, and temperature jump can be observed in the first order phase transition.     

Tricritical and Critical Exponents in Microcanonical Ensemble of Systems with Long-Range Interactions

We explore the tricritical points and the critical lines of both Blume-Emery-Griffiths and Ising model within long-range interactions in the microcanonical ensemble. For K=K_MTP, the tricritical exponents take the values β=1/4, 1=γ^-≠γ⁺=1/2 and 0=α^-≠α⁺=-1/2, which disagree with classical (mean field) values. When K>K_MTP, the phase transition becomes second order and the critical exponents have classical values except close to the canonical tricritical parameters (K_CTP), where the values of the critical expoents become β=1/2, 1=γ^-≠γ⁺=2 and 0=α⁺≠α⁺=1.     

Long-Range Order in Nonequilibrium Systems of Interacting Brownian Linear Oscillators

Long-range order (lro) is established with the help of a generalized Peierls argument for non-equilibrium lattice systems of one-dimensional (linear) interacting oscillators whose equation of motion (for a finite number of them) is the Smolouchowski equation for the density of a probability distribution. Interaction is mediated through the pair nearest-neighbor quadratic translation invariant potential. The initial density is Gibbsian with a potential energy satisfying the Ruelle superstability and regularity conditions.     

Efficiency of Interacting Brownian Motors: Improved Mean-Field Treatment

The “reversible ratchet” model of interacting Brownian motors, introduced by us earlier, is investigated using a one-site approximation of a mean-field type. We confirm the effect of enhanced efficiency due to repulsive interaction and we provide arguments suggesting that the enhancement is of energetic, rather than entropic, origin. We also check the validity of the fluctuation theorem for stationary particle current.     

Macroscopic Determinism in Interacting Systems Using Large Deviation Theory

We consider the quasi-deterministic behavior of systems with a large number, n, of deterministically interacting constituents. This work extends the results of a previous paper [J. Statist. Phys. 99:1225–1249 (2000)] to include vector-valued observables on interacting systems. The approach used here, however, differs markedly in that a level-1 large deviation principle (LDP) on joint observables, rather than a level-2 LDP on empirical distributions, is employed. As before, we seek a mapping _t on the set of (possibly vector-valued) macrostates such that, when the macrostate is given to be a ₀ at time zero, the macrostate at time t is _t (a ₀) with a probability approaching one as n tends to infinity. We show that such a map exists and derives from a generalized dynamic free energy function, provided the latter is everywhere well defined, finite, and differentiable. We discuss some general properties of _t relevant to issues of irreversibility and end with an example of a simple interacting lattice, for which an exact macroscopic solution is obtained.     

Resonant Continuum in the Relativistic Mean-Field Theory

Energies, widths and wave functions of the single-particle resonant continuum are determined by solvingscattering states of the Dirac equation with proper asymptotic conditions for the continuous spectrum in the relativisticmean-field theory. The relativistic regular and irregular Coulomb wave functions are calculated numerically. Theresonance states in the continuum for some closed- or sub-closed-shell nucleus in Sn-isotopes, such as 1 14Sn, 1 16Sn, 1 18Sn,and 120Sn are calculated. Results show that the S-matrix method is a reliable and straightforward way in determiningenergies and widths of resonant states.     

Slave-Fermion Mean-Field Theory of Heisenberg Model

A nearly half-filled two-dimensional Heisenberg model is investigated. A slave-fermion method with fermions as the charge carriers and bosons as the spin carriers is proposed. The ground state shows antiferromagnetic long range order at T = 0. The spin-spin correlation and static susceptibility are also obtained.     

Slave-Fermion Mean-Field Theory of Heisenberg Model

A nearly half-filled two-dimensional Heisenberg model is investigated. A slave-fermion method with fermions as the charge carriers and bosons as the spin carriers is proposed. The ground state shows antiferromagnetic long range order at T = 0. The spin-spin correlation and static susceptibility are also obtained.     

Long-Range Orders in Models of Itinerant Electrons Interacting with Heavy Quantum Fields

The Falicov–Kimball model consists of itinerant lattice fermions interacting with Ising spins by an on-site potential of strength U. Kennedy and Lieb proved that at half filling there is a low temperature phase with chessboard long range order on ^d , d2, for all non-zero values of U. Here we investigate the stability of this phase when small quantum fluctuations of the Ising spins are introduced in two different ways. The first one corresponds to replace the classical spins by quantum two level systems attached to each site of the lattice. In the second one we interpret the spins as occupation numbers of localized f-electrons or heavy ions which have a small kinetic energy. This leads to the so-called asymmetric Hubbard model. For both models we prove that for all non-zero values of U the long range order of the original Falicov–Kimball model remains stable if the additional quantum fluctuations are small enough. This result is proved by non-perturbative methods based on a chessboard estimate and the principle of exponential localisation. In order to derive the chessboard estimate the phase factors in the kinetic energy of fermions must have a flux equal to . We also investigate the models where the fermions are replaced by hard-core bosons and prove the same result for large U. For hard core bosons the kinetic term is the conventional one with zero phase factors. For small U and hard-core bosons we find that there is an off-diagonal long range order for low enough temperature and any strength of the additional quantum fluctuations. Open problems are discussed.     

Mean-Field Theory for Percolation Models of the Ising Type

The q=2 random cluster model is studied in the context of two mean-field models: the Bethe lattice and the complete graph. For these systems, the critical exponents that are defined in terms of finite clusters have some anomalous values as the critical point is approached from the high-density side, which vindicates the results of earlier studies. In particular, the exponent ~ which characterizes the divergence of the average size of finite clusters is 1/2, and ~, the exponent associated with the length scale of finite clusters, is 1/4. The full collection of exponents indicates an upper critical dimension of 6. The standard mean field exponents of the Ising system are also present in this model (=1/2, =1), which implies, in particular, the presence of two diverging length-scales. Furthermore, the finite cluster exponents are stable to the addition of disorder, which, near the upper critical dimension, may have interesting implications concerning the generality of the disordered system/correlation length bounds.     

Hydrodynamic Limit of Brownian Particles Interacting with Short- and Long-Range Forces

We investigate the time evolution of a model system of interacting particles moving in a d-dimensional torus. The microscopic dynamics is first order in time with velocities set equal to the negative gradient of a potential energy term plus independent Brownian motions: is the sum of pair potentials, V(r)+ ^d J(r); the second term has the form of a Kac potential with inverse range . Using diffusive hydrodynamic scaling (spatial scale ^–1, temporal scale ^–2) we obtain, in the limit 0, a diffusive-type integrodifferential equation describing the time evolution of the macroscopic density profile.     

Quasi-Stationary States for Particle Systems in the Mean-Field Limit

We prove that the N particles approximation of a class of stable stationary solutions of the Vlasov equation is uniformly valid on a time scale N ^β for β>0 (explicitly given in various cases) much longer than the usual log N scale. The vortex blob method in dimension 2 is also discussed. The result applies to a class of stationary solutions more general than in a previous work.     

Understanding the Nature of the Long-Range Memory Phenomenon in Socioeconomic Systems

In the face of the upcoming 30th anniversary of econophysics, we review our contributions and other related works on the modeling of the long-range memory phenomenon in physical, economic, and other social complex systems. Our group has shown that the long-range memory phenomenon can be reproduced using various Markov processes, such as point processes, stochastic differential equations, and agent-based models—reproduced well enough to match other statistical properties of the financial markets, such as return and trading activity distributions and first-passage time distributions. Research has lead us to question whether the observed long-range memory is a result of the actual long-range memory process or just a consequence of the non-linearity of Markov processes. As our most recent result, we discuss the long-range memory of the order flow data in the financial markets and other social systems from the perspective of the fractional Lèvy stable motion. We test widely used long-range memory estimators on discrete fractional Lèvy stable motion represented by the auto-regressive fractionally integrated moving average (ARFIMA) sample series. Our newly obtained results seem to indicate that new estimators of self-similarity and long-range memory for analyzing systems with non-Gaussian distributions have to be developed.     

Self-consistent Mean-Field Theory of Two-Dimensional Anisotropic Heisenberg Antiferrornagnet

A self-consistent mean-field theory for the two-dimensional spin-1/2 XY-Heisenberg an tiferromagnet without long-range-order is developed. Within this theoretical framework, the thermodynamic properties, such as the internal energy, specific heat and parallel susceptibility, are discussed, and the results are in rough agreement with the numerical simulation.