Exponential wave functions for atomic-molecular systems

Geometric properties of correlated exponential basis functions for n-particle Coulomb systems are studied. Using a system of model Schrödinger equations, the relations between the average values of Coulomb interaction energies of pairs of the particles and average values of cosines of the angles of mutual tilt of interparticle bonds are derived. The use of these relations significantly simplifies calculations of the energy operator matrix of many-particle systems by reducing them to evaluation of the Coulomb and normalization integrals. Geometric inequalities are established that allow one to estimate the many-particle Coulomb interaction integrals that are hard to evaluate. The results obtained can be used in calculations of atomic-molecular systems.     

The thermodynamic limit for long-range random systems

Long-range spin systems with random interactions are considered. A simple argument is presented showing that the thermodynamic limit of the free energy exists and depends neither on the specific random configuration nor on the sample shape, provided there is no external field. The argument is valid for both classical and quantum spin systems, and can be applied to (a) spins randomly distributed on a lattice and interacting via dipolar interactions; and (b) spin systems with potentials of the formJ(x ₁,x ₂)/|x ₁ -x ₂| ^αd , where theJ(x ₁,x ₂) are independent random variables with mean zero,d is the dimension, and α > 1/2. The key to the proof is a (multidimensional) subadditive ergodic theorem. As a corollary we show that, for random ferromagnets, the correlation length is a nonrandom quantity.     

Analyticity for one-dimensional systems with long-range superstable interactions

We consider unbounded spin systems and classical continuous particle systems in one dimension. We assume that the interaction is described by a superstable two-body potential with a decay at large distances at least asr ^?2(lnr)^?(2+ε), ε > 0. We prove the analyticity of the free energy and of the correlations as functions of the interaction parameters. This is done by using a “renormalization group technique” to transform the original model into another, physically equivalent, model which is in the high-temperature (small-coupling) region.     

Off-diagonal long-range order and meissner effect for lattice systems

We study some general properties of a strongly correlated electron system defined on a lattice. Assuming that the system exhibits off-diagonal long-range order, we show that this assumption implies the Meissner effect. This extends to lattice systems previous results obtained for the continuous case.     

Incomplete equilibrium in long-range interacting systems

We use Hamiltonian dynamics to discuss the statistical mechanics of long-lasting quasistationary states particularly relevant for long-range interacting systems. Despite the presence of an anomalous single-particle velocity distribution, we find that the central limit theorem implies the Boltzmann expression in Gibbs' Gamma space. We identify the nonequilibrium submanifold of Gamma space characterizing the anomalous behavior and show that by restricting the Boltzmann-Gibbs approach to this submanifold we obtain the statistical mechanics of the quasistationary states.     

Methods for analysing transport in disordered systems with long-range traps

The continuous time random walk (CTRW) methods and the homomorphic cluster coherent potential approximation (HCCPA) are known to be useful approximate methods for the analysis of transport between localised states in ordinary disordered systems. In order to examine their applicability to disordered systems in the presence of long-range traps, calculations of the trapping probability were performed by these methods for ordered systems containing such traps, and the results compared with those obtained by the usual coherent potential approximation. It is found that the HCCPA method is totally unreliable for such systems, and the reasons for this are discussed. The CTRW methods are found to be unreliable for quantitative results, but the results that they give have the correct qualitative behaviour.     

Finite-size scaling in systems with long-range interaction

The finite-size critical properties of the (n) vector ϕ⁴ model, with long-range interaction decaying algebraically with the interparticle distance r like r ^{-d - σ}, are investigated. The system is confined to a finite geometry subject to periodic boundary condition. Special attention is paid to the finite-size correction to the bulk susceptibility above the critical temperature T _c. We show that this correction has a power-law nature in the case of pure long-range interaction i.e. 0 < σ < 2 and it turns out to be exponential in case of short-range interaction i.e.σ = 2. The results are valid for arbitrary dimension d, between the lower ( d _< = σ) and the upper ( d _> = 2σ) critical dimensions. Received 2 July 2001 and Received in final form 4 Septembre 2001     

Entropy of classical systems with long-range interactions

We discuss the form of the entropy for classical Hamiltonian systems with long-range interaction using the Vlasov equation which describes the dynamics of a N particle in the limit N-->infinity. The stationary states of the Hamiltonian system are subject to infinite conserved quantities due to the Vlasov dynamics. We show that the stationary states correspond to an extremum of the Boltzmann-Gibbs entropy, and their stability is obtained from the condition that this extremum is a maximum. As a consequence, the entropy is a function of an infinite set of Lagrange multipliers that depend on the initial condition. We also discuss in this context the meaning of ensemble inequivalence and the temperature.     

Asymptotic completeness for long-range many-particle systems with Stark effect. II

We prove the existence and the asymptotic completeness of the Dollard-type modified wave operators for many-particle Stark Hamiltonians with long-range potentials.     

Absence of long-range order in certain random systems

It is proved that a classical X-Y model where the field conjugate to the order parameter is static and random shows no long-range order at any temperature for spatial dimensions d ? 4. The proof can be extended to systems with higher order parameter dimension n.     

Absence of long-range order in one-dimensional spin systems

For a one-dimensional Ising model with interaction energy $$E\left\{ \mu \right\} = - \sum\limits_{1 \leqslant i< j \leqslant N} {J(j - i)} \mu _\iota \mu _j \left[ {J(k) \geqslant 0,\mu _\iota = \pm 1} \right]$$ it is proved that there is no long-range order at any temperature when $$S_N = \sum\limits_{k = 1}^N {kJ\left( k \right) = o} \left( {[\log N]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right)$$ The same result is shown to hold for the corresponding plane rotator model when $$S_N = o\left( {\left[ {{{\log N} \mathord{\left/ {\vphantom {{\log N} {\log \log N}}} \right. \kern-\nulldelimiterspace} {\log \log N}}} \right]} \right)$$      

Binary N-step Markov chains and long-range correlated systems

A theory of systems with long-range correlations based on the consideration of binary N-step Markov chains is developed. In our model, the conditional probability that the ith symbol in the chain equals zero (or unity) is a linear function of the number of unities among the preceding N symbols. The correlation and distribution functions as well as the variance of number of symbols in the words of arbitrary length L are obtained analytically and numerically. If the persistent correlations are not extremely strong, the variance is shown to be nonlinearly dependent on L. A self-similarity of the studied stochastic process is revealed. The applicability of the developed theory to the coarse-grained written and DNA texts is discussed.     

Critical behavior in random-spin systems with long-range interactions

Effects of the potential range of the interaction to critical behaviors of quenched random-spin systems are investigated in the limit M → 0 of the MN-component models by means of the renormalization-group theories. As static critical phenomena the stability of the fixed points is investigated and the critical exponents (η, γ, , crossover index) and the equation of state are derived. These phenomena are different from those in pure systems, for the positive specific heat exponent of the pure Heisenberg system.