Equilibrium states for classical systems

It is shown that the Dobrushin-Lanford-Ruelle equations for the probability measure μ, and the Kirkwood-Salsburg type equations for the lattice or continuum correlation functions ?, and for the spin correlation functions σ, are essentially equivalent for all ?, σ, and μ satisfying certain boundedness conditions. It is also noted that the lattice equations are identical to the equations for the stationary states of a certain Markoff process. This extends previous results of Ruelle, Brascamp and Holley who proved some of these equivalences for states.     

Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems

We consider quantum unbounded spin systems (lattice boson systems) in -dimensional lattice space Z. Under appropriate conditions on the interactions we prove that in a region of high temperatures the Gibbs state is unique, is translationally invariant, and has clustering properties. The main methods we use are the Wiener integral representation, the cluster expansions for zero boundary conditions and for general Gibbs state, and explicitly -dependent probability estimates. For one-dimensional systems we show the uniqueness of Gibbs states for any value of temperature by using the method of perturbed states. We also consider classical unbounded spin systems. We derive necessary estimates so that all of the results for the quantum systems hold for the classical systems by straightforward applications of the methods used in the quantum case.     

Passive states for finite classical systems

Starting from a passivity condition based on the second law of thermodynamics [12], we show that ground states and Gibbs states (0<<) are essentially the only passive states.Research supported by M. Skodowska-Curie Fund Grant No. OIP74-01416.     

Fluctuation susceptibility relations for classical spin systems

We prove identities between integrated Ursell functions and derivatives of the pressure in the thermodynamic limit, for multicomponent classical spin systems which obey the Lee-Yang theorem and some form of Gaussian domination, when the susceptibility is finite (T>T _c). Following Refs. 3 and 4, we view the moment generating function of the magnetization as the inverse of an infinitely divisible characteristic function. Fluctuation susceptibility relations of all orders then follow by bounding the corresponding cumulants, taken in zero external field. High-order cumulants are bounded in terms of the susceptibility using Gaussian and Simon's inequalities for short-range interactions.     

Solvable spin systems with random interactions

It is shown that if the bonds connecting spins on a lattice are separable functions of random variables, the thermodynamic and magnetic parameters may be obtained using the known properties of a spin system with non-random bonds.     

Thermodynamic duality for classical systems of arbitrary spin

Given a classical spins system, namely, a set of spin sites of maximum spins inv-dimensional space along with a Hamiltonian defined on the possible spin configurations, a general method is described for constructing a large class of dual lattices of the same spin. The method utilizes the commutative group structure with which the configuration space is endowed.On leave from: Department of Mathematics, Indiana University, Bloomington, Indiana.     

Thermodynamic states of classical systems

The existence of a unique thermodynamic state for dilute classical systems is proved for a class of regular multi-particle potentials. The method relies on integral equations for modified correlation functions.Supported in part by U.S. Air Force Office of Scientific Research Contract No. 53-296-08.     

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.     

Topology on interactions for classical continuous systems

For classical continuous systems with interactions without hard core a metric space of interactions is formed. To each type of thermodynamic limit there exist classes of interactions for which the pressure exists and is a continuous functional. Metric spaces are created with respect to each kind of ensemble, such that the sets of local thermodynamic functions (pressure and the densities of free energy and entropy) are equicontinuous.     

Analyticity and uniqueness of the invariant equilibrium states for general spin 1/2 classical lattice systems

Asano-Ruelle-Slawny method is generalized to discuss analyticity and uniqueness of the correlation functions in terms of the group structure associated with any lattice systems. The use of Poisson formula for abelian groups gives a simple method to obtain explicit domains where the above properties are verified.     

Ornstein-Zernike and analyticity properties for classical lattice spin systems

The leading Ornstein-Zernike behavior of the truncated two-point function G(x) is obtained from information on the poles of the Fourier transform G~(p). Analyticity of G~(p) as a function of the complex variables p and the inverse temperature β is studied in detail for |β| sufficiently small.     

Longitudinal magnetic excitations in classical spin systems

Using spin dynamics simulations we predict the splitting of the longitudinal spin-wave peak in all antiferromagnets with single site anisotropy into two peaks separated by twice the energy gap at the Brillouin zone center. This phenomenon has yet to be observed experimentally but can be easily investigated through neutron scattering experiments on MnF2 and FeF2. We have also determined that for all classical Heisenberg models the longitudinal propagative excitations are entirely multiple spin wave in nature.     

FT NMR of nonequilibrium states of complex spin systems. II. Weakly coupled systems

Since the Fourier transform NMR spectrum of a nonequilibrium state may be different from the slow-passage spectrum, a theory is developed which describes the phenomenon in any weakly coupled system, including those with magnetic equivalence. Liouville space techniques are used to derive simple expressions for the line intensities as a function of flip angle, in terms of Legendre polynomials. The theory uses F² representations directly and is applicable to any weakly coupled system. The results are tabulated for A_mX_n systems, m,n = 1, 2, 3, and 6. The theory is still applicable when. the weak coupling approximation starts to break down and line intensities are perturbed. It is shown that if the nonequilibrium intensities are corrected by a simple factor, as in the equilibrium case, these systems may be treated as if they were weakly coupled. In practice, the theory may be used to extrapolate to the slow-passage, zero flip angle spectrum. Furthermore, different parts of a degenerate line may be separated by means of the flip angle dependence. As an illustration, a nonequilibrium state was created among the protons in ethyl acetate. The experimental flip angle dependence of its spectra is well represented by the theory.     

The classical limit of quantum spin systems

We derive a classical integral representation for the partition function,Z ^Q , of a quantum spin system. With it we can obtain upper and lower bounds to the quantum free energy (or ground state energy) in terms of two classical free energies (or ground state energies). These bounds permit us to prove that when the spin angular momentumJ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality isZ ^C (J)Z ^Q (J)Z ^C (J+1).On leave from the Department of Mathematics, M.I.T., Cambridge, Mass. 02139, USA. Work partially supported by National Science Foundation Grant GP-31674X and by a Guggenheim Memorial Foundation Fellowship.     

Ground states in classical lattice systems with hard core

In this paper we prove the existence of translation invariant ground states in an infinite classical lattice system with hard core and give a characterization of their support. Some examples are discussed.     

Tricritical point in classical spin systems with random transverse fields

The existence of a tricritical point is demonstrated for a classical n-vector model with annealed random transverse fields, and the corresponding tricritical temperature T_t and distribution width σ_t are evaluated within Landau theory for continuous spin dimensionality -2 < n < 10.     

Ground states of quantum spin systems

We prove that ground states of quantum spin systems are characterized by a principle of minimum local energy and that translationally invariant ground states are characterized by the principle of minimum energy per unit volume.     

Study of classical mechanical systems with complex potentials

We apply the factorization technique developed by Kuru and Negro [Ann. Phys. 323 (2008) 413] to study complex classical systems. As an illustration we apply the technique to study the classical analogue of the exactly solvable PT symmetric Scarf II model, which exhibits the interesting phenomenon of spontaneous breakdown of PT symmetry at some critical point. As the parameters are tuned such that energy switches from real to complex conjugate pairs, the corresponding classical trajectories display a distinct characteristic feature — the closed orbits become open ones.     

Equilibrium and metastable states of classical systems

We formulate a characterization of equilibrium and metastable states of classical hard-core continuous systems in terms of certain global and local stability conditions. The equilibrium states are assumed to be those that are both locally and globally stable; the metastable states are assumed to be those that are locally, but not globally, stable, and that possess also a certain restricted global stability. It is found that a certain specified class of systems with appropriately weakly tempered, or long range forces, can support metastable states, possessing bona fide thermodynamic properties, whose pressure functions are real analytic continuations in the chemical potential of those of some equilibrium phases. This result is complementary to that of Lanford and Ruelle, concerning the absence of metastable states of systems with strongly tempered forces.