Statistical mechanics of quantum spin systems. III

In the algebraic formulation the thermodynamic pressure, or free energy, of a spin system is a convex continuous functionP defined on a Banach space of translationally invariant interactions. We prove that each tangent functional to the graph ofP defines a set of translationally invariant thermodynamic expectation values. More precisely each tangent functional defines a translationally invariant state over a suitably chosen algebra of observables, i. e., an equilibrium state. Properties of the set of equilibrium states are analysed and it is shown that they form a dense set in the set of all invariant states over . With suitable restrictions on the interactions, each equilibrium state is invariant under time-translations and satisfies the Kubo-Martin-Schwinger boundary condition. Finally we demonstrate that the mean entropy is invariant under time-translations.     

Convergence in Energy-Lowering (Disordered) Stochastic Spin Systems

We consider stochastic processes, with finite, in which spin flips (i.e., changes of S ^t _x ) do not raise the energy. We extend earlier results of Nanda–Newman–Stein that each site x has almost surely only finitely many flips that strictly lower the energy and thus that in models without zero-energy flips there is convergence to an absorbing state. In particular, the assumption of finite mean energy density can be eliminated by constructing a percolation-theoretic Lyapunov function density as a substitute for the mean energy density. Our results apply to random energy functions with a translation-invariant distribution and to quite general (not necessarily Markovian) dynamics.     

Relaxation in two-level long-range-memory systems

Recently, new rigorous results concerning integrals of exact memory functions in the time-convolution Generalized Master Equations (GME) for state occupation probabilities, governing relaxation of open quantum systems, have been obtained. They include that a) time integrals of exact memories and b) memories w _ij(t) have tails unobtainable by perturbational arguments which cause that does not exist or is infinite. For a two-level system, a simple model for such memories is considered and solved. It is concluded that GME may yield that with increasing time, the system unphysically more and more deviates from equilibrium, indicating thus instability of the equilibrium distribution. Thus, in contrast to, e.g., the famous Boltzmann equation, the mathematical structure of GME alone does not guarantee the stability of the equilibrium state.     

Entropy andp-particle observables. I. The general program

An information-theoretic notion of entropy is proposed for a system ofN interacting particles which assesses an observer's limited knowledge of the state of the system, assuming that he or she can measure with arbitrary precision all one-particle observables and correlations involving some numberp of the particles but is completely ignorant of the form of any higher-order correlations involving more thanp particles. The idea is to define a generic measure of entropyS[ ] = –Tr log for an arbitrary density matrix or distribution function , and then, given the trueN-particle, to define a reduced _R ^P which reflects the observer's partial knowledge. The result, at any timet, is a chain of inequalitiesS[ _R ¹ ]S[ _R ² ]...S[ _R ^N ]S[], with true equalityS[ _R ^p ]=S[ _R ^p+1 ] if and only if the true factorizes exactly into a product of contributions involving all possiblep-particle groupings. It follows further than (1) if, at some initial timet ₀, the true factorizes in this way, thenS[ _R ^p (]S[ _R ^p (t ₀)] for all finite timest>t ₀, with equality if and only if the factorization is restored, and (2) the initial response of the system must be to increase itsp-particle entropy.     

Rigorous information-theoretic derivation of quantum-statistical thermodynamics. II

Part I of the present work outlined the rigorous application of information theory to a quantum mechanical system in a thermodynamic equilibrium state. The general formula developed there for the best-guess density operator \(\hat \rho\) was indeterminate because it involved in an essential way an unspecified prior probability distribution over the continuumD _H of strong equilibrium density operators. In Part II mathematical evaluation of \(\hat \rho\) is completed after an epistemological analysis which leads first to the discretization ofD _H and then to the adoption of a suitable indifference axiom to delimit the set of admissible prior distributions. Finally, quantal formulas for information-theoretic and thermodynamic entropies are contrasted, and the entire work is summarized.     

Gauge covariant theory of the generating operator. I

A gauge covariant formulation of the generating operator (-operator) theory for the Zakharov-Shabat system is proposed. The operator , corresponding to the gauge equivalent system in the pole gauge is explicitly calculated. Thus the unified approach to the nonlinear Schrödinger-type equations based on is automatically reformulated with the help of for the Heisenberg ferromagnet-type equations. Consequently, it is established that the conserved densities for the Heisenberg-ferromagnet-type equations are polynomial inS(x) and itsx-derivatives. Special attention is paid to the interrelation between the hierarchies of symplectic structures corresponding to the above mentioned families of gauge-equivalent equations. It is shown that the geometrical properties of the conjugated operator * are gaugeindependent.     

A theoretical study on electron transfer in low-energy collisions of a collinear meta-stable with a $ {\rm{Na}}_3^ - $ {\rm{Na}}_3^ - " border="0"> Na

Electron transfer in the collisions of a with a Na is theoretically studied. It is assumed that the target is collinear (D _h ) and that its electronic state is meta-stable triplet state. Adiabatic potential energy surfaces and non-adiabatic couplings of the system are calculated by using a semi-empirical diatomics-in-molecules (DIM) method. The positions of (avoided)-crossings of potential surfaces are investigated and the non-adiabatic couplings between two different electronic states are calculated. An avoided crossing is found in the region where the separation between the target and projectile is relatively large (10–15 bohr). A dynamical calculation demonstrates that this crossing causes charge transfer between the target and projectile. Another intersection at a smaller separation changes the targets spin state (from triplet state to singlet state or vice versa). The cross-sections for charge and spin transfer reaction are estimated at the collision energy of 6.8 keV. It is found that the charge transfer cross-section is extremely enhanced when the target cluster ion is in its meta-stable triplet state comared to the case where the cluster is the ground singlet state.     

Equilibrium states of gravitational systems

We formulate the equilibrium correlation functions for local observables of an assembly of non-relativistic, neutral gravitating fermions in the limit where the number of particles becomes infinite, and in a scaling where the region , to which they are confined, remains fixed. We show that these correlation functions correspond, in the limit concerned, to states on the discrete tensor product , where the are copies of the gauge invariantC*-algebra of the CAR overL ²(R ³). The equilibrium states themselves are then given by , where , is the Gibbs state on for an infinitely extended ideal Fermi gas at density , and where ₀ is the normalised density function that minimises the Thomas-Fermi functional, obtained in [2], governing the equilibrium thermodynamics of the system.     

B c and $$b\bar bc\bar c$$ at theZ 0-boson pole

The cross-section for the production of quarks ine ⁺ e ^– annihilation, that proves to be at a level of for is calculated within the frames of the QCD perturbation theory. The cross sections for the associated production of 1S-and 2S-wave states ofB _c-meson in the reaction were calculated in the nonrelativistic model of a heavy quarkonium. The number of _bc -hyperons to be expected at LEP is estimated on the basis of the assumption on quark-hadron duality.     

Equilibrium state of aggregation in suspensions comprising linear clusters

The final state of aggregation in a suspension containing nonbranching, one-dimensional (linear) flocs is investigated. Allowing coagulation to occur at a finite secondary minimum of magnitudeV* rather than the infinite primary well, it is realized that the system eventually reaches steady state where a relatively large number of flocs coexist at equilibrium. It is shown that, under this condition, the number of flocs¯N expected to result is exponentially related to –V* and directly proportional to , whereN _o is the initial number of individual particles in the suspension.     

Kondo effect,spin dynamics and magnetism in anomalous rare earth and actinide compounds

The influence of spin dynamics on the Kondo effect manifestations in the Kondo lattices is investigated within perturbation theory with respect to thes-f interaction. It may give rise to Kondo-like divergencies in the electron self-energy already in the second order, resulting in an appreciable effective mass enhancement. As for usual Kondo contributions to thermodynamic and transport properties, the effect of spin dynamics reduces roughly to the replacement ln , with the characteristic spin-fluctuation energy. The thermoelectric power of dense Kondo systems is discussed. Singular contributions to the electron self-energies in the ferro-and antiferromagnetic state are considered. Kondo-like corrections to the intersite exchange interactions, saturation magnetic moment and total energy in a magnetically ordered state are calculated. The strong-coupling regionT<T _K is investigated within the Anderson lattice model. A decrease ofT _K due to spin fluctuations is demonstrated.     

Measurement of a complete set of analyzing powers of the fusion reactions D(\vec d,p)3He atE d=28 keV

Angular distributions of the four analyzing powers of the mirror fusion reactions D( ,p)³H and D( ,n)³He have been measured at the very low energy of 28±3 keV. Differences between both reactions are visible. The data are compared with anR-matrix parametrization and used as input to a comprehensive reaction-matrix fit.D-wave contributions are strongly suppressed in contrast toP- andS-waves. However, entrance-channel quintetS-waves are not suppressed. The possibility of a neutron-lean fusion reactor is therefore excluded.     

Isospectral Hamiltonian flows in finite and infinite dimensions

The approach to isospectral Hamiltonian flow introduced in part I is further developed to include integration of flows with singular spectral curves. The flow on finite dimensional Ad^*-invariant Poisson submanifolds of the dual of the positive part of the loop algebra is obtained through a generalization of the standard method of linearization on the Jacobi variety of the invariant spectral curveS. These curves are embedded in the total space of a line bundleTP ₁(C), allowing an explicit analysis of singularities arising from the structure of the image of a moment map from the space of rank-r deformations of a fixedN×N matrixA. It is shown how the linear flow of line bundles over a suitably desingularized curve may be used to determine both the flow of matricial polynomialsL() and the Hamiltonian flow in the spaceM _N,r×M_N,r in terms of -functions. The resulting flows are proved to be completely integrable. The reductions to subalgebras developed in part I are shown to correspond to invariance of the spectral curves and line bundles under certain linear or anti-linear involutions. The integration of two examples from part I is given to illustrate the method: the Rosochatius system, and the CNLS (coupled non-linear Schrödinger) equation.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and by U.S. Army grant DAA L03-87-K-0110     

On General Plane Fronted Waves. Geodesics

A general class of Lorentzian metrics, , , with any Riemannian manifold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic study of their main geodesic properties: geodesic completeness, geodesic connectedness and multiplicity causal character of connecting geodesics. These results are independent of the possibility of a full integration of geodesic equations. Variational and geometrical techniques are applied systematically. In particular, we prove that the asymptotic behavior of H(x,u) with x at infinity determines many properties of geodesics. Essentially, a subquadratic growth of H ensures geodesic completeness and connectedness, while the critical situation appears when H(x,u) behaves in some direction as , as in the classical model of exact gravitational waves.     

Hearing the zero locus of a magnetic field

We investigate the ground state of a two-dimensional quantum particle in a magnetic field where the field vanishes nondegenerately along a closed curve. We show that the ground state concentrates on this curve ase/h tends to infinity, wheree is the charge, and that the ground state energy grows like (e/h)^2/3. These statements are true for any energy level, the level being fixed as the charge tends to infinity. If the magnitude of the gradient of the magnetic field is a constantb ₀ along its zero locus, then we get the precise asymptotics(e/h) ^2/3 (b ₀) ^2/3 E _* +O(1) for every energy level. The constantE _* .5698 is the infimum of the ground state energiesE() of the anharmonic oscillator family .     

Systems of covariance in quantum probability

Symmetry groups and systems of covariance are investigated in the framework of quantum probability theory. It is shown that a measurementX can be represented by a positive operator-valued measure on a sectorS of the amplitude space. Moreover, provides a generalized system of covariance for the generalized unitary representation of a symmetry group.     

SimpleC*-algebra generated by isometries

We consider theC*-algebra generated byn2 isometriesS ₁,...,S _n on an infinite-dimensional Hilbert space, with the property thatS ₁ S*₁+...+S _n S* _n =1. It turns out that has the structure of a crossed product of a finite simpleC*-algebra by a single endomorphism scaling the trace of by 1/n. Thus, is a separableC*-algebra sharing many of the properties of a factor of typeIII with =1/n. As a consequence we show that is simple and that its isomorphism type does not depend on the choice ofS ₁,...,S _n .     

A unified quantum theory of mechanics and thermodynamics. Part I. Postulates

A unified axiomatic theory that embraces both mechanics and thermodynamics is presented in three parts. It is based on four postulates; three are taken from quantum mechanics, and the fourth is the new disclosure of the existence of quantum states that are stable (Part I). For nonequilibrium and equilibrium states, the theory provides general original results, such as the relation between irreducible density operators and the maximum work that can be extracted adiabatically (Part IIa). For stable equilibrium states, it shows for the first time that the canonical and grand canonical distributions are the only stable distributions (Part IIb). The theory discloses the incompleteness of the equation of motion of quantum mechanics not only for irreversible processes but, more significantly, for reversible processes (Part IIb). It establishes the operational meaning of an irreducible density operator and irreducible dispersions associated with any state, and reveals the relationship between such dispersions and the second law (Part III).     

Quantum conformal superspace

For a compact connected orientablen-manifoldM, n 3, we study the structure ofclassical superspace ,quantum superspace ,classical conformal superspace , andquantum conformal superspace . The study of the structure of these spaces is motivated by questions involving reduction of the usual canonical Hamiltonian formulation of general relativity to a non-degenerate Hamiltonian formulation, and to questions involving the quantization of the gravitational field. We show that if the degree of symmetry ofM is zero, thenS,S ₀,C, andC ₀ areilh orbifolds. The case of most importance for general relativity is dimensionn=3. In this case, assuming that the extended Poincaré conjecture is true, we show that quantum superspaceS ₀ and quantum conformal superspaceC ₀ are in factilh-manifolds. If, moreover,M is a Haken manifold, then quantum superspace and quantum conformal superspace arecontractible ilh-manifolds. In this case, there are no Gribov ambiguities for the configuration spacesS ₀ andC ₀. Our results are applicable to questions involving the problem of thereduction of Einstein's vacuum equations and to problems involving quantization of the gravitational field. For the problem of reduction, one searches for a way to reduce the canonical Hamiltonian formulation together with its constraint equations to an unconstrained Hamiltonian system on a reduced phase space. For the problem of quantum gravity, the spaceC ₀ will play a natural role in any quantization procedure based on the use of conformal methods and the reduced Hamiltonian formulation.