Exact gravitational field of the infinitely long rotating hollow cylinder

The vacuum line element inside an infinitely long rotating hollow cylinder is the usual flat space line element. It is fitted in a most general way to the general cylindrical vacuum field outside at the singular hypersurfaceR ₀=const, representing the infinitely thin hollow cylinder. With the use of the jump conditions atR ₀=const the surface densities , of which the energy-momentum-stress tensor of the shell consists, are calculated. The physical properties of the cylinder, as derived from the eigenvalues and -vectors of , and the generated gravitational field are discussed in full detail.     

Space-time ergodic properties of systems of infinitely many independent particles

We investigate the ergodic properties of the equilibrium states of systems of infinitely many particles with respect to the group generated by space translations and time evolution. The particles are assumed to move independently in a periodic external field. We show that insofar as good thermodynamic behavior is concerned these properties provide much sharper discrimination than the ergodic properties of the time evolution alone. In particular, we show that though the infinite ideal gas is mixing in the space-time framework, it has vanishing space-time entropy and fails to be a space-timeK-system. On the other hand, if the particles interact with fixed convex scatterers (the Lorentz gas) the system forms a space-timeK-system. Also, the space-time entropy of a system of the type we consider is shown to equal its time entropy per unit volume.Research supported in part by the National Science Foundation Grant No. GP-16147 A No. 1.     

The classical mechanics of one-dimensional systems of infinitely many particles

We apply the existence theorem for solutions of the equations of motion for infinite systems to study the time evolution of measures on the set of locally finite configurations of particles. The set of allowed initial configurations and the time evolution mappings are shown to be measurable. It is shown that infinite volume limit states of thermodynamic ensembles at low activity or for positive potentials are concentrated on the set of allowed initial configurations and are invariant under the time evolution. The total entropy per unit volume is shown to be constant in time for a large class of states, if the potential satisfies a stability condition.On leave from: Department of Mathematics, University of California, Berkeley, California.     

The classical mechanics of one-dimensional systems of infinitely many particles

We prove a global existence and uniqueness theorem for solutions of the classical equations of motion for a one-dimensional system of infinitely many particles interacting by finite-range two-body forces which satisfy a Lipschitz condition.     

Thermodynamic limit of theq-state Potts-Hopfield model with infinitely many patterns

We prove the almost sure convergence of the free energy and of the overlap order parameters in aq-state version of the Hopfield neural network model. We compute explicitly these limits for all temperatures different from some critical value. The number of stored patterns is allowed to grow with the size of the systemN like (/lnq) lnN. We study the limiting behavior of the extremal states of the model that are the measures induced on the Gibbs measures by the overlap parameters.     

Time evolution of infinitely many particles: An existence theorem

In this paper we deal with systems of infinitely many particles in ³, given by a two-body, short-range potential and an external potential, depending on the position of the particles. We show the existence of dynamics for a set of initial configurations, which has measure one with respect to the Gibbs measure induced by a suitable family of Hamiltonians.     

Escape from the unstable equilibrium in a random process with infinitely many interacting particles

We consider a one-dimensional version of the model introduced in Ref. l. At each site of Z there is a particle with spin ± 1. Particles move according to the Stirring Process and spins change according to the Glauber dynamics. In the hydrodynamical limit, with the stirring process suitably speeded up, the local magnetic densitym _t(r) is proven in Ref. 1 to satisfy the reaction-diffusion equation (*) $$\partial _t m_t (r) = \tfrac{1}{2}\partial _r^2 m_t (r) - V'(m_t )$$ \(V(m) = - \tfrac{1}{2}\alpha m^2 + \tfrac{1}{4}\beta m^4 \) ,α andβ being determined by the parameters of the Glauber dynamics. In the present paper we consider an initial state with zero magnetization,m ₀(r)=0. We then prove that at long times, before taking the hydrodynamical limit, the evolution departs from that predicted by (*) and that the microscopic state becomes a nontrivial mixture of states with different magnetizations.     

On one-dimensional integrable quantum systems with infinitely many degrees of freedom

We study one-dimensional quantum systems whose S-operator conserves the incoming momenta and particle identities. A survey of systems with known S-operators and an approach for solving them rigorously are presented, and several problems and conjectures are formulated. Subsequently, this approach is used to arrive at relativistic dynamics whose S-operators are those of the Ising model in the scaling limit and of the Federbush model. An invariance property of the wave and scattering operators is discovered and argued to hold at the classical level, too.     

Universality class of nonequilibrium phase transitions with infinitely many absorbing states

We consider systems whose steady states exhibit a nonequilibrium phase transition from an active state to one-among an infinite number-absorbing state, as some control parameter is varied across a threshold value. The pair contact process, stochastic fixed-energy sandpiles, activated random walks, and many other cellular automata or reaction-diffusion processes are covered by our analysis. We argue that the upper-critical dimension below which anomalous fluctuation driven scaling appears is d(c)=6, in contrast to a widespread belief. We provide the exponents governing the critical behavior close to or at the transition point to first order in an epsilon =6-d expansion.     

The solutions of the classical KMS-equation for infinitely many non-interacting particles

The complete solution of the classical KMS-equation for quasi-free evolutions is given under two different conditions.     

Existence of infinitely many complex eigenvalues for the mono-energetic neutron transport operator

In this Letter it is proved that the mono-energetic neutron transport operator for the case of a spherically-symmetric, isotropically-scattering sphere with a central cavity, has infinitely many complex eigenvalues.     

A Nekhoroshev-type theorem for Hamiltonian systems with infinitely many degrees of freedom

We study the propagation of lattice vibrations in models of disordered, classical anharmonic crystals. Using classical perturbation theory with an optimally chosen remainder term (i.e. a Nekhoroshev-type scheme), we are able to show that vibrations corresponding to localized initial conditions do essentially not propagate through the crystal up to times larger than any inverse power of the strength of the anharmonic couplings.     

A novel class of two-dimensional chaotic maps with infinitely many coexisting attractors

We study a novel class of two-dimensional maps with infinitely many coexisting attractors. Firstly, the mathematical model of these maps is formulated by introducing a sinusoidal function. The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable. In particular, a computer searching program is employed to explore the chaotic attractors in these maps, and a simple map is exemplified to show their complex dynamics.Interestingly, this map contains infinitely many coexisting attractors which has been rarely reported in the literature. Further studies on these coexisting attractors are carried out by investigating their time histories, phase trajectories, basins of attraction, Lyapunov exponents spectrum, and Lyapunov(Kaplan–Yorke) dimension. Bifurcation analysis reveals that the map has periodic and chaotic solutions, and more importantly, exhibits extreme multi-stability.     

Concrete Hilbert spaces for quantum systems with infinitely many degrees of freedom

We demonstrate a method to describe quantum systems with infinitely many degrees of freedom in concrete Hilbert spaces, using the electromagnetic radiation field as a well-known example of such a system. Since our method is not only applicable to the case of countably many but even to the case of uncountably many degrees of freedom, there is no need for a finite quantization volume in radiation theory.     

Composition independence of the possible finite-range gravitational force

A scalar-tensor theory of gravitation with scale invariance broken spontaneously is examined to show if and how the finite-range force added to the ordinary Newtonian force can be composition independent in the Eötvös experiment.     

A two dimensional Lagrangian model with extended supersymmetry and infinitely many constants of motion

The reduction of the supersymmetric gradedSU(2|1) /S(U ₂×U ₁) -model is discussed. If no extra constraint is imposed, one gets a set of two coupled equations (involving two scalar superfields) which generalizes the supersymmetric sine-Gordon equation. It is shown that these equations, which can be derived by a supersymmetric Lagrangian, reproduce in the bosonic limit the reduced version of theO(4) -model (Pohlmeyer, Lund Regge, Getmanov model). Moreover the associate linear set and an infinite set of local conservation laws for this new supersymmetric model are exhibited. It turns out that, beyond the spinorial charge which generates the supersymmetry transformations, another unexpected spinorial charge is conserved; then the model appears to be invariant underN=2 extended supersymmetry.     

On a possible unification of gravitational and weak interactions

Within the framework of Cartan's generalization of Einstein's theory of gravitation one can achieve a unification of gravitational and weak interactions by appropriate choice of the parameter that couples spin and torsion. The proposed spin-torsion coupling has negligible cosmic effects except at stages of evolution when 10⁸¹ nucleons are confined to a sphere with a radius of about one astronomical unit. For a single nuclear particle the gravitational effects of mass and spin balance at a radius of about 1% of its Compton wavelength, thus stabilizing it against gravitational collapse.This essay received the fourth award from the Gravity Research Foundation for 1975 (Editor).     

Ground state energies of bound exciton for all mass ratios

We report on a calculation of the ground state energy of bound excitons using states from a model Hamiltonian and correcting the variational results by treating the difference between the model and actual Hamiltonian in second order perturbation theory. The resulting procedure is valid for all mass ratios.