The projection operator approach to the Fokker-Planck equation. II. Dichotomic and nonlinear Gaussian noise

Two models for the Freedericksz transition in a fluctuating magnetic field are considered: one is based on a dichotomic and the other on a nonlinear Gaussian noise. Both noises are characterized by a finite correlation time. It is shown that the linear response assumption leading to the best Fokker-Planck approximation in the dichotomic and nonlinear Gaussian cases can be trusted only up to the order ¹ and ⁰, respectively. The role of the corrections to the linear response approximation is discussed and it is shown how to replace the non-Fokker-Planck terms stemming from these corrections with equivalent terms of standard type. This technique is shown to produce perfect agreement with the exact analytical results (dichotomic noise) and to satisfactorily fit the results of analog simulation (nonlinear Gaussian noise).     

The unitary bogoliubov transformation for a quantized scalar field in a friedman space-time

The time-dependent creation and annihilation operators for a complex scalar field, in a Friedmann space-time, defining particle states with respect to which the Hamiltonian is diagonal, are related by a Bogoliubov transformation to the creation and annihilation operators defined in strict analogy with the procedure carried out in Minkowski space. The Bogoliubov transformation is here written in terms of a unitary operator,U, and an expression for that operator is found via the generating functionF=i InU. The properties of the representation obtained by makingU act upon the state vector , to give a new state _U, are discussed. It is shown that the particle-number operator remains constant in such a picture so that the evolution of the system with time is clearly seen to depend upon the energy _k on the one hand, and upon the state vector _U on the other. Also, it is pointed out that this new representation permits the in and out states to be defined unambiguously.On leave of absence from Istituto de Fisica G. Galilei (Padova) and Istituto Nazionale di Fisica Nucleare (Sezione di Padova).     

Representations of Lie algebras from representations of quantum groups

In paper [*] (P. Moylan: Czech. J. Phys., Vol. 47 (1997), p. 1251) we gave an explicit embedding of the three dimensional Euclidean algebra (2) into a quantum structure associated with U _q(so(2, 1)). We used this embedding to construct skew symmetric representations of (2) out of skew symmetric representations of U _q(so(2, 1)). Here we consider generalizations of the results in [*] to a more complicated quantum group, which is of importance to physics. We consider U _q(so(3, 2)), and we show that, for a particular representation, namely the Rac representation, many of the results in [*] carry over to this case. In particular, we construct representations of so(3, 2), P(2, 2), the Poincaré algebra in 2+2 dimensions, and the Poincaré algebra out of the Rac representation of U _q(so(3, 2)). These results may be of interest to those working on exploiting representations of U _q(so(3, 2)), like the Rac, as an example of kinematical confinement for particle constituents such as the quarks.     

Small representations of quantum affine algebras

We characterize the finite-dimensional representations of the quantum affine algebra U _q ( _n+1) (whereq ^× is not a root of unity) which are irreducible as representations of U _q (sl _n+1). We call such representations small. In 1986, Jimbo defined a family of homomorphismsev _a from U _q (sl _n+1) to (an enlargement of) U _q (sl_,n+1), depending on a parametera ^·. A second family,ev ^a can be obtained by a small modification of Jimbo's formulas. We show that every small representation of U _q ( _n+1) is obtained by pulling back an irreducible representation of U _q (sl _n+1) byev _a orev ^a for somea ^·.     

On the global behaviour of symmetric Skyrmions

It is shown that the chiral angle, (r), of the hedgehog (symmetric) Skyrmions with an arbitrary baryon number, is a strictly decreasing or increasing function. For large values of r>0, (r) is strictly convex or concave. As r, (r) and (r) approach their limit values at the rate Or ^- for any (0,2).     

Phase transitions on fractal lattices with long-range interactions

A fractal latticeF is defined here to comprise all points of the forma + ma+ m² a+ ... +m^qa^(q), whereq is a nonnegative integer anda, a,..., a^(q)A, whereA is a finite set of points in some Euclidean space. Providedm is not too small (in particular,m must be at least 2), the dimension ofF is shown to beD = log n/logm, wheren is the number of points inA. It is shown further that an Ising model onF, with a ferromagnetic pair interaction r^– between spins separated by a distancer, has a phase transition ifD < < 2D. On the other hand, for > 2D, provided a certain condition which rules out periodic lattices is satisfied, there can be no finite-temperature transition leading to spontaneous magnetization.     

Entropy,the central limit theorem and the algebra of the canonical commutation relation

The central limit theorem of Cushen and Hudson is reformulated on the algebra of the CCR. Namely, for a gauge invariant state , the weighted convolutions _n of the central limit tend to the quasi-free reduction _Q of pointwise. It is proved that if the initial relative entropy S(, _Q ) is finite, then S( _n , _Q ) goes to 0 and so _n – _Q 0. No restriction on the dimension of the test function space is made.     

Determination of the low ω cusp in the velocity autocorrelation spectrum of liquid sodium

Recently performed high resolution incoherent neutron scattering experiments on liquid sodium at smallA and low make the precise evaluation of the frequency distribution of the liquid feasible. An appropriate extension of the well known Egelstaff formula for the evaluation of the frequency distribution in a liquid is deduced. The new analysis is shown to be exact in the smallQ low region, removing the singularity of the above formula at 0 andQ0. The easy applicability of the evaluation is demonstrated on neutron scattering data of liquid sodium, where the long discussed low frequency cusp ofz() could be revealed.     

Nonlinear neural networks. II. Information processing

Information processing in nonlinear neural networks with a finite numberq of stored patterns is studied. Each network is characterized completely by its synaptic kernelQ. At low temperatures, the nonlinearity typically results in 2^q–2–q metastable, pure states in addition to theq retrieval states that are associated with theq stored patterns. These spurious states start appearing at a temperature , which depends onq. We give sufficient conditions to guarantee that the retrieval states bifurcate first at a critical temperatureT _c and that /T _c 0 asq. Hence, there is a large temperature range whereonly the retrieval states and certain symmetric mixtures thereof exist. The latter are unstable, as they appear atT _c . For clipped synapses, the bifurcation and stability structure is analyzed in detail and shown to approach that of the (linear) Hopfield model asq. We also investigate memories that forget and indicate how forgetfulness can be explained in terms of the eigenvalue spectrum of the synaptic kernelQ.     

Non-Markovian reversible Chapman-Kolmogorov measures on subshifts of finite type

We consider shift-invariant probability measures on subshift dynamical systems with a transition matrixA which satisfies the Chapman-Kolmogorov equation for some stochastic matrix compatible withA. We call them Chapman-Kolmogorov measures. A nonequilibrium entropy is associated to this class of dynamical systems. We show that ifA is irreducible and aperiodic, then there are Chapman-Kolmogorov measures distinct from the Markov chain associated with and its invariant row probability vectorq. If, moreover, (q, ) is a reversible chain, then we construct reversible Chapman-Kolmogorov measures on the subshift which are distinct from (q, ).     

On the determination of χl,η+−0 AND η000 from bubble chamber measurements

Formulae for time distribution of neutral kaon decay in material medium are derived and discussed. Essential effects of the medium are emphasized which should be taken into account in the discussion of the bubble chamber experiments in which theS=Q rule and CP-violation are studied.     

Renormalization group approach for the site-bond percolation in structured stochastic environments

The quenched averaged percolation problem of a lattice with a given structure is analyzed. The structure is described by the static structure factorS(q)q ^–ain the regionq 0. As a result of the renormalization group, it follows that the critical behavior fora < 2 is the same as in the random percolation. In the case ofa=2 second universality class with=0 and=1/2+/8+ ²/32 is predicted.     

Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem

We prove that theq-state Potts antiferromagnet on a lattice of maximum coordination numberr exhibits exponential decay of correlations uniformly at all temperatures (including zero temperature) wheneverq>2r. We also prove slightly better bounds for several two-dimensional lattices: square lattice (exponential decay forq7), triangular lattice (q11), hexagonal lattice (q4), and Kagomé lattice (q6). The proofs are based on the Dobrushin uniqueness theorem.     

Spectra of Hamiltonians with generalized single-site dynamical disorder

Starting from the deformed commutation relationsa _q (t) a ^q (s)–q a ^q (s) a _q (t)=(t–s)1, –1q1 with a covariance (t–s) and a parameterq varying between –1 and 1, a stochastic process is constructed which continuously deforms the classical Gaussian and classical compound Poisson process. The moments of these distinguished stochastic processes are identified with the Hilbert space vacuum expectation values of products of with fixed parametersq, and . Thereby we can interpolate between dichotomic, random matrix and classical Gaussian and compound Poisson processes. The spectra of Hamiltonians with single-site dynamical disorder are calculated for an exponential covariance (coloured noise) by means of the time convolution generalized master equation formalism (TC-GME) and the partial cumulants technique. The final result for the spectral function is given as aq-dependent infinite continued fraction. In the case of the random matrix processes the infinite continued fraction can be summed up yielding a self-consistent equation for the one-particle Green function.     

Quantization of space-time and the corresponding quantum mechanics

An axiomatic framework for describing general space-time models is presented. Space-time models to which irreducible propositional systems belong as causal logics are quantum (q) theoretically interpretable and their event spaces are Hilbert spaces. Such aq space-time is proposed via a canonical quantization. As a basic assumption, the time t and the radial coordinate r of aq particle satisfy the canonical commutation relation [t,r]=±i . The two cases will be considered simultaneously. In that case the event space is the Hilbert space L²(³). Unitary symmetries consist of Poincaré-like symmetries (translations, rotations, and inversion) and of gauge-like symmetries. Space inversion implies time inversion. Thisq space-time reveals a confinement phenomenon: Theq particle is confined in an size region of Minkowski space at any time. One particle mechanics overq space-time provides mass eigenvalue equations for elementary particles. Prugoveki's stochasticq mechanics andq space-time offer a natural way for introducing and interpreting consistently such aq space-time andq particles existing in it. The mass eigenstates ofq particles generate Prugoveki's extended elementary particles. When 0, these particles shrink to point particles and is recovered as the classical (c) limit ofq space-time. Conceptual considerations favor the case [t,r]=+i , and applications in hadron physics give the fit 2/5 fermi/GeV.This paper is a revised version of the author's work, Quantization of Space-time and the Corresponding Quantum Mechanics (Part I), report KFKI-1981-48.     

Analyticity properties of the surface free energy of the Ising model

For an Ising ferromagnet with nearest-neighbour interactions of strengthK and surface magnetic fieldh, the surface free energy in the presence of a positively (or negatively) magnetized zero-field bulk phase is shown to be analytic inh for Reh<K–/, where =2.96 ... and is the inverse temperature. This puts the lower boundK–/ on the values ofh at which wetting and layering transitions can take place.     

The shape resonance

For a class of Schrödinger operatorsH:=–(²/2m)+V onL ²( ⁿ ), with potentials having minima embedded in the continuum of the spectrum and non-trapping tails, we show the existence of shape resonances exponentially close to the real axis as 0. The resonant energies are given by a convergent perturbation expansion in powers of a parameter exhibiting the expected exponentially small behaviour for tunneling.     

Bound on the Q-Factor of a G-Constant Measuring Pendulum Device from Gravitational Stochastic Background

We discuss the possible effects of gravitational-wave stochastic background on the performance of a pendulum device such those used in precise measurements of the gravitational constant G. The variation Q of the quality factor Q of the pendulum induced by the stochastic background is evaluated, by using as numerical input the results obtained in gravitational antennas experiments. It is found |q|10^–10, completely negligible with respect to a typical value Q10⁵.     

Empty bin effect and multiple bin phenomenon in dynamical fluctuations at high energies

Intermittency and multifractality characterizing the presence of dynamical fluctuations are analysed in terms of self-similar processes of Cantor type involving the continuous spectrum of singularities. The fundamental properties of the intermittency slopes _q as well as of the strength _q and the densityf _q , of singularities depending on the order parameterq are established. It is also argued here that the intermittency arisen owing to those self-similar processes leads to such anomalous fractal dimensions and density of singularities which hardly satisfy the assumptions allowing the interpretation of intermittency in terms of thermal as well as non-thermal phase transitions. Four concrete cases are discussed in the present contribution: the e⁺ e^– annihilations at s=29 GeV, the ¯pp interactions at s=1800 GeV, the collisions of 200 GeV/ nucleon¹⁶0 beams with the¹²C nuclei and the¹⁹⁷Au nuclei of energy 1 GeV/ nucleon break up into fragments when interacting with the nuclear emulsion. The corresponding data published so far reveal the fundamental properties following from the aforementioned processes with great accuracy.The author expresses his thanks to M. Markytan, H. Dibon and W. Kittel for fruitful discussions and F. Rimondi for providing him with the preliminary ¯pp data.     

Radiation losses in completely ionized plasma

The losses caused by bremsstahlung during electron-ion and electron-electron collisions in a completely ionized plasma (in Born's approximation) are calculated. The calculation can be carried out analytically for a sufficiently dilute plasma (plasma with infinitely large Debay-Hückel radius). This assumption is satisfied very well by the known classification to actual controlled thermonuclear reaction. A dependence of the formI=an ². (1+3/2), wherea=0·73×10^–16 MeV.cm³.sec^–1,n is the number of ions per cm³ and =kT/mc ², is determined for the amount of energy radiated from 1 cm³ of plasma per sec. In the relativistic temperature regionkTmc ² the influence of electron-electron collisions begins to predominate. A number of important results concerning radiation losses in relativistic plasma will be published in a later paper.

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, ( ). ( -). . , I=an ². (1+3/2), =0,73. 10^–16 MeVcm²s^–1,n — =kT/mc ². kTmc ² . , .