Nature of interacting electron states of Coulomb glass in local energy minima

Non-equilibrium relaxation of Coulomb glass in disordered thin films is investigated by kinetic Monte Carlo simulation. We numerically confirm aging phenomena in the autocorrelation function C(t, t_W ) in a quasi-two-dimensional system with finite thickness and clarify the effect of an external electric field on the elongated relaxation time due to aging. We also study the statistical properties of electron states belonging to local energy minima in random site models. Our results highlight the difference in the properties of energy landscape between two different models to describe Coulomb glass, called the random site model and the lattice model.     

Solvable models of temporally correlated random walk on a lattice

We seek the conditional probability functionP(m,t) for the position of a particle executing a random walk on a lattice, governed by the distributionW(n, t) specifying the probability ofn jumps or steps occurring in timet. Uncorrelated diffusion occurs whenW is a Poisson distribution. The solutions corresponding to two different families of distributionsW are found and discussed. The Poissonian is a limiting case in each of these families. This permits a quantitative investigation of the effects, on the diffusion process, of varying degrees of temporal correlation in the step sequences. In the first part, the step sequences are regarded as realizations of an ongoing renewal process with a probability densityψ(t) for the time interval between successive jumps.W is constructed in terms ofψ using the continuous-time random walk approach. The theory is then specialized to the case whenψ belongs to the class of special Erlangian density functions. In the second part,W is taken to belong to the family of negative binomial distributions, ranging from the geometric (most correlated) to the Poissonian (uncorrelated). Various aspects such as the continuum limit, the master equation forP, the asymptotic behaviour ofP, etc., are discussed.     

A Markov Process Associated with a Boltzmann Equation Without Cutoff and for Non-Maxwell Molecules

Tanaka,⁽¹⁸⁾ showed a way to relate the measure solution {P _t } _t of a spatially homogeneous Boltzmann equation of Maxwellian molecules without angular cutoff to a Poisson-driven stochastic differential equation: {P _t } is the flow of time marginals of the solution of this stochastic equation. In the present paper, we extend this probabilistic interpretation to much more general spatially homogeneous Boltzmann equations. Then we derive from this interpretation a numerical method for the concerned Boltzmann equations, by using easily simulable interacting particle systems.     

A new method for determining excited states of quantum systems

Summary A new method for computing excited states of a given operatorH is here presented. Our procedure is of particular value when its representation requires an orthonormal basis set of large dimension. In order to obtain the excited state ofH nearest in energy to any chosen trial energyE _t, we consider the auxiliary operatorA=(H−E _t)². We show that a reasonable number of relaxations on appropriately generated low-order Krylov subspaces forA is sufficient to produce better and better approximations of its ground state; a high-accuracy final refinement of the ground state ofA is then possible through the standard Lanczos procedure. An important feature of our method is that storage memory limitations, encountered in the conventional determination of all eigenvalues of large systems, are here overcome. As an illustration of the method two significant examples are discussed. In honour of Prof. Fausto Fumi on the occasion of his retirement from teaching.     

Numerical study of local and global persistence in directed percolation

The local persistence probability P _l (t) that a site never becomes active up to time t, and the global persistence probability P _g (t) that the deviation of the global density from its mean value does not change its sign up to time t are studied in a (1+1)-dimensional directed percolation process by Monte-Carlo simulations. At criticality, starting from random initial conditions, P _l (t) decays algebraically with the exponent . The value is found to be independent of the initial density and the microscopic details of the dynamics, suggesting is an universal exponent. The global persistence exponent is found to be equal or larger than . This contrasts with previously known cases where . It is shown that in the special case of directed-bond percolation, P _l (t) can be related to a certain return probability of a directed percolation process with an active source (wet wall). Received: 15 December 1997 / Revised: 6 April 1998 / Accepted: 29 May 1998     

2H-NMR study of ammonium ion rotational tunneling and reorientation in (ND4)2SnCl6 single crystal: I. Tunneling frequency measurements

Theory of²H-NMR spectra is developed for a ND ₄ ⁺ ion. Quadrupole and dipole-dipole interactions for four deuterons of the tetrahedral ion as well as the rotational tunneling are taken into account. Features in the single crystal spectra at the high magnetic field of 6.7 T are analyzed. Their central doublets (±5 kHz range around the Larmor frequency) are attributed to theA symmetry species and exhibit sensitivity to theA–T tunneling frequencyv _t . This allows the measurement ofv _t in a wide rangev _t <10 MHz. The shape of sidebands (±140 kHz range) is related to the T levels structure. Theoretical predictions are verified on a (ND₄)₂SnCl₆ single crystal at 44 MHz. The tunneling frequency at temperatures below 10 K equals 7.5 MHz, which is about 100 times smaller than for (NH₄)₂SnCl₆.v _t was measured up to 40 K. At still higher temperatures motional narrowing effects were observed for theA spectral components preventing the determination ofv _t .     

Phase transitions in random uniaxial systems with dipolar interactions

The critical behaviour of random uniaxial ferromagnetic (ferroelectric) systems with both short range and long range dipolar interactions is investigated, using the field theoretic renormalization method of Brézin et al. for the free energy above and below the transition pointT _c. The randomness is due to externally introduced fluctuations in the short range interactions (quenched case) or (and) magneto-elastic coupling to the lattice (annealed case). Strong deviations in the critical behaviour with respect to the pure systems are found. In the quenched case e.g. the specific heatC and the coefficientf ₂ (ofM ³ in the equation of state, whereM is the magnetization) change fromC |ln|t^1/3,f ₂ |ln|t^–1 in the pure system toC is the reduced temperature andA _±,C _± are constants) in the random situation. This change should e.g. be experimentally observable by deuterization of the ferroelectric tri-glycine sulfate where the logarithmic behaviour off ₂ has already been detected in the pure case. For nonvanishing magnetoelastic coupling a complex critical behaviour is obtained and discussed. We find the interesting result that if both quenched randomness and a weak magnetoelastic coupling are present the quenched random critical behaviour dominates in the close vicinity ofT _c. Finally the influence of the magnetoelastic coupling on the longitudinal phonons in investigated and it is found that the relative changes in the corresponding elastic constant and structure factor are proportional to the specific heat and the wavevector dependent energy-energy correlation function respectively, suggesting new experiments.     

On retroactive occurrence and twin events in quantum mechanics

Bohr's well-known claim that only a registered phenomenon is a true phenomenon is further elaborated into occurrence in the past: If ideal occurrence of an eventP ((1–P)) is a state at a timet _i makes another eventQ ((1–Q))certain at a later timet _f, and, finallyU is the evolution operator fromt _i tot _f, then, it is proved that the final collapsed stateQ(U U ⁺)Q/TrQU U ⁺, which comes about in ideal occurrence ofQ att _f,equals the initial collapsed stateU(P P/TrP)U ⁺, which evolves from the state resulting from the ideal occurrence ofP in att _i. Utilizing the latter state is called theretroactive apparent ideal occurrence (RAIO) ofP in. A number of consequences, including the general notion of twin events (the case whent _f=t _i, andU=1) is derived. It is pointed out that RAIO is relevant in second-kind quantum measurement, in Wheeler's delayed-choice experiments in second-kind (or conditional) quantum preparators.     

Direct gauging of the Poincaré group V. Group scaling,classical gauge theory,and gravitational corrections

Homogeneous scaling of the group space of the Poincaré group,P ₁₀, is shown to induce scalings of all geometric quantities associated with the local action ofP ₁₀. The field equations for both the translation and the Lorentz rotation compensating fields reduce toO(1) equations if the scaling parameter is set equal to the general relativistic gravitational coupling constant 8Gc ^–4. Standard expansions of all field variables in power series in the scaling parameter give the following results. The zeroth-order field equations are exactly the classical field equations for matter fields on Minkowski space subject to local action of an internal symmetry group (classical gauge theory). The expansion process is shown to breakP ₁₀-gauge covariance of the theory, and hence solving the zeroth-order field equations imposes an implicit system ofP ₁₀-gauge conditions. Explicit systems of field equations are obtained for the first- and higher-order approximations. The first-order translation field equations are driven by the momentum-energy tensor of the matter and internal compensating fields in the zeroth order (classical gauge theory), while the first-order Lorentz rotation field equations are driven by the spin currents of the same classical gauge theory. Field equations for the first-order gravitational corrections to the matter fields and the gauge fields for the internal symmetry group are obtained. Direct Poincaré gauge theory is thus shown to satisfy the first two of the three-part acid test of any unified field theory. Satisfaction of the third part of the test, at least for finite neighborhoods, seems probable.     

Normal and antinormal ordering of field operators in quantum optics

The relations between the probability distributionp n _vt of the number of photonsn _vt in a volumeV at timet and the probability distributions P_N(W) and P_A(W) of the integrated intensityW related to the normal and antinormal ordering of field operators and the relations between corresponding moments are studied for a field containingM modes. As the normally ordered correlations are measured with photodetectors and the antinormally ordered correlations are measured with quantum counters the relations betweenn _vt , P_N(W) and P_A(W) permit the statistical behaviour of light to be determined from measurements with photodetectors and quantum counters. The results obtained here are used for the superposition of coherent and thermal fields. It is also shown that the antinormal correlations depend explicitly on the number of modes and that in the classical limit, when the average photon occupation number per mode becomes large, the distributionsn _vt , P_N(W) and P_A(W) become equal.     

Transverse momenta of secondaries from 200 GeV/c proton-nucleus interactions

A study of 100 interactions, produced by secondary particles from 200 GeV/c proton interactions in nuclear emulsions, has been made to estimate the transverse momenta of the secondary particles. The data have been analysed by different methods of energy estimation and the weighted average values ofp _t have been compared as estimated from various methods. An average value ofp _t equal to 0.38 ± 0.03 GeV/c, in proton-nucleus interactions at 200 GeV/c, has been obtained from the production mechanism method.     

Vicious random walkers in the limit of a large number of walkers

The vicious random walker problem on a line is studied in the limit of a large number of walkers. The multidimensional integral representing the probability that thep walkers will survive a timet (denotedP _t ^(p) ) is shown to be analogous to the partition function of a particular one-component Coulomb gas. By assuming the existence of the thermodynamic limit for the Coulomb gas, one can deduce asymptotic formulas forP _t ^(p) in the large-p, large-t limit. A straightforward analysis gives rigorous asymptotic formulas for the probability that after a timet the walkers are in their initial configuration (this event is termed a reunion). Consequently, asymptotic formulas for the conditional probability of a reunion, given that all walkers survive, are derived. Also, an asymptotic formula for the conditional probability density that any walker will arrive at a particular point in timet, given that allp walkers survive, is calculated in the limittp.     

The spin of the electron according to stochastic electrodynamics

By making use of the method of moments we study some aspects of the statistical behavior of the nonrelativistic harmonic oscillator according to stochastic electrodynamics. We show that the random rotations induced on the particle by the zero-point field account for the magnitude of the spin of the electron, the result differing from the correct one(3/4)h ² by a factor of2. Assuming that the measurement of a spin projection may be effectively taken into account by considering the action of only the subensemble of the field with the corresponding circular polarization, the calculated value of the spin projection comes out to be the correct one within a factor of order unity. The radiative corrections give rise to both the Lamb shift and the anomalous magnetic moment of the electron, the latter being evaluated to within a factor of2. The magnetic and gyromagnetic properties of the electron come out to be in agreement with quantum mechanics. Interference effects are shown to occur when evaluating the average value of the square of the angular momentum.Fellow of CONACYT, México.     

Larc: A State Reduction Theory of Quantum Measurement

This proposes a new theory of Quantum measurement; a state reduction theory in which reduction is to the elements of the number operator basis of a system, triggered by the occurrence of annihilation or creation (or lowering or raising) operators in the time evolution of a system. It is from these operator types that the acronym ‘LARC’ is derived. Reduction does not occur immediately after the trigger event; it occurs at some later time with probability P _t per unit time, where P _t is very small. Localisation of macroscopic objects occurs in the natural way: photons from an illumination field are reflected off a body and later absorbed by another body. Each possible absorption of a photon by a molecule in the second body generates annihilation and raising operators, which in turn trigger a probability per unit time P _t of a state reduction into the number operator basis for the photon field and the number operator basis of the electron orbitals of the molecule. Since all photons in the illumination field have come from the location of the first body, wherever that is, a single reduction leads to a reduction of the position state of the first body relative to the second, with a total probability of mP _t , where m is the number of photon absorption events. Unusually for a reduction theory, the larc theory is naturally relativistic.     

Temporal correlations and persistence in the kinetic Ising model: the role of temperature

We study the statistical properties of the sum S _t = dt'σ _t', that is the difference of time spent positive or negative by the spin σ _t, located at a given site of a D-dimensional Ising model evolving under Glauber dynamics from a random initial configuration. We investigate the distribution of S_t and the first-passage statistics (persistence) of this quantity. We discuss successively the three regimes of high temperature ( T > T _c), criticality ( T = T _c), and low temperature ( T < T _c). We discuss in particular the question of the temperature dependence of the persistence exponent , as well as that of the spectrum of exponents (x), in the low temperature phase. The probability that the temporal mean S _t/t was always larger than the equilibrium magnetization is found to decay as t ^{- - ?}. This yields a numerical determination of the persistence exponent in the whole low temperature phase, in two dimensions, and above the roughening transition, in the low-temperature phase of the three-dimensional Ising model. Received 4 December 2000     

Nonequilibrium transitions driven by external dichotomous noise

The stationary probability densityP _s for a class of nonlinear one-dimensional models driven by a dichotomous Markovian process (DMP)I _t , can be calculated explicitly. For the specific case of the Stratonovich model, x=ax –-x ³ +I _t x, the qualitative shape ofP _s and its support is discussed in the whole parameter region. The location of the maxima ofP _s shows a behavior similar to order parameters in continuous phase transitions. The possibility of a noiseinduced change from continuous to a discontinuous transition in an extended model, in which the DMP couples also to the cubic term, is discussed. The time-dependent moments x_t ⁿ can be represented as an infinite series of terms, which are determined by a recursion formula. For negative even moments the series terminates and the long-time behavior can be obtained analytically. As a function of the physical parameters, qualitative changes of this behavior may occur which can be partially related to the behavior ofP _s . All results reproduce those for Gaussian white noise in the corresponding limit. The influence of the finite correlation time and the discreteness of the space of states of the DMP are discussed. An extensive list of references is contained in U. Behn, K. Schiele, and A. Teubel,Wiss. Z. Karl-Marx-Univ. Leipzig, Mathem.-Naturwiss. R. 34:602 (1985).Contribution to the symposium on the Statistical Mechanics of Phase Transitions —Mathematical and Physical Aspects, Tebo, Czechoslovakia, September 1–6, 1986.     

Stochastic Dissipative PDE's and Gibbs Measures

We study a class of dissipative nonlinear PDE's forced by a random force η^{omega( t} , ^x ), with the space variable x varying in a bounded domain. The class contains the 2D Navier–Stokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in statistical hydrodynamics: they are random fields smooth in t and stationary, short-correlated in time t. In this paper, we confine ourselves to “kick forces” of the form

Time evolution of initially localized states in two-dimensional quantum dot arrays in a magnetic field

The evolution of non-stationary localized states |Ψ(t=0) is investigated in two-dimensional tight binding systems of N potential wells with and without a homogeneous field perpendicular to the plane. Most results are presented in analytical form, what is almost imperative if the patterns are as complex as for rings in a magnetic field, where the qualitatively different features arise depending on rational or irrational numbers. The systems considered comprise finite linear chains (N=2,3), finite rings (N=3–6), infinite chains, finite rings (N=3–6) in a magnetic field, and rings with leads attached to each ring site. The position of the particle at time t is described by the projection of the wave function P_m(t)=|m|Ψ(t)|² onto the localized basis function at site m. For finite chains and rings with N=3,4,6 the time evolution is periodic, whereas it is non-periodic for N=5 and N greater then 6. Rings in a magnetic field show a rich spectrum of different features depending on N and the number of flux quanta through the ring, including periodic oscillation and rotation of the charge as well as non-periodic charge fluctuations.     

An analysis of energetic cosmic ray interactions in graphite

Fifty two high energy cosmic ray interactions in graphite have been analysed. The analysis strongly suggests that some of the characteristics of high energy interactions are dependent on the total transverse momentum,Σp _t, of allγ-rays in an interaction. Out of the 52 events analysed, 29 haveΣp _t⩽2.5 GeV/c and the rest haveΣp _t>2.5 GeV/c; the former are called smallp _t events while the latter are designated as largep _t events. For these two types of events, the characteristics investigated are: (a) fractional energy distribution ofγ-rays, (b) the invariant mass ofγ-rays and (c) the energy distribution ofγ-rays in the emission system.