Extrema statistics of Wiener-Einstein processes in one,two, and three dimensions

The maxima and first-passage-time statistics of Wiener-Einstein processes are evaluated analytically in one, two, and three dimensions. We show that the mean square maximum displacement has the same time dependence as the mean square displacement, i.e., it grows linearly with time. The ratio of the mean square maximum to the mean square displacement is shown to decrease with increasing dimensionality. We also calculate the mean first passage time for the process to attain a given absolute displacement and find that it grows as the square of the displacementand is independent of the dimensionality of the process. In addition, we evaluate the dispersion of maxima and of first passage times and discuss their dependence on dimensionality.Supported in part by the National Science Foundation under Grant CHE 75-20624.     

Information and Distinguishability of Ensembles of Identical Quantum States

We consider a fixed quantum measurement performed over n identical copies of quantum states. Using a rigorous notion of distinguishability based on Shannon’s 12th theorem, we show that in the case of a single qubit, the number of distinguishable states is , where (α₁,α₂) is the angle interval from which the states are chosen. In the general case of an N-dimensional Hilbert space and an area Ω of the domain on the unit sphere from which the states are chosen, the number of distinguishable states is . The optimal distribution is uniform over the domain in Cartesian coordinates.     

Local quantum statistics in arbitrary curved space-time

A local quantum statistics based on a finite temperature field theory in an arbitrary Riemann space-time is considered. The expressions have been derived for the partition functions, the grand thermodynamic potential and the particle distributions 〈n _k〉 of massive scalar gas and fermion gas in arbitrary space-time. It is shown that the chemical potential depends on the geometry of manifold.     

The effect of center-of-mass motion on photon statistics

We analyze the photon statistics of a weakly driven cavity quantum electrodynamics system and discuss the effects of photon blockade and photon-induced tunneling by effectively utilizing instead of avoiding the center-of-mass motion of a two-level atom trapped in the cavity. With the resonant interaction between atom, photon and phonon, it is shown that the bunching and anti-bunching of photons can occur with properly driving frequency. Our study shows the influence of the imperfect cooling of atom on the blockade and provides an attempt to take advantage of the center-of-mass motion.     

Gentile statistics and restricted partitions

In a recent paper (Tranet al, Ann. Phys.311, 204 (2004)), some asymptotic number theoretical results on the partitioning of an integer were derived exploiting its connection to the quantum density of states of a many-particle system. We generalise these results to obtain an asymptotic formula for therestricted or coloured partitionsp _k ^s (n), which is the number of partitions of an integern into the summand of sth powers of integers such that each power of a given integer may occur utmostk times. While the method is not rigorous, it reproduces the well-known asymptotic results fors = 1 apart from yielding more general results for arbitrary values ofs.     

A solvable problem in poisson statistics

Among the wide field of interest of Pierre Résibois, the exact solution of various problems in nonequilibrium statistical mechanics took a large place. Recently he used¹ the model of hard rods moving on a line to study some properties in kinetic theory. As a tribute to his memory, I present in this paper the derivation of the exact solution of a problem of Poisson noise.     

Level spacing statistics for two-dimensional massless Dirac billiards

Classical-quantum correspondence has been an intriguing issue ever since quantum theory was proposed. The searching for signatures of classically nonintegrable dynamics in quantum systems comprises the interesting field of quantum chaos. In this short review, we shall go over recent efforts of extending the understanding of quantum chaos to relativistic cases. We shall focus on the level spacing statistics for two-dimensional massless Dirac billiards, i.e., particles confined in a closed region. We shall discuss the works for both the particle described by the massless Dirac equation(or Weyl equation)and the quasiparticle from graphene. Although the equations are the same, the boundary conditions are typically different,rendering distinct level spacing statistics.     

Full counting statistics of renormalized dynamics in open quantum transport system

The internal dynamics of a double quantum dot system is renormalized due to coupling respectively with transport electrodes and a dissipative heat bath. Their essential differences are identified unambiguously in the context of full counting statistics. The electrode coupling caused level detuning renormalization gives rise to a fast-to-slow transport mechanism, which is not resolved at all in the average current, but revealed uniquely by pronounced super-Poissonian shot noise and skewness. The heat bath coupling introduces an interdot coupling renormalization, which results in asymmetric Fano factor and an intriguing change of line shape in the skewness.     

The interpolation approach to nonextensive quantum statistics

Recently it has been shown by the present author [H. Hasegawa, Phys. Rev. E 80 (2009) 011126] that the interpolation approximation (IA) to the generalized Bose-Einstein and Fermi-Dirac distributions yields good results in agreement with the exact ones within the O(q−1) and in high- and low-temperature limits, where (q−1) expresses the nonextensivity: the case of q=1 corresponding to the conventional quantal distributions. This paper reports applications of the IA to four nonextensive quantum subjects: (i) the black-body radiation, (ii) the Bose-Einstein condensation, (iii) the BCS superconductivity and (iv) itinerant-electron (metallic) ferromagnetism. Effects of the nonextensivity on physical quantities in these nonextensive quantum systems have been investigated. Comparisons between the calculated results and available observed data are made for the subjects (ii) and (iii). It has been pointed out that the factorization approximation (FA) which has been so far applied to many nonextensive systems, overestimates the nonextensivity and that it leads to inappropriate results for fermion systems like the subjects (iii) and (iv).     

Role of quantum statistics in multi-particle decay dynamics

The role of quantum statistics in the decay dynamics of a multi-particle state, which is suddenly released from a confining potential, is investigated. For an initially confined double particle state, the exact dynamics is presented for both bosons and fermions. The time-evolution of the probability to measure two-particle is evaluated and some counterintuitive features are discussed. For instance, it is shown that although there is a higher chance of finding the two bosons (as oppose to fermions, and even distinguishable particles) at the initial trap region, there is a higher chance (higher than fermions) of finding them on two opposite sides of the trap as if the repulsion between bosons is higher than the repulsion between fermions. The results are demonstrated by numerical simulations and are calculated analytically in the short-time approximation. Furthermore, experimental validation is suggested.     

Energy-level statistics of model quantum systems: Universality and scaling in a lattice-point problem

We investigate the statistics of the numberN(R, S) of lattice pointsnZ ², in an annular domain (R, w)=(R+w)A\RA, whereR, w>0. HereA is a fixed convex set with smooth boundary andw is chosen so that the area of (R, w) isS. The statistics comes fromR being taken as random (with a smooth density) in some interval [c ₁ T,c ₂,T],c ₂>c ₁>0. We find that in the limitT the variance and distribution of N=N(R; S)–S depend strongly on howS grows withT. There is a saturation regimeS/T, asT, in which the fluctuations in N coming from the two boundaries of are independent. Then there is a scaling regime,S/Tz, 0<z<, in which the distribution depends onz in an almost periodic way going to a Gaussian asz0. The variance in this limit approachesz for genericA, but can be larger for degenerate cases. The former behavior is what one would expect from the Poisson limit of a distribution for annuli of finite area.     

Behavior of general one-dimensional diffusion processes

We develop simple rigorous techniques to estimate the behavior of general one-dimensional diffusion processes. Any one-dimensional diffusion process (with drift) can be mapped onto a symmetric diffusion through an explicit change of variable. For such processes we can estimate explicitly the diffusion exponent, the recurrence properties, and the large fluctuations. In a second part, we apply these results to different models (including the Sinaï random walk: diffusion in a random drift) and we show how the main features of the diffusion can be readily handled.     

Probing the effective nuclear-spin magnetic field in a single quantum dot via full counting statistics

We study theoretically the full counting statistics of electron transport through a quantum dot weakly coupled to two ferromagnetic leads, in which an effective nuclear-spin magnetic field originating from the configuration of nuclear spins is considered. We demonstrate that the quantum coherence between the two singly-occupied eigenstates and the spin polarization of two ferromagnetic leads play an important role in the formation of super-Poissonian noise. In particular, the orientation and magnitude of the effective field have a significant influence on the variations of the values of high-order cumulants, and the variations of the skewness and kurtosis values are more sensitive to the orientation and magnitude of the effective field than the shot noise. Thus, the high-order cumulants of transport current can be used to qualitatively extract information on the orientation and magnitude of the effective nuclear-spin magnetic field in a single quantum dot.     

Energy level statistics in disordered metals with an anderson transition

We present numerical scaling results for the energy level statistics in orthogonal and symplectic tight-binding Hamiltonian random matrix ensembles defined on disordered two and three-dimensional electronic systems with and without spinorbit coupling (SOC), respectively. In the metallic phase for weak disorder the nearest level spacing distribution functionP(S), the number variance <(N)²>, and the two-point correlation functionK ₂(), are shown to be described by the Gaussian random matrix theories. In the insulating phase, for strong disorder, the correlations vanish for large scales and the ordinary Poisson statistics is asymptotically recovered, which is consistent with localization of the corrosponding eigenstates. At the Anderson metal-insulator transition we obtain new universal scale-invariant distribution functions describing the critical spectral density fluctuations.     

The relation between properties of Gentile statistics and fractional statistics of anyon

In this paper, we discuss the relationship of two kinds of intermediate-statistics, the Gentile statistics and the fractional statistics of anyons. The anyon winding number representation is introduced. We construct the transformation between anyon winding number representation and the occupation number representation of particles of Gentile statistics. We study intermediate-statistics quantum bracket and coherent states for anyons in the winding number representation. We demonstrate that anyons can be simulated by Gentile statistics with a geometric phase.     

The extremal independence problem

Consider a finite sequence of independent-though not, necessarily, identically distributed-real-valued random scores. If the scores are absolutely continuous random variables, the sequence possesses a unique maximum (minimum). We say that “maximal (minimal) independence” holds if the value and the identity of the sequence’s unique maximal (minimal) score are independent random variables. In this research we study the class of statistics for which maximal (minimal) independence holds, and: (i) establish explicit characterizations of this class; (ii) connect this class with the class of Lévy processes; (iii) unveil the underlying spatial Poissonian structure of this class.     

Generalized Fock spaces,new forms of quantum statistics and their algebras

We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as ‘infinite’, Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot be mapped into single-indexed systems are studied. Our theory is based on a three-tiered structure consisting of Fock space, statistics and algebra. This general formalism not only unifies the various forms of statistics and algebras, but also allows us to construct many new forms of quantum statistics as well as many algebras of creation and destruction operators. Some of these are: new algebras for infinite statistics,q-statistics and its many avatars, a consistent algebra for fractional statistics, null statistics or statistics of frozen order, ‘doubly-infinite’ statistics, many representations of orthostatistics, Hubbard statistics and its variations.     

Average position in quantum walks with a U（2） coin

We investigated discrete-time quantum walks with an arbitary unitary coin.Here we discover that the average position x=max(x) sin(α+γ),while the initial state is 1/2～(1/2)(|0L+i|0R).We verify the result,and obtain some symmetry properties of quantum walks with a U(2) coin with |0L and |0R as the initial state.     

Foundations of the quantum statistics of systems in equilibrium: A measuretheoretic approach

The fundamental equations of equilibrium quantum statistical mechanics are derived in the context of a measure-theoretic approach to the quantum mechanical ergodic problem. The method employed is an extension, to quantum mechanical systems, of the techniques developed by R. M. Lewis for establishing the foundations of classical statistical mechanics. The existence of a complete set of commuting observables is assumed, but no reference is made a priori to probability or statistical ensembles. Expressions for infinite-time averages in the microcanonical, canonical, and grand canonical ensembles are developed which reduce to conventional quantum statistical mechanics for systems in equilibrium when the total energy is the only conserved quantity. No attempt is made to extend the formalism at this time to deal with the difficult problem of the approach to equilibrium.     

Orbital effect of a magnetic field on two-leg Hubbard ladder

By setting up the relevant recursion relations and by doing exact and approximate calculations, we show that there is no critical dimension in a self-avoiding random walk on a simplex fractal. Received: 6 April 1998 / Revised: 4 August 1998 / Accepted: 26 August 1998