Weakly Non-Ergodic Statistical Physics

For weakly non ergodic systems, the probability density function of a time average observable is where is the value of the observable when the system is in state j=1,…L. p _j ^eq is the probability that a member of an ensemble of systems occupies state j in equilibrium. For a particle undergoing a fractional diffusion process in a binding force field, with thermal detailed balance conditions, p _j ^eq is Boltzmann’s canonical probability. Within the unbiased sub-diffusive continuous time random walk model, the exponent 0<α<1 is the anomalous diffusion exponent 〈x ²〉∼t ^α found for free boundary conditions. When α→1 ergodic statistical mechanics is recovered . We briefly discuss possible physical applications in single particle experiments.     

Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation

We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on the real axis, i.e. the probability that these polynomials have no real root in a given interval. For generalized Kac polynomials, indexed by an integer d, of large degree n, one finds that the probability of no real root in the interval [0,1] decays as a power law n ^−θ(d) where θ(d)>0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n≫1 even, the probability that they have no real root on the full real axis decays like n ^{−2(θ(2)+θ(d))}. For Weyl polynomials and Binomial polynomials, this probability decays respectively like and where θ _∞ is such that in large dimension d. We also show that the probability that such polynomials have exactly k roots on a given interval [a,b] has a scaling form given by where N _ab is the mean number of real roots in [a,b] and a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors, and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has a universal algebraic tail with exponent −2. These analytical results are confirmed by detailed numerical computations. Some of these results were announced in a recent letter (Schehr and Majumdar in Phys. Rev. Lett. 99:060603, 2007).     

The Invariant Eigen-Operator Method for?N?Harmonically Coupled Identical Oscillators

In this paper, we apply the method of “invariant eigen-operator” to study the Hamiltonian of arbitrary number of coupled identical oscillators and derive their invariant eigen-operator. The results show that, (1) for the system of arbitrary number of identical harmonic oscillators by coordinate coupling or momentum coupling, the invariant eigen operator of system always has the form of or ; (2) the energy level gap of the system has two kinds of possibilities: one is that gap only related to ω that the frequency of oscillators; another one is that gap not only related to ω that the frequency of oscillators, but also related to the number of the coupling oscillators.     

Condensation of the Roots of Real Random Polynomials on the Real Axis

We introduce a family of real random polynomials of degree n whose coefficients a _k are symmetric independent Gaussian variables with variance , indexed by a real α≥0. We compute exactly the mean number of real roots 〈N _n 〉 for large n. As α is varied, one finds three different phases. First, for 0≤α<1, one finds that . For 1<α<2, there is an intermediate phase where 〈N _n 〉 grows algebraically with a continuously varying exponent, . And finally for α>2, one finds a third phase where 〈N _n 〉∼n. This family of real random polynomials thus exhibits a condensation of their roots on the real line in the sense that, for large n, a finite fraction of their roots 〈N _n 〉/n are real. This condensation occurs via a localization of the real roots around the values , 1≪k≤n.     

Detailed Examination of Transport Coefficients in Cubic-Plus-Quartic Oscillator Chains

We examine the thermal conductivity and bulk viscosity of a one-dimensional (1D) chain of particles with cubic-plus-quartic interparticle potentials and no on-site potentials. This system is equivalent to the FPU-α β system in a subset of its parameter space. We identify three distinct frequency regimes which we call the hydrodynamic regime, the perturbative regime and the collisionless regime. In the lowest frequency regime (the hydrodynamic regime) heat is transported ballistically by long wavelength sound modes. The model that we use to describe this behaviour predicts that as ω→0 the frequency dependent bulk viscosity, , and the frequency dependent thermal conductivity, , should diverge with the same power law dependence on ω. Thus, we can define the bulk Prandtl number, , where m is the particle mass and k _B is Boltzmann’s constant. This dimensionless ratio should approach a constant value as ω→0. We use mode-coupling theory to predict the ω→0 limit of Pr _ζ . Values of Pr _ζ obtained from simulations are in agreement with these predictions over a wide range of system parameters. In the middle frequency regime, which we call the perturbative regime, heat is transported by sound modes which are damped by four-phonon processes. This regime is characterized by an intermediate-frequency plateau in the value of . We find that the value of in this plateau region is proportional to T ⁻² where T is the temperature; this is in agreement with the expected result of a four-phonon Boltzmann-Peierls equation calculation. The Boltzmann-Peierls approach fails, however, to give a nonvanishing bulk viscosity for all FPU-α β chains. We call the highest frequency regime the collisionless regime since at these frequencies the observing times are much shorter than the characteristic relaxation times of phonons.     

Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon

As Bleher (J. Stat. Phys. 66(1):315–373, 1992) observed the free flight vector of the planar, infinite horizon, periodic Lorentz process {S _n ∣n=0,1,2,…} belongs to the non-standard domain of attraction of the Gaussian law—actually with the $\sqrt{n\log n}As Bleher (J. Stat. Phys. 66(1):315–373, 1992) observed the free flight vector of the planar, infinite horizon, periodic Lorentz process {S _n ∣n=0,1,2,…} belongs to the non-standard domain of attraction of the Gaussian law—actually with the scaling. Our first aim is to establish his conjecture that, indeed, converges in distribution to the Gaussian law (a Global Limit Theorem). Here the recent method of Bálint and Gou?zel (Commun. Math. Phys. 263:461–512, 2006), helped us to essentially simplify the ideas of our earlier sketchy proof (Szász, D., Varjú, T. in Modern dynamical systems and applications, pp. 433–445, 2004). Moreover, we can also derive (a) the local version of the Global Limit Theorem, (b) the recurrence of the planar, infinite horizon, periodic Lorentz process, and finally (c) the ergodicity of its infinite invariant measure. Dedicated to Ya.G. Sinai on the occasion of his seventieth birthday. Research supported by the Hungarian National Foundation for Scientific Research grants No. T046187, NK 63066 and TS 049835, further by Hungarian Science and Technology Foundation grant No. A-9/03.     

An Algebraic Derivation of the Eigenspaces Associated with an Ising-Like Spectrum of the Superintegrable Chiral Potts Model

In terms of the loop algebra and the algebraic Bethe-ansatz method, we derive the invariant subspace associated with a given Ising-like spectrum consisting of 2 ^r eigenvalues of the diagonal-to-diagonal transfer matrix of the superintegrable chiral Potts (SCP) model with arbitrary inhomogeneous parameters. We show that every regular Bethe eigenstate of the τ ₂-model leads to an Ising-like spectrum and is an eigenvector of the SCP transfer matrix which is given by the product of two diagonal-to-diagonal transfer matrices with a constraint on the spectral parameters. We also show in a sector that the τ ₂-model commutes with the loop algebra, , and every regular Bethe state of the τ ₂-model is of highest weight. Thus, from physical assumptions such as the completeness of the Bethe ansatz, it follows in the sector that every regular Bethe state of the τ ₂-model generates an -degenerate eigenspace and it gives the invariant subspace, i.e. the direct sum of the eigenspaces associated with the Ising-like spectrum.     

Exponential Control of Overlap in the Replica Method for <Emphasis Type="Italic">p</Emphasis>-Spin Sherrington-Kirkpatrick Model

In Talagrand (J. Stat. Phys. 126(4–5):837–894, 2007) the large deviations limit for the moments of the partition function Z _N in the Sherrington-Kirkpatrick model (Sherrington and Kirkpatrick in Phys. Rev. Lett. 35:1792–1796, 1972) was computed for all real a≥0. For a≥1 this result extends the classical physicist’s replica method that corresponds to integer values of a. We give a new proof for a≥1 in the case of the pure p-spin SK model that provides a strong exponential control of the overlap. This work is partially supported by NSF grant.     

Dilatonic Cosmology Model in the <Emphasis Type="Italic">ω</Emphasis>–<Emphasis Type="Italic">ω</Emphasis>′ Plane

In dilatonic cosmology model, we study the behavior of attractor solution in ω–ω′ plane, which is defined by the equation of state parameter for the dark energy and its derivative with respect to N (the logarithm of the scale factor a). This is a good method which is useful to the study of classifying the dynamical dark energy models including “freezing” and “thawing” model. We find that our model belongs to “freezing” type model classified in ω–ω′ plane. We show mathematically the property of attractor solutions which correspond to ω _σ =−1, Ω _σ =1. The present values of energy density parameter , and are 0.715001, 0.284972 and 0.00002706 respectively, which meet the current observations well. Finally, we can obtain that the coupling between dilaton and matter affects the evolutive process of the Universe, but not the fate of the Universe.     

L e e-Yang circl e analysis of e+ e− and $p\ov erlin e{p}$ g en eraliz ed multiplicity distribution

We study the evolution of Lee-Yang zeros structure of generalized multiplicity distribution (GMD) in high energy collision. Starting our study with electron-positron e ⁺ e ⁻ scattering data, we extend the study by Chan and Chew (Z. Phys. C 55:503, 1992) on TASSO and AMY multiplicity data for , 22, 34.8, 43.6 and 57 GeV to the ones from DELPHI and OPAL Collaboration for , 133, 161, 172, 183 and 189 GeV. We compare the results with the Lee-Yang structure for proton-antiproton at , 546 and 900 GeV from UA5 Collaboration. Our preliminary result shows that there is indeed a change in the shape and size of the Lee-Yang zeros with increasing energy, accompanied by the development of the so-called “ear”-like structure in the Lee-Yang plot. We expect that the development of this “ear”-like structure is related to the “shoulder” structure in the multiplicity data, which further indicates an ongoing phase transition from soft to semihard scattering. We also extend our prediction to LHC’s  TeV. Insert your abstract here.     

A Note on the Extreme Points of Positive Quantum Operations

In this paper, an error in the proof of Theorem 4.9 in Gudder’s paper (Int. J. Theor. Phys. 47(1):268–279, 2008) is pointed out and it is proved that if such that E _i ∈ℂI∖{0} and E _j ∉ℂI for some i,j in {1,2,…,n}, then . This subject is supported by the NNSF of China (No. 10571113, 10871224).     

Di-jet production in γ γ collisions at LEP2

The production of two high-p _T jets in the interactions of quasi-real photons in e ⁺ e ⁻ collisions at from 189 GeV to 209 GeV is studied with data corresponding to an integrated e ⁺ e ⁻ luminosity of 550 pb⁻¹. The jets reconstructed by the k _⊥ -cluster algorithm are defined within the pseudo-rapidity range −1<η<1 and with jet transverse momentum, p _T , above 3 GeV/c. The differential di-jet cross-section is measured as a function of the mean transverse momentum of the jets and is compared to perturbative QCD calculations. Deceased     

Random Walk Weakly Attracted to a Wall

We consider a random walk X _n in ℤ₊, starting at X ₀=x≥0, with transition probabilities

First-Order Intertwining Operators with Position Dependent Mass and <Emphasis Type="Italic">η</Emphasis>-Weak-Pseudo-Hermiticity Generators

A Hermitian and an anti-Hermitian first-order intertwining operators are introduced and a class of η-weak-pseudo-Hermitian position-dependent mass (PDM) Hamiltonians are constructed. A corresponding reference-target η-weak-pseudo-Hermitian PDM—Hamiltonians’ map is suggested. Some η-weak-pseudo-Hermitian -symmetric Scarf II and periodic-type models are used as illustrative examples. Energy-levels crossing and flown-away states phenomena are reported for the resulting Scarf II spectrum. Some of the corresponding η-weak-pseudo-Hermitian Scarf II- and periodic-type-isospectral models ( -symmetric and non- -symmetric) are given as products of the reference-target map.     

Identified hadron production at high transverse momenta in p+ p collisions at $\sqrt{s_{NN}}=200$ GeV in STAR

We report the transverse momentum (p _T ) distributions for identified charged pions, protons and anti-protons using events triggered by high deposit energy in the Barrel Electro-Magnetic Calorimeter (BEMC) from p+p collisions at  GeV. The spectra are measured around mid-rapidity (|y|<0.5) over the range of 3<p _T <15 GeV/c with particle identification (PID) by the relativistic ionization energy loss (rdE/dx) in the Time Projection Chamber (TPC) of the Solenoidal Tracker at RHIC (STAR). The charged pion, proton and anti-proton spectra at high p _T are compared with published results from minimum bias triggered events and the Next-Leading-Order perturbative quantum chromodynamic (NLO pQCD) calculations (DSS, KKP and AKK 2008). In addition, we present the particle ratios of π ⁻/π ⁺, , p/π ⁺ and in p+p collisions.     

Asymptotic Behavior of Inflated Lattice Polygons

We study the inflated phase of two dimensional lattice polygons with fixed perimeter N and variable area, associating a weight exp [pA−Jb] to a polygon with area A and b bends. For convex and column-convex polygons, we calculate the average area for positive values of the pressure. For large pressures, the area has the asymptotic behaviour , where , and ρ<1. The constant K(J) is found to be the same for both types of polygons. We argue that self-avoiding polygons should exhibit the same asymptotic behavior. For self-avoiding polygons, our predictions are in good agreement with exact enumeration data for J=0 and Monte Carlo simulations for J≠0. We also study polygons where self-intersections are allowed, verifying numerically that the asymptotic behavior described above continues to hold.     

Spherical-separability of Non-Hermitian Hamiltonians and Pseudo- PT\mathcal{PT} -symmetry

Non-Hermitian but -symmetrized spherically-separable Dirac and Schr?dinger Hamiltonians are considered. It is observed that the descendant Hamiltonians H _r , H _θ , and H _φ play essential roles and offer some “user-feriendly” options as to which one (or ones) of them is (or are) non-Hermitian. Considering a -symmetrized H _φ , we have shown that the conventional Dirac (relativistic) and Schr?dinger (non-relativistic) energy eigenvalues are recoverable. We have also witnessed an unavoidable change in the azimuthal part of the general wavefunction. Moreover, setting a possible interaction V(θ)≠0 in the descendant Hamiltonian H _θ would manifest a change in the angular θ-dependent part of the general solution too. Whilst some -symmetrized H _φ Hamiltonians are considered, a recipe to keep the regular magnetic quantum number m, as defined in the regular traditional Hermitian settings, is suggested. Hamiltonians possess properties similar to the -symmetric ones (here the non-Hermitian -symmetric Hamiltonians) are nicknamed as pseudo- -symmetric.     

Multiple Intersection Exponents for Planar Brownian Motion

Let p≥2, n ₁≤⋅⋅⋅≤n _p be positive integers and be independent planar Brownian motions started uniformly on the boundary of the unit circle. We define a p-fold intersection exponent ς _p (n ₁,…,n _p ), as the exponential rate of decay of the probability that the packets , i=1,…,p, have no joint intersection. The case p=2 is well-known and, following two decades of numerical and mathematical activity, Lawler et al. (Acta Math. 187:275–308, 2001) rigorously identified precise values for these exponents. The exponents have not been investigated so far for p>2. We present an extensive mathematical and numerical study, leading to an exact formula in the case n ₁=1, n ₂=2, and several interesting conjectures for other cases.     

Fragmentation function and hadronic production of the heavy supersymmetric hadrons

The light top-squark may be the lightest squark, and its lifetime may be ‘long enough’ in a kind of SUSY models that have not been ruled out yet experimentally, so colorless ‘supersymmetric hadrons (superhadrons)’ (q is a quark excluding the t-quark) may be formed as long as the light top-squark can be produced. The fragmentation function of into heavy ‘supersymmetric hadrons (superhadrons)’ (Q̄=c̄ or b̄) and hadronic production of the superhadrons are investigated quantitatively. The fragmentation function is calculated precisely. Due to the difference in spin of the SUSY component, the asymptotic behavior of the fragmentation function is different from those of the existing ones. The fragmentation function is also applied to compute the production of heavy superhadrons at the hadronic colliders Tevatron and LHC in the so-called fragmentation approach. The resultant cross-section for the heavy superhadrons is too small to observe at Tevatron, but large enough at LHC, when all the relevant parameters in the SUSY models are taken within the favored region for the heavy superhadrons. The production of ‘light superhadrons’ (q=u,d,s) is also roughly estimated with the same SUSY parameters. It is pointed out that the production cross-sections of the light superhadrons may be much greater than those of the heavy superhadrons, so that even at Tevatron the light superhadrons may be produced in great quantities. PACS 12.38.Bx; 13.87.Fh; 12.60.Jv; 14.80.Ly     

On the Rate of Explosion for Infinite Energy Solutions of the Spatially Homogeneous Boltzmann Equation

Let μ ₀ be a probability measure on ℝ³ representing an initial velocity distribution for the spatially homogeneous Boltzmann equation for pseudo Maxwellian molecules. As long as the initial energy is finite, the solution μ _t will tend to a Maxwellian limit. We show here that if , then instead, all of the mass “explodes to infinity” at a rate governed by the tail behavior of μ ₀. Specifically, for L0, define