Probability distribution functions for a single-particle vibrating in one dimension: experimental study and theoretical analysis

We consider the form of the rebound velocity, ν₀, particle velocity, ν, and height, h, probability density functions (PDFs) for the one-dimensional motion of a single particle on a sinusoidally oscillating base. The motion is considered in the limit of high excitation (vibration frequency ⪢ collision rate). Experimentally, we find that these PDFs are well-approximated by P_ν0(ν₀) ∞ ν₀ exp(− αν₀²), a Gaussian P_ν(ν) ∞ exp(− αν²) and a Boltzmann-type function P_h(h) ∞ exp(− 2αgh), where α is a constant and g is the acceleration due to gravity. We develop an analytical model which accurately predicts the general form for the rebound velocity PDF; the other two PDFs are then analytically shown to follow as a consequence. Scaling laws for the particle granular temperature with peak base velocity and particle-base restitution coefficient, determined from previous work, can also be predicted from the PDF. A fine scale “spiky” structure in the rebound velocity PDF is found, using numerical simulations, to be a consequence of resonance phenomena between the particle and vibrating base. Good agreement between scaling laws from the theory and simulation is found but insufficient data is obtainable to derive accuracy exponents experimentally.     

The effect of an external field on an interface,entropic repulsion

We consider the effects of an external potential -h∑f(S _x ) withh>0,f increasing, on the equilibrium state of a system with a Hamiltonian of the form $$H^0 (S) = \sum\limits_{\left\langle {xy} \right\rangle } {\Phi (S_x - S_y )} ,S_x \in R,x \in Z^d ,d \geqslant 3$$ Φ even and convex, e.g.,Φ(t)=1/2t ² andf(t)=signt. This can be thought of as a model of the interactions between a random interface S _x and a “soft” wall. We show that the random surface is (entropically) repelled to infinity for allh>0, i.e., with probability one,S _x ≥K, for anyK ε R.     

Dynamical theory for photon and photoelectric pulse distributions in single molecule fluorescence

A single two-level molecule driven by CW-laser field and a photomultiplier tube (PMT) are considered as two parts of the united dynamical system connected with each other by photons of molecular fluorescence. Each PMT is characterized by a rate α of photo-effect and by a rate β of PMT recovery. A theory for the photon distribution function w_N(t) and for the photoelectric pulse distribution function f_n(t) for such a system is built up. If times 1/ α and 1/ β characterizing PMT are much shorter as compared to the average time interval 1/ k between two successively emitted photons of fluorescence, the photon and the photoelectron distribution functions coincide with each other, i.e. f_n(t) ≅ w_N(t). A relation between w_N(t) and f_n(t) is studied in detail for the case in which PMT works slower as compared to the rate k of photon emission, i.e. at 1/ α, 1/ β ≥ 1/ k.     

Controlled vortex motion in multiple interpenetrating pinning arrays

We study the effect of multiple interpenetrating pinning arrays on the vortex motion in the presence of an ac driving force, f _d (t), by using extensive molecular dynamics (MD) simulations. Firstly, the response to a square ac wave f _d (t) has been explored for the vortices interacting with a periodic square pinning array which has different pinning strengths and sizes. The effect of the type of an ac drive and its amplitude on the oscillatory dynamics of vortices have been investigated in detail. For very low displacements of the vortices, we have found that the single-particle model can produce results analytically similar to the ones obtained by the MD simulations. It is shown that the collective motion of vortices can be controlled easily by varying the number of multiple interpenetrating square pinning lattices (N _SPSL). A regular sequence of peaks has been observed for N _SPSL = 3 in the time evolution of the average velocity of the vortices (i.e., V? _{x - t} curves). The number of peaks (N _peak) strongly depends on the magnitude of f _d (t), and increases with increasing the magnitude of f _d . The close relation between N _peak and f _d is considered as an indication of controlling vortex motion in a multiple periodic pinning structure. Finally, the variation of the power spectrum of noise S(ν) with N _SPSL has been investigated. For N _SPSL = 3, it has been found that the plastic motion of the row of vortices evolves at low frequencies, i.e., 1/ν behavior, whereas, at high frequencies, S(ν) shows a typical behavior of Gaussian white noise.     

Study off o (995) resonance in the π0π0 decay channel

A high-statistics study of the π⁰π⁰-system produced in the π^- p→π⁰π⁰ n reaction at 38GeV/c has been carried out at the IHEP accelerator using the GAMS-2000 multiphoton spectrometer. A partial-wave analysis which includesS, D ₀, D_? andD ₊ waves has been performed in the |t|-range up to 1 (GeV/c)². TheS ^*/f ₀(980) resonance is seen as a dip in theS-wave amplitude at small |t|. A distinct peak with a mass of 997±5MeV and a width of 48±10MeV is observed in theS-wave at |t|>0.3 (GeV/c)². The production cross sections are measured.     

Study of the K − → μ − νπ 0 decay

The decay K ^? → μ ^? νπ ⁰ has been studied using in-flight decays detected with the ISTRA+ setup operating in the 25-GeV negative secondary beam of the U-70 PS. About 112K events were used for the analysis. The λ ₊ and λ ₀ slope parameters of the decay form factors f ₊(t) and f ₀(t) have been measured: λ ₊=0.0321±0.004(stat.)±0.002(syst.), λ ₀=0.0209±0.004(stat.)±0.002(syst.); the correlation dλ ₀/dλ ₊=?0.46. The limits on the possible tensor and scalar couplings have been derived: f _T/f ₊(0)=?0.021±0.028(stat.)±0.014(theor.), f _S/f ₊(0)=0.004±0.005(stat.)±0.005(theor.).     

Closed geodesics on Riemannian manifolds via the heat flow

The main topic discussed in this paper is the following question: Given a Riemannian manifold M and a closed C¹ curve f: S¹ → M does there exist a (unique) solution of the heat equation ?_tf_t = τ(f_t) defined for all t ≧ 0 which is continuous at t = 0 along with its first S¹-derivative and which coincides with f at t = 0.     

Kinetic Limit of N-Body Description of Wave–Particle Self-Consistent Interaction

A system of N particles $\xi ^N = x_1 ,\upsilon_1,...,x_N ,\upsilon _N )$ interacting self-consistently with one wave Z = A exp(iφ) is considered. Given initial data (Z ^(N)(0), ξ ^N (0)), it evolves according to Hamiltonian dynamics to (Z ^(N)(t), ξ ^N (t)). In the limit N → ∞, this generates a Vlasov-like kinetic equation for the distribution function f(x, v, t), abbreviated as f(t), coupled to the envelope equation for Z: initial data (Z ^(∞)(0), f(0)) evolve to (Z ^(∞)(t), f(t)). The solution (Z, f) exists and is unique for any initial data with finite energy. Moreover, for any time T>0, given a sequence of initial data with N particles distributed so that the particle distribution f ^N(0) → f(0) weakly and with Z ^(N)(0) → Z(0) as N → ∞, the states generated by the Hamiltonian dynamics at all times 0 ≤ t ≤ T are such that (Z ^(N)(t), f ^N(t)) converges weakly to (Z ^(∞)(t), f(t)).     

The exact solution of an elimination problem in kinetic theory: I. The one-dimensional hard sphere gas

A class of initial value problems for a one-dimensional hard sphere gas is considered where a specified particle has a given distribution f⁽¹⁾(z₁; 0) and the rest are in equilibrium at t=0. An exact expansion is obtained for a certain n-particle reduced distribution function f⁽ⁿ⁾(z₁;…;z_n; t) in terms of the 1-particle reduced distribution function f⁽¹⁾(z₁; t) for the specified particle by starting with separate expressions for these functions in terms of f⁽¹⁾(z₁; 0). Expansions for the corresponding cluster functions are first obtained and then graph theoretic methods applied to obtain a solution.     

A meson-theoretical explanation of the f 0 (980)-production puzzle in the reaction π− p → π0π0 n

A meson-theoretical model for the reaction π^? p → π⁰π⁰ n for large π^? momenta is developed which treats the final-state interaction between the two produced pions microscopically. For small momentum transfers, the squares of the S-wave amplitudes of the produced two-pion system show a dip for invariant twopion masses in the vicinity of the f ₀(980) meson, while for momentum transfers ?t>0.2(GeV/c)², the f ₀(980) meson appears as a bump. The model provides a microscopic explanation of the long-standing puzzle seen by both the GAMS and the BNL experimental collaborations.     

Radiative transfer: Gaussian quadrature formulas for integrals with the weight functions exp(-|x−t|) and En(|x−t|)

One to six point Gaussian quadrature formulas have been computed for the roots and weight coefficients of the integrals, from zero to infinity, of the integrands f(x) exp(-|x?t|) and f(x)E_n(|x?t|). These are then tabulated as functions of t. Analytic formulas are given explicitly for one and two point quadratures for any n. The asymptotic behavior of the roots and weight coefficients is discussed. Estimates of errors for the quadratures are also given.     

Diffusion activation energy versus the favourable energy in two-order-parameter model:A molecular dynamics study of liquid Al

In the present work, we find that both diffusion activation energy E_a(D) and E_a(S^ex) increase linearly with pressure and have the same slope (0.022±0.001 eV/GPa) for liquid Al. The temperature and pressure dependence of excess entropy is well fitted by the expression -S^ex(T,P)/k_B=a(P)+b(P)T+c(P)exp(E_f/k_BT), which together with the small ratio of E_f/k_BT leads to the relationship of excess entropy to temperature and pressure, i.e. S^ex≈-cE_f/T, where c is about 12 and E_f (=Δ E-PΔV) is the favourable energy. Therefore, there exists a simple relation between E_a(S^ex) and E_f, i.e. E_a(S^ex)≈cE_f.     

Adiabatic theorem in axiomatic quantum field theory

It is shown that Møller matricesS _± and scattering matrixS in axiomatic field theory can be expressed through their adiabatic analogs. In particular, it is proved under certain conditions that \(S_ - = \mathop {s\lim }\limits_{\alpha \to 0} S_\alpha (0,\infty )W_\alpha \) whereW _α is a trivial phase factor [i.e. a unitary operator of the form exp i / α ∝r(k)a ⁺ (k)a(k)dk]. Corresponding results in Hamiltonian approach are discussed.     

Investigation of the ηη system in π − p interactions at a momentum of 32.5 GeV/c at the GAMS-4π spectrometer

The ηη system produced in charge-exchange π ^? p interaction at a momentum of 32.5 GeV/c is studied in an experiment performed with the GAMS-4π spectrometer at the 70-GeV accelerator of the Institute for High-Energy Physics (IHEP, Protvino). A partial-wave analysis is performed in the mass range between 1.1 and 3.9 GeV for ?t <0.2 (GeV/c)², S, D, G, and J waves being taken into account in this analysis. The S wave has a complicated structure, displaying peaks at about 1.5 and 1.7 GeV. These peaks are associated with the f ₀(1500) and f ₀(1710) mesons. One of the solutions (preferable one) involves the f ₀(2200) and f ₂(1950) resonances. The mass region above 2.4 GeV is dominated by the G wave. A broad state of mass about 3 GeV and width 0.7 GeV is found in the J wave. The parameters of the resonances in question and their production cross sections are measured.     

K1-Dominance of the Ke3 form factor and Sirlin's relation

We discuss the precise momentum dependence of the K_e3 form factor f₊(t) by studying some of the recent experimental results on K⁰_L → π^± e^? ν decays. The parametrization of f₊(t) based on the assumption of K¹-dominance can considerably improve the agreement of Sirlin's relation with existing data.     

Boltzmann'sH theorem and the Loschmidt and the Zermelo paradoxes

The Umkehreinwand of Loschmidt and the Wiederkehreinwand of Zermelo have been reexamined. The former paradox depends on the augument that for a dynamical system, upon the reversal of the velocities of all the molecules, theH function retraces its sequence of values so thatdH/dt will change its sign. The latter paradox depends on the argument that theH function returns infinitely close to its value after a Poincare' quasi-period and therefore cannot be decreasing all the time. While the main contention of the two paradoxes is correct, that theH theorem is inconsistent with classical dynamical laws, the arguments there can be considerably simplified and the “paradoxes” answered more directly. If the distribution functionf(q _K ,p _K ,t) is governed by an equation which is time-reversal invariant (such as the Liouville equation for a closed dynamical system), then it can be shown immediately thatdH/dt=0,H=cons. In this case, both paradoxes disappear, but together with them, thedH/dt<0 part of theH theorem also has disappeared, i.e., there is no second law of thermodynamics. Iff(q _K ,p _K ,t) is governed by an equation which is not time-reversal invariant (such as the Boltzmann equation, or the Master Equation for Markovian processes), then (1) there is no argument forf andH(t) to retrace their sequence of values upon the reversal of all the velocities of the system, (2) there is no quasiperiod in whichf andH(t) return to their earlier values. In this case, both paradoxes disappear also, but then one must go beyond classical dynamics in order to maintain theH theorem.     

Entropy andP-particle observables. II. The two-particle entropyS 2

This paper defines, and then evaluates perturbatively, an information-theoretic notion of entropyS ₂ for a system ofN interacting particles which assesses an observer's limited knowledge of the state of the system, assuming that he or she can measure with arbitrary precision all one-particle observables and correlations involving pairs of particles, but is completely ignorant of the form of any higher-order correlations involving three or more particles. By construction, thisS ₂(t) involves only the reduced two-particle distribution functions, or density matrices,f ₂(i,j) at timet, and, though the implementation of a subdynamics,dS ₂ (t)/dt can be realized in terms of thef ₂(i, j)'s at retarded timest–. A similar line of reasoning demonstrates that the most probable three-particlef ₃(i,j, k) consistent with a knowledge of thef ₂'s is precisely thatf ₃ suggested by the Kirkwood, or cluster, decomposition.     

Time evolution of coherent ground-state correlations and the TDHF approach

We present a formalism and some of its applications aimed at describing RPA-like correlations on top of a time-dependent mean field. These correlations correspond to the zero-point oscillations of collective modes orthogonal to the TDHF trajectory. They are not the result of some instantaneous local RPA equations but they evolve freely from some given initial conditions. A generalized time-dependent generator coordinate (TDGCM) variational formulation leads to explicit equations of motion for both f(q, t) and |φ_q(t〉 in |ψ(t)〉 = ∝dqf(q, t)|φ_q(t)〉, the correlated time-dependent wave functions. The q-distorted Slater determinants |φ_q(t)〉, obey well known TDHF equations of motion while f(q, t) obey a time-dependent generator coordinate integral equation. This last one is further simplified by the assumption of gaussian overlaps for the correlations, so that one can then solve for the time dependence of the correlations by evaluating a few TDHF trajectories differing by small amounts in the collective variables q considered. This TDGCM method is applied in a one-dimensional TDHF scheme. Momentum, energy and breathing fluctuations are studied, as well as their mutual interaction. We find that the fluctuations can be enlarged by an order of magnitude over the values predicted by pure TDHF.     

Diffusion Effects on the Breakdown of a Linear Amplifier Model Driven by the Square of a Gaussian Field

We investigate solutions to the equation ? _t ?? $\mathcal{D}$ Δ?=λS ²?, where S(x, t) is a Gaussian stochastic field with covariance C(x?x′, t, t′), and x∈ $\mathbb{R}$ ^d . It is shown that the coupling λ _cN (t) at which the N-th moment <? ^N (x, t)> diverges at time t, is always less or equal for $\mathcal{D}$ >0 than for $\mathcal{D}$ =0. Equality holds under some reasonable assumptions on C and, in this case, λ _cN (t)=Nλ _c (t) where λ _c (t) is the value of λ at which <exp[λ ∫ ^t ₀ S ²(0, s) ds]> diverges. The $\mathcal{D}$ =0 case is solved for a class of S. The dependence of λ _cN (t) on d is analyzed. Similar behavior is conjectured when diffusion is replaced by diffraction, $\mathcal{D}$ →i $\mathcal{D}$ , the case of interest for backscattering instabilities in laser-plasma interaction.