Supersymmetric graphene on squashed hemisphere

Depolarized Rayleigh wing scattering data are reported for liquid chloroform, methyl iodide and methylene chloride. The calculation of orientation correlation functions from these results and from infrared vibration-rotation bands is discussed. Fourier transforms of the infrared bands are corrected for shape changes due to variable refractive index, isotope splitting and hot bands. The resulting correlation functions are primarily a measure of orientational and collision-induced changes in polarizability or dipole moment. It is argued that the collision-induced contributions are important only at short times. The long-time behavior of the correlation functions is then compared with several models for molecular reorientation in liquids, and it is concluded that the motion of these molecules is probably best characterized by models involving random walks in angular momentum space.

Molecular thermodynamic model for equilibria in solution: III. Equilibrium constants and correlation functions in probability, thermodynamic, and kinetic energy space

We have investigated the redshift space distortions in the optically selected Durham/UKST Galaxy Redshift Survey using the 2-point galaxy correlation function perpendicular and parallel to the observer’s line of sight, �(�,�). We present results for the real space 2-point correlation function, �(r), by inverting the optimally estimated projected correlation function, which is obtained by integration of �(�,�), and find good agreement with other real space estimates. On small, non-linear scales we observe an elongation of the constant �(�,�) contours in the line of sight direction. This is due to the galaxy velocity dispersion and is the common “Finger of God” effect seen in redshift surveys. Our result for the one-dimensional pairwise rms velocity dispersion is 1/2 = 416 ± 36kms 1 which is consistent with those from recent redshift surveys and canonical values, but inconsistent with SCDM or LCDM models. On larger, linear scales we observe a compression of the �(�,�) contours in the line of sight direction. This is due to the infall of galaxies into overdense regions and the Durham/UKST data favours a value of ( 0.6 /b)�0.5, where is the mean mass density of the Universe and b is the linear bias factor which relates the galaxy and mass distributions. Comparison with other optical estimates yield consistent results, with the conclusion that the data does not favour an unbiased critical-density universe.

Within the framework of probability distributions on projective Hilbert space a scheme for the calculation of multitime correlation functions is developed. The starting point is the Markovian stochastic wave function description of an open quantum system coupled to an environment consisting of an ensemble of harmonic oscillators in arbitrary pure or mixed states. It is shown that matrix elements of reduced Heisenberg picture operators and general time-ordered correlation functions can be expressed by time-symmetric expectation values of extended operators in a doubled Hilbert space. This representation allows the construction of a stochastic process in the doubled Hilbert space which enables the determination of arbitrary matrix elements and correlation functions. The numerical efficiency of the resulting stochastic simulation algorithm is investigated and compared with an alternative Monte Carlo wave function method proposed first by Dalibard et al. [Phys. Rev. Lett. {\bf 68}, 580 (1992)]. By means of a standard example the suggested algorithm is shown to be more efficient numerically and to converge faster. Finally, some specific examples from quantum optics are presented in order to illustrate the proposed method, such as the coupling of a system to a vacuum, a squeezed vacuum within a finite solid angle, and a thermal mixture of coherent states.

On the brownian movement of a rotator in a cosine potential

The deconfinement phase transition of large N gauge theories has been recognized as the Hawking-Page transition through AdS/CFT correspondence. To derive the phase transition, the calculation of the partition function is restricted to the singlet sector. This thesis examines whether the singlet constraint of gauge theory at the deconfinement phase was a physical consequence. A model of multiple matrices quantum harmonic oscillators is considered in this thesis as a toy model of large N, four dimensional SU(N) gauge theory. This thesis aims to compare the partition function restricted to singlet sector to the known partition function of a group of non-interacting harmonic oscillators, which represent the model without imposing the singlet constraint. We compute the singlet partition function analytically in the large m limit for a model with m matrices, and we can correct the result to higher orders in the large m expansion by the resolvent method in a random matrix model. We also develop numerical methods using a quadratic programming algorithm to solve the partition function at small m. We assure that the eigenvalues are frozen at the saddle point at high temperature. However the singlet partition function has the free energy converging to (m − 1)N2 ln(1 − q) in the large m limit while the free energy of harmonic oscillators is mN2 ln(1− q). Changing the integration variables from group elements to eigenvalues explains how to derive such discrepancy of the free energy between imposing and relaxing the single constraint. Gauge theory cannot simply relax the singlet constraint at high temperature without adding a normalization factor.

On the calculation of correlation functions in nuclear matter

A method for the quantification of synchrony and oscillatory properties of neuronal activity

Relations between quantum integrable models solvable by the quantum inverse scattering method and some aspects of enumerative combinatorics and partition theory are discussed. The main example is the Heisenberg spin chain in the limit cases of zero or infinite anisotropy. Form factors and some thermal correlation functions are calculated, and it is shown that the resulting form factors in a special -parametrization are the generating functions for plane partitions and self-avoiding lattice paths. The asymptotic behaviour of the correlation functions is studied in the case of a large number of sites and a moderately large number of spin excitations. For sufficiently low temperature a relation is established between the correlation functions and the theory of matrix integrals. Bibliography: 125 titles.

The partition function and spin pair correlation functions have been calculated exactly for a classical linear chain model with alternate next-nearest-neighbor (nnn) interactions, in which the interaction energies between pairs of nearest (nn) and next-nearest (nnn) neighbor spins are arbitrary functions of the angles between the relevant spins. Of special interest is the cosine interaction model described by the Hamiltonian ℋ=−ΣNi=1[J1(cosϑ2i−1,2i +cosϑ2i,2i+1)+J2cosϑ2i−1,2i+1]. When the nnn interaction is antiferromagnetic (J2<0) it competes with the nn interaction J1, and there can be disorder point(s) at which nnn correlations change from monotonic to oscillatory. The ground state is ferromagnetic when the interaction ratio r≡J2/‖J1‖ ≳−1/2=rc, but is disordered for more negative values. The disorder point locus has been determined. It terminates at zero temperature at rD=−1/21/2, at which point the ground state energy is a maximum. The result that rD differs from rc is thought to be peculiar to one-dimensional models. Over a limited range of values of r there can be two disorder points. The low temperature asymptotic behavior of the partition and correlation functions is analyzed in detail. Also a novel summation formula for spherical Bessel functions is obtained.

We discuss the stability of semiclassical gravity solutions with respect to small quantum corrections by considering the quantum fluctuations of the metric perturbations around the semiclassical solution. We call the attention to the role played by the symmetrized 2-point quantum correlation function for the metric perturbations, which can be naturally decomposed into two separate contributions: intrinsic and induced fluctuations. We show that traditional criteria on the stability of semiclassical gravity are incomplete because these criteria based on the linearized semiclassical Einstein equation can only provide information on the expectation value and the intrinsic fluctuations of the metric perturbations. By contrast, the framework of stochastic semiclassical gravity provides a more complete and accurate criterion because it contains information on the induced fluctuations as well. The Einstein–Langevin equation therein contains a stochastic source characterized by the noise kernel (the symmetrized 2-point quantum correlation function of the stress tensor operator) and yields stochastic correlation functions for the metric perturbations which agree, to leading order in the large N limit, with the quantum correlation functions of the theory of gravity interacting with N matter fields. These points are illustrated with the example of Minkowski space-time as a solution to the semiclassical Einstein equation, which is found to be stable under both intrinsic and induced fluctuations.

The Helmholtz free energy, A, and the Gibbs free energy, G, of polyethylene (PE) were determined based on its empirical pressure–volume–temperature–entropy equation of state, P-V-T-S Eos, and the empirical partition function, which we reported previously. The PA-VA-TA-SA Eos for the iso-Helmholtz free energy, iso-A, was determined based on the P-V-T-S Eos where the pressure (PA ), volume (VA ), temperature (TA ), and entropy (SA ) satisfied the condition that A is constant. The PG-VG-TG-SG Eos for the G being constant was also determined based on the P-V-T-S Eos. The three-dimensional (3D) plots, 3D plot, with the three axes, x = TA, y = PA, z = VA expressed by the 3D (TA , PA , VA ) and the 3D (TA, VA , SA ) at constant A in PE were determined, where the iso-A processes were characterized assuming that the volume and the entropy in the iso-A process decreased with increasing pressure and temperature, while in the 3D (TG, PG , VG ) and 3D (TG, VG , SG ) plots at constant G in PE, the volume and the entropy in the iso-G process were increasing with increasing pressure and temperature. The empirical partition function for the Helmholtz free energy was expressed by the summation of the terms of T, T 2, and T 4 , the Boltzmann factors with i = 1, 2, 3, and 6 and and the U(V,T = 0 K) term for the zero Kelvin internal energy where the T 4 term and the Boltzmann factors with i = 6 and 3 were dominant negative factors in the A-T relation at the constant V = 1.04 cm3/g. In case of the Gibbs free energy where the partition function was expressed by the summation of the terms of T, T 2, and T 4 , the Boltzmann factors with i = 1, 2, 3, and 6, the U(V,T = 0 K) term, PV and PV(T = 0) terms, the T2 term and the Boltzmann factors with i = 6 and 3 were dominant negative factors in the G-T relation at 1 atm (1.01 × 10−4 GPa), while at high pressure, 1.0 GPa, the PV and PV(T = 0) terms were dominant positive factors.

Multiscaling is shown to be a consequence of multifractality when a lower cutoff \ensuremath{\varepsilon} is introduced in calculations of correlation functions. After a suitable rescaling, the correlation function data for different values of \ensuremath{\varepsilon} seem to fall onto a single curve. In the multiscaling regime, however, we show that there is not a unique functional form at varying \ensuremath{\varepsilon}, but a spread very close to a single curve. For each \ensuremath{\varepsilon}, this curve can be computed analytically in terms of the f(\ensuremath{\alpha}) spectrum which characterizes the multifractal. Part of this spectrum can thus be obtained by computing only one moment of the weights at \ensuremath{\varepsilon}\ensuremath{\ne}0.

A theory of magnetic nuclear relaxation, providing a calculation of the correlation functions of complex motion of methyl groups is presented. The complex motion consists of jumps over the barrier (classical motion) and jumps through the barrier (tunneling). The Schrödinger equation has been applied in the calculation of the rate constant of tunneling jumps through the barrier. The equations for the spectral densities (ω), where ω = (ωI ± ωT), and ω = (mωI), where m = 1 or 2, ωI is the resonance angular frequency, and ωT is the angular frequency of the tunnel splitting, are derived. These spectral densities concern the motion of spin pairs inside methyl groups ("intra") and outside methyl groups ("inter"). The calculated spectral densities are applied to analyze the temperature dependencies of the spin−lattice relaxation rate in solids containing methyl groups. A wide regime of temperatures from 0 K up to the melting point is considered. The temperature at which the tunneling process ceases is discussed. The theory proposed explains the different temperature dependencies of T1 for CH3COOK obtained in the experiments caused by small amounts of CH3COOH or water impurities. The theoretical equations derived in this paper are compared to those known in the literature.

The Schwinger-Keldysh diagram technique is usually involved in the calculation of real-time in-in correlation functions. In the case of a thermal state, one can analytically continue imaginary-time Matsubara correlation functions to real times. Nevertheless, not all real-time correlation functions usually can be obtained by such procedure. Moreover, numerical analytic continuation is an ill-posed problem. Thus, even in the case of a thermal state one may need for the Schwinger-Keldysh formalism. If the potential of a system admits degenerate minima, instantonic effects enter the game, so one should also integrate over the instantonic moduli space, including the one, corresponding to the imaginary time translational invariance. However, the Schwinger-Keldysh closed time contour explicitly breaks such invariance. We argue, that this invariance must be recovered, and show, how it can be done. After that, we construct an extension of the Schwinger-Keldysh diagram technique to instantonic systems and demonstrate it on the example of the first few-point correlation functions.