The canonical partition function for relativistic hadron gases

Particle production in high-energy collisions is often addressed within the framework of the thermal (statistical) model. We present a method to calculate the canonical partition function for the hadron resonance gas with exact conservation of the baryon number, strangeness, electric charge, charmness and bottomness. We derive an analytical expression for the partition function which is represented as series of Bessel functions. Our results can be used directly to analyze particle production yields in elementary and in heavy ion collisions. We also quantify the importance of quantum statistics in the calculations of the light particle multiplicities in the canonical thermal model of the hadron resonance gas.     

Microcanonical mean-field thermodynamics of self-gravitating and rotating systems

We derive the global phase diagram of a self-gravitating N-body system enclosed in a finite three-dimensional spherical volume V as a function of total energy and angular momentum, employing a microcanonical mean-field approach. At low angular momenta (i.e., for slowly rotating systems) the known collapse from a gas cloud to a single dense cluster is recovered. At high angular momenta, instead, rotational symmetry can be spontaneously broken and rotationally asymmetric structures (double clusters) appear.     

Statistical hadronization and hadronic microcanonical ensemble I

We present a full treatment of the microcanonical ensemble of the ideal hadron-resonance gas starting from a quantum-mechanical formulation which is appropriate for the statistical model of hadronization. By using a suitable transition operator for hadronization we are able to recover the results of the statistical theory, particularly the expressions of the rates of different channels. Explicit formulae are obtained for the phase space volume or density of states of the ideal relativistic gas in quantum statistics as a cluster decomposition, generalizing previous ones in the literature. The problem of the computation of averages in the hadron gas microcanonical ensemble and the comparison with canonical ones will be the main subject of a forthcoming second paper.Received: 8 July 2003, Revised: 17 October 2003, Published online: 5 May 2004     

Statistical properties of chaotic wavefunctions in two and more dimensions

Starting with Berry's hypothesis for fixed energy waves in a classically chaotic system, and casting it in a Green function form, we derive wavefunction correlations and density matrices for few or many particles. Universal features of fixed energy (microcanonical) random wavefunction correlation functions appear which reflect the emergence of the canonical ensemble as N↦∞. This arises through a little known asymptotic limit of Bessel functions. The Berry random wave hypothesis in many dimensions may be viewed as an alternative approach to quantum statistical mechanics, when extended to include constraints and potentials.     

Induced quantum numbers of a magnetic vortex at non-zero temperature

The phenomenon of the finite-temperature induced quantum numbers in fermionic systems with topological defects is analyzed. We consider an ideal gas of two-dimensional relativistic massive electrons in the background of a defect in the form of a pointlike magnetic vortex with arbitrary flux. This system is found to acquire, in addition to fermion number, also orbital angular momentum, spin, and induced magnetic flux, and we determine the functional dependence of the appropriate thermal averages and correlations on the temperature, the vortex flux, and the continuous parameter of the boundary condition at the location of the defect. We find that non-negativeness of thermal quadratic fluctuations imposes a restriction on the admissible range of values of the boundary parameter. The long-standing problem of the adequate definition of total angular momentum for the system considered is resolved.     

Spin dynamics in a one-dimensional ferromagnetic bose gas

We investigate the propagation of spin excitations in a one-dimensional ferromagnetic Bose gas. While the spectrum of longitudinal spin waves in this system is soundlike, the dispersion of transverse spin excitations is quadratic, making a direct application of the Luttinger liquid theory impossible. By using a combination of different analytic methods we derive the large time asymptotic behavior of the spin-spin dynamical correlation function for strong interparticle repulsion. The result has an unusual structure associated with a crossover from the regime of trapped spin wave to an open regime and does not have analogues in known low-energy universality classes of quantum 1D systems.     

Effect of temperature on $$\mathbf{In}_{{\varvec{x}}} \mathbf{Ga}_{1-{{\varvec{x}}}} \mathbf{As}/\mathbf{GaAs}$$ quantum dots

Analytical solution of the Dirac equation for the modified Pöschl–Teller potential and trigonometric Scarf II non-central potential for spin symmetry is studied using asymptotic iteration method. One-dimensional Dirac equation consisting of the radial and angular parts can be obtained by the separation of variables. By using asymptotic iteration method, the relativistic energy equation and orbital quantum number (l) equation can be obtained, where both are interrelated. Relativistic energy equation is calculated numerically by the Matlab software. The increase in the radial quantum number n _r causes a decrease in the energy value, and the wave functions of the radial and the angular parts are expressed in terms of hypergeometric functions. Some thermodynamical properties of the system can be determined by reducing the relativistic energy equation to the non-relativistic energy equation. Thermodynamical properties such as vibrational partition function, vibrational specific heat function and vibrational mean energy function are expressed in terms of error function.     

A spin observable for a Dirac particle

We discuss the form of the spin operator in relativistic quantum mechanics. We derive the form of the spin operator in the case when the states with negative energies are admitted. It appears that for a Dirac particle the spin operator reduces to the so called mean-spin operator introduced by Foldy and Wouthuysen. We show that the spin operator transforms under Lorentz group action according to an operator Wigner rotation, analogously as a Bloch vector describing polarization of a particle in momentum representation.     

Classical and Quantum Theories of Spin

A great effort has been devoted to formulating a classical relativistic theory of spin compatible with quantum relativistic wave equations. The main difficulty in connecting classical and quantum theories rests in finding a parameter that plays the role of proper time at a purely quantum level. We present a partial review of several proposals of classical and quantum spin theories from the pioneering works of Thomas and Frenkel, revisited in the classical BMT work, to the semiclassical model of Barut and Zanghi. We show that the last model can be obtained from a semiclassical limit of the Feynman proper time parametrization of the Dirac equation. At the quantum level, we derive spin precession equations in the Heisenberg picture. Analogies and differences with respect to classical theories are discussed in detail.     

The ideal relativistic spinning gas: Polarization and spectra

We study the physics of the ideal relativistic rotating gas at thermodynamical equilibrium and provide analytical expressions of the momentum spectra and polarization vector for the case of massive particles with spin 1/2 and 1. We show that the finite angular momentum J entails an anisotropy in momentum spectra, with particles emitted orthogonally to J having, on average, a larger momentum than along its direction. Unlike in the non-relativistic case, the proper polarization vector turns out not to be aligned with the total angular momentum with a non-trivial momentum dependence.     

A statistical mechanical problem in Schwarzschild spacetime

We use Fermi coordinates to calculate the canonical partition function for an ideal gas in a circular geodesic orbit in Schwarzschild spacetime. To test the validity of the results we prove theorems for limiting cases. We recover the Newtonian gas law subject only to tidal forces in the Newtonian limit. Additionally we recover the special relativistic gas law as the radius of the orbit increases to infinity. We also discuss how the method can be extended to the non ideal gas case.     

Fluctuation and dissipation of work in a Joule experiment

We elucidate the connection between various fluctuation theorems by a microcanonical version of the Crooks relation. We derive the microscopically exact expression for the work distribution in an idealized Joule experiment, namely, for a convex object moving at constant speed through an ideal gas. Analytic results are compared with molecular dynamics simulations of a hard disk gas.     

Nonlocality in Relativistic Dynamics

Recent experiments have renewed interest in nonlocal interpretations of quantum mechanics. The experimental observation of the violation of Bell's inequalities implies the existence of nonlocality. Bohm expressed the nonlocal connection between quantum particles through the wave function and the quantum potential. This paper shows that a similar connection exists in a relativistic dynamical theory known as parametrized relativistic quantum theory (PRQT). We present an introduction to PRQT, derive the quantum potential for a system of relativistic scalar particles, and discuss alternative interpretations of nonlocality.     

The O(n) Model on the Annulus

We use Coulomb gas methods to derive an explicit form for the scaling limit of the partition function of the critical O(n) model on an annulus, with free boundary conditions, as a function of its modulus. This correctly takes into account the magnetic charge asymmetry and the decoupling of the null states. It agrees with an earlier conjecture based on Bethe ansatz and quantum group symmetry, and with all known results for special values of n. It gives new formulae for percolation (the probability that a cluster connects the two opposite boundaries) and for self-avoiding loops (the partition function for a single loop wrapping non-trivially around the annulus.) The limit n→0 also gives explicit examples of partition functions in logarithmic conformal field theory.     

The relativistic fermi gas in a nonuniform velocity field

We review the relativistic Fermi gas in an inhomogeneous velocity field and calculate components of the current vector and the energy-momentum tensor in a comoving frame. We derive quantum corrections to the classical distribution function in the framework of the grand canonical ensemble.     

Inozemtsev's hyperbolic spin model and its related spin chain

In this paper we study Inozemtsev's su(m) quantum spin model with hyperbolic interactions and the associated spin chain of Haldane–Shastry type introduced by Frahm and Inozemtsev. We compute the spectrum of Inozemtsev's model, and use this result and the freezing trick to derive a simple analytic expression for the partition function of the Frahm–Inozemtsev chain. We show that the energy levels of the latter chain can be written in terms of the usual motifs for the Haldane–Shastry chain, although with a different dispersion relation. The formula for the partition function is used to analyze the behavior of the level density and the distribution of spacings between consecutive unfolded levels. We discuss the relevance of our results in connection with two well-known conjectures in quantum chaos.     

Rescattering of electrons at laser-cluster interactions

The goal of this work is to derive the angular distributions of electrons irradiated at the outer ionization of large atomic clusters from Xe atoms by relativistic laser pulses taking into account rescattering processes. Both the magnetic field of the laser pulse and the Coulomb field of the ionized cluster significantly influence the rescattering of ejected electrons. The multiply inner ionization of atoms occurs at the leading edge of the laser pulse. The atomic ions with charge multiplicities up to Z = 26 are subsequently produced (each atomic ion with the next charge multiplicity appears in 3–5 fs) when the laser intensity increases. The measurements of the angular distributions of electrons allow us to reproduce the imaging dynamics of outer ionization of the cluster at the leading edge of the relativistic femtosecond laser pulse.