Classical Equilibrium Thermostatistics, "Sancta sanctorum of Statistical Mechanics", From Nuclei to Stars

Equilibrium statistics of Hamiltonian systems is correctly described by the microcanonical ensemble. Classically this is the manifold of all points in the N-body phase space with the given total energy. Due to Boltzmann-Planck's principle, e^S=tr(\delta(E-H)), its geometrical size is related to the entropy S(E,N,V,...). This definition does not invoke any information theory, no thermodynamic limit, no extensivity, and no homogeneity assumption. Therefore, it describes the equilibrium statistics of extensive as well of non-extensive systems. Due to this fact it is the fundamental definition of any classical equilibrium statistics. It addresses nuclei and astrophysical objects as well. S(E,N,V,...) is multiply differentiable everywhere, even at phase-transitions. All kind of phase transitions can be distinguished harply and uniquely for even small systems. What is even more important, in contrast to the canonical theory, also the region of phase-space which corresponds to phase-separation is accessible, where the most interesting phenomena occur. No deformed q-entropy is needed for equilibrium. Boltzmann-Planck is the only appropriate statistics independent of whether the system is small or large, whether the system is ruled by short or long range forces.Comment: Invited paper for NEXT2003, 10pages, 6 figures Reference 1 correcte

Two-dimensional dynamics of QCD_3

Exact loop-variables formulation of pure gauge lattice QCD_3 is derived from the Wilson version of the model. The observation is made that the resulting model is two-dimensional. This significant feature is shown to be a unique property of the gauge field. The model is defined on the infinite genus surface which covers regularly the original three-dimensional lattice. Similar transformation applied to the principal chiral field model in two and three dimensions for comparison with QCD.Comment: 6 pages, LaTeX (revision: references added

Negative heat-capacity at phase-separations in microcanonical thermostatistics of macroscopic systems with either short or long-range interactions

Conventional thermo-statistics address infinite homogeneous systems within the canonical ensemble. However, some 170 years ago the original motivation of thermodynamics was the description of steam engines, i.e. boiling water. Its essential physics is the separation of the gas phase from the liquid. Of course, boiling water is inhomogeneous and as such cannot be treated by conventional thermo-statistics. Then it is not astonishing, that a phase transition of first order is signaled canonically by a Yang-Lee singularity. Thus it is only treated correctly by microcanonical Boltzmann-Planck statistics. This was elaborated in the talk presented at this conference. It turns out that the Boltzmann-Planck statistics is much richer and gives fundamental insight into statistical mechanics and especially into entropy. This can be done to a far extend rigorously and analytically. The deep and essential difference between ``extensive'' and ``intensive'' control parameters, i.e. microcanonical and canonical statistics, was exemplified by rotating, self-gravitating systems. In the present paper the necessary appearance of a convex entropy $S(E)$ and the negative heat capacity at phase separation in small as well macroscopic systems independently of the range of the force is pointed out.Comment: 6 pages, 1 figure, 1 table; contribution to the international conference "Next Sigma Phi" on news, expectations, and trends in statistical physics, Crete 200

Mastering the Master Field

The basic concepts of non-commutative probability theory are reviewed and applied to the large $N$ limit of matrix models. We argue that this is the appropriate framework for constructing the master field in terms of which large $N$ theories can be written. We explicitly construct the master field in a number of cases including QCD$_2$. There we both give an explicit construction of the master gauge field and construct master loop operators as well. Most important we extend these techniques to deal with the general matrix model, in which the matrices do not have independent distributions and are coupled. We can thus construct the master field for any matrix model, in a well defined Hilbert space, generated by a collection of creation and annihilation operators---one for each matrix variable---satisfying the Cuntz algebra. We also discuss the equations of motion obeyed by the master field.Comment: 46 pages plus 11 uuencoded eps figure

Zero-norm states and High-energy Symmetries of String Theory

We derive stringy Ward identities from the decoupling of two types of zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string. These Ward identities are valid to all energy and all loop orders in string perturbation theory. The high-energy limit of these stringy Ward identities can then be used to fix the proportionality constants between scattering amplitudes of different string states algebraically without referring to Gross and Mende's saddle point calculation of high-energy string-loop amplitudes. As examples, all Ward identities for the mass level 4 and 6 are derived, their high-energy limits are calculated and the proportionality constants between scattering amplitudes of different string states are determined. In addition to those identified before, we discover some new nonzero components of high-energy amplitudes not found previously by Gross and Manes. These components are essential to preserve massive gauge invariances or decouple massive zero-norm states of string theory. A set of massive scattering amplitudes and their high energy limits are calculated explicitly for each mass level to justify our results

Stringy Symmetries and Their High-energy Limits

We derive stringy symmetries with conserved charges of arbitrarily high spins from the decoupling of two types of zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string. These symmetries are valid to all energy and all loop orders in string perturbation theory. The high-energy limit of these stringy symmetries can then be used to fix the proportionality constants between scattering amplitudes of different string states algebraically without referring to Gross and Mende's saddle point calculation of high-energy string-loop amplitudes. These proportionality constants are, as conjectured by Gross, independent of the scattering angle and the order of string perturbation theory. However, we also discover some new nonzero components of high-energy amplitudes not found previously by Gross and Manes. These components are essential to preserve massive gauge invariances or decouple massive zero-norm states of string theory. A set of massive scattering amplitudes and their high energy limit are calculated explicitly to justify our results.Comment: 10 pages. A corrected version of hep-th/0303012. Final version to appear in Phys. Lett.

Folds in 2D String Theories

We study maps from a 2D world-sheet to a 2D target space which include folds. The geometry of folds is discussed and a metric on the space of folded maps is written down. We show that the latter is not invariant under area preserving diffeomorphisms of the target space. The contribution to the partition function of maps associated with a given fold configuration is computed. We derive a description of folds in terms of Feynman diagrams. A scheme to sum up the contributions of folds to the partition function in a special case is suggested and is shown to be related to the Baxter-Wu lattice model. An interpretation of folds as trajectories of particles in the adjoint representation of $SU(N)$ gauge group in the large $N$ limit which interact in an unusual way with the gauge fields is discussed.Comment: 56 pages, latex, followed by epsf, 13 uuencoded epsf figure

Freeze-out Configuration in Multifragmentation

The excitation energy and the nuclear density at the time of breakup are extracted for the $\alpha + ^{197}Au$ reaction at beam energies of 1 and 3.6 GeV/nucleon. These quantities are calculated from the average relative velocity of intermediate mass fragments (IMF) at large correlation angles as a function of the multiplicity of IMFs using a statistical model coupled with many-body Coulomb trajectory calculations. The Coulomb component $\vec{v}_{c}$ and thermal component $\vec{v}_{0}$ are found to depend oppositely on the excitation energy, IMFs multiplicity, and freeze-out density. These dependencies allow the determination of both the volume and the mean excitation energy at the time of breakup. It is found that the volume remained constant as the beam energy was increased, with a breakup density of about $\rho_{0}/7$, but that the excitation energy increased $25\%$ to about 5.5 MeV/nucleon.Comment: 12 pages, 2 figures available upon resues

Electromagnetic interactions for the two-body spectator equations

This paper presents a new non-associative algebra which is used to (i) show how the spectator (or Gross) two-body equations and electromagnetic currents can be formally derived from the Bethe-Salpeter equation and currents if both are treated to all orders, (ii) obtain explicit expressions for the Gross two-body electromagnetic currents valid to any order, and (iii) prove that the currents so derived are exactly gauge invariant when truncated consistently to any finite order. In addition to presenting these new results, this work complements and extends previous treatments based largely on the analysis of sums of Feynman diagrams.Comment: 44 pages, 14 figure

Is there an Ay problem in low-energy neutron-proton scattering?

We calculate Ay in neutron-proton scattering for the interactions models WJC-1 and WJC-2 in the Covariant Spectator Theory. We find that the recent 12 MeV measurements performed at TUNL are in better agreement with our results than with the Nijmegen Phase Shift Analysis of 1993, and after reviewing the low-energy data, conclude that there is no Ay problem in low-energy np scattering.Comment: 5 pages, 2 figures, accepted by PL