Microcanonical Density Of States Research Articles

Exact and approximate methods of calculating the sum of states for noninteracting classical and quantum particles occupying a finite number of modes

We present exact expressions for the sum of states of noninteracting classical and quantum particles occupying a finite number of modes with arbitrary spacings. Exploiting a probabilistic analogy, we derive an analytic fourth-order approximation to the density of states, which captures its variance and kurtosis, and is superior to the previous, commonly used methods for all three particle statistics. Our approach employs a simple exact method of calculating the moments of the microcanonical density of states for quantum particles, which requires less computational effort than the commonly used saddle-point approximation. We test our methods numerically and discuss their applicability to various physical systems.

Microcanonical statistics of black holes and the bootstrap condition

The microcanonical statistics of the Schwarzschild black holes as well as the Reissner-Nordstr$\sf \ddot{o}$m black holes are analyzed. In both cases we set up the inequalities in the microcanonical density of states. These are then used to show that the most probable configuration in the gases of black holes is that one black hole acquires all of the mass and all of the charge at high energy limit. Thus the black holes obey the statistical bootstrap condition and, in contrast to the other investigation, we see that U(1) charge does not break the bootstrap property.

MC-simulation of the 3D,q=3 Potts model

We consider theq=3 Potts model in three dimensions by Monte Carlo simulations. The microcanonical density of states Ω is calculated as a function of the internal energy e of the system. We extrapolate the data for the simulated finite systems to the thermo-dynamic limit and find a discontinuous phase transition. This method is checked in the two-dimensional case, where exact results are known.

First order phase transitions in the canonical and the microcanonical ensemble

In the canonical ensemble any singularity of a thermodynamic function at a temperatureTc is smeared over a temperature range of orderTT/N. Therefore it is rather difficult to distinguish between a discontinuous and a continuous phase transition on the basis of numerical data obtained for finite systems in the canonical ensemble. It is demonstrated for four model systems that this problem cannot be circumvented by considering higher cumulants of the energy distribution or cumulant ratios. On the other hand, the distinction between first and a second order phase transition is rather direct if based on the microcanonical density of states which is readily obtainable in the dynamical ensemble.

Controlling unboundedness in the gravitational path integral.

New results are presented on the Euclidean path-integral formulation for the partition function and density of states pertinent to spherically symmetric black-hole systems in thermodynamic equilibrium. We extend the path-integral construction of Halliwell and Louko which has already been used by one of us (Louko and Whiting), and investigate further a lack of uniqueness in our previous formulation of the microcanonical density of states and in the canonical partition function. In that work, the method chosen for removing the ambiguity resulted in two specific path-integral contours having finite extent. Physically motivated criteria exercised a dominant influence on that choice, as did the need to overcome the unboundedness from below of the gravitational action. The new results presented here satisfy the same physical criteria, but differ in ways which are physically significant. The unboundedness is not now eliminated directly but, for positive temperatures only, it is dealt with by what may be viewed as the introduction of an effective measure, which nevertheless may be of exponential order. Having chosen to investigate alternative contours which, in fact, have infinite extent, we find that imposing the Wheeler-DeWitt equation automatically selects out particular finite end points for the contours, at which the singularity in the action is canceled. A further important outcome of this work is the emergence of a variational principle for the black hole entropy, which has already proved useful at the level of a zero-loop approximation to the coupling of a shell of quantum matter in equilibrium around a Schwarzschild black hole (Horwitz and Whiting). In the course of enquiring into the nature of the variables in which the path integral is constructed and evaluated, we were able to see how to give a unifying description of several previous results in the literature. A concise review of these separate approaches forms an integral part of our new synthesis, relating their various underlying ideas on Hamilton-Jacobi theory and Hamiltonian reduction in the context of path integration. The new insight we gain finally helps motivate the choice of the integration variables, identification of which has played an important role in our whole analysis.

Implications of Bose-Einstein statistics within an uncorrelated-jet model

The framework of an uncorrelated-jet model (UJM) is used to investigate the implications of Bose-Einstein statistics in multiplicity moments and inclusive spectra. An asymptotic expansion for the grand microcanonical density of states is derived. The choice of the transverse-momentum cutoff function near ${p}_{T}=0$ is found to be of utmost importance for obtaining significant Bose-Einstein-type correlations between particles. A strong enhancement of average particle multiplicity and correlation moments with energy is found in some cases, in contrast to the conventional UJM with Boltzmann statistics. Several rapidity distributions showing the violation of Feynman scaling (plateau behavior) are presented.