Precision of approximate kramers formulas for the fission rate: Canonical and microcanonical ensembles

The precision of the Kramers formulas, which are frequently used to calculate the rate of fission of excited nuclei, is studied quantitatively bymeans of a numerical simulation. The deformation dependence of the level-density parameter is taken here into account in contrast to what was done in previous studies on this subject. It turns out that, as the excitation energy becomes lower, the error in the integral Kramers formula decreases monotonically, reaching 2% (precision of dynamical simulation) for a potential-barrier height three to four times greater than the temperature. The approximate Kramers formula obtained from the aforementioned integral formula is in substantially poorer agreement with the results of the dynamical simulation. In the same region of the controlling parameter, its typical error may change sign and reaches 5 to 15%.     

Optimized Monte Carlo analysis for generalized ensembles

We present a new, maximum-likelihood based method to combine data from a multiple number of Monte Carlo simulations performed within any type of ensemble. The method offers an efficient iterative scheme to obtain the density of states of a wide range of energies as well as of other macroscopic variables. It should in particular be useful for the study of systems with a rough energy landscape. Received 4 June 2002 Published online 14 October 2002 RID="a" ID="a"e-mail: borg@alf.nbi.dk     

Conditional equilibrium and the equivalence of microcanonical and grandcanonical ensembles in the thermodynamic limit

Equivalence (allowing for convex combinations) of microcanonical, canonical and grandcanonical ensembles for states of classical systems is established under very mild assumptions on the limiting state. We introduce the notion of conditional equilibrium (C.E.), a property of states of infinite systems which characterizes convex combinations of limits of microcanonical ensembles. It is shown that C.E. states are, under quite general conditions, mixtures of Gibbs states.Supported in part by NSF Grant No. MCS 75-21684 A02Supported in part by NSF Grant No. MPS 72-04534Supported in part by NSF Grant No. Phy 77-22302     

Quantum corrected cell model for an anharmonic generalized Lennard-Jones solid

A simple method is proposed to dispose the quantum effect and anharmonic effect at the same time. Considering the quantum effect is remarkable only at low temperature, and tends to zero at high temperature, the potential energy of an atom is expanded harmonically to consider the quantum effect of solids within the harmonic oscillator framework. The anharmonic effect is remarkable only at high temperature, and tends to zero at low temperature, it was disposed by using a classical approximation. The universal formalism is applied to the generalized Lennard-Jones solid. The comparison shows that the results with and without anharmonic effect are in agreement with each other at some low temperature, to which the Einstein model is applicable. The results without anharmonic effect become divergent at slightly higher temperatures; however, the results including anharmonic effect are in good agreement with the experimental data of solid xenon. The method proposed in this paper can be extended to other potentials to develop practical molecular thermodynamic equations of state for solids.     

A microcanonical model for interface formation

We describe a new cellular automaton model which allows us to simulate separation of phases. The model is an extension of existing cellular automata for the Ising model, such as Q2R. It conserves particle number and presents the qualitative features of spinodal decomposition. The dynamics is deterministic and does not require random number generators. The spins exchange energy with small local reservoirs or demons. The rate of relaxation to equilibrium is investigated, and the results are compared to the Lifshitz-Slyozov theory.     

Thermodynamics with generalized ensembles: The class of dual orthodes

We address the problem of the foundation of generalized ensembles in statistical physics. The approach is based on Boltzmann's concept of orthodes. These are the statistical ensembles that satisfy the heat theorem, according to which the heat exchanged divided by the temperature is an exact differential. This approach can be seen as a mechanical approach alternative to the well established information-theoretic one based on the maximization of generalized information entropy. Our starting point are the Tsallis escort ensembles which have been previously proved to be orthodes, and have been proved to interpolate between canonical and microcanonical ensembles. Here we shall see that the Tsallis escort ensembles belong to a wider class of orthodes that include the most diverse types of ensembles. All such ensembles admit both a microcanonical-like parametrization (via the energy), and a canonical-like one (via the parameter β). For this reason we name them “dual”. One central result used to build the theory is a generalized equipartition theorem. The theory is illustrated with a few examples and the equivalence of all the dual orthodes is discussed.     

Path integrals for the quantum microcanonical ensemble

Path integral representations for the quantum microcanonical ensemble are presented. In the quantum microcanonical ensemble, two operators are of primary interest. First, rhoinsertion mark=delta(E-Hinsertion mark) corresponds to the microcanonical density matrix and can be used to calculate expectation values. Second, Ninsertion mark=Theta(E-Hinsertion mark) can give the number of states with energy E(n) and Theta(x,x('),E)=. A path integral formalism leads to exact integral representations for Omega(x,x('),E) and Theta(x,x('),E). We present both phase space and configuration space forms. For simple systems, such as the free particle and harmonic oscillator, exact solutions are possible. For more complicated systems, expansion schemes or numerical evaluations are required. A perturbative calculation and numerical integration results are presented for the quantum anharmonic oscillator.     

Inequivalence of dynamical ensembles in a generalized driven diffusive lattice gas

We generalize the driven diffusive lattice gas model by using a combination of Kawasaki and Glauber dynamics. We find via Monte Carlo simulations and perturbation studies that the simplest possible generalization of the equivalence of the canonical and grand-canonical ensembles, which holds in equilibrium, does not apply for this class of nonequilibrium systems.     

Canonical and microcanonical calculations for fermi systems

Recursion relations are presented that allow exact calculation of canonical and microcanonical partition functions of degenerate Fermi systems, assuming no explicit two-body interactions. Calculations of the level density, sorted by angular momentum, are presented for 56Ni. The issue of treating unbound states is also addressed.     

Gauge-invariant lattice gas for the microcanonical Ising model

We introduce a lattice gas for particles with discrete momenta (1, 0, –1) and local deterministic microdynamics, which exactly reproduces Creutz's microcanonical algorithm for the ferromagnetic Ising model. However, because of the manifest gauge invariance of our variables, both the Ising ferromagnetic and spin-glass systems share precisely the same dynamics with different initial conditions. Additional conservation laws in the 1D Ising case result in a completely integrable system in the limit of zero or unbounded demon energy cutoff. Numerical investigations of ergodicity are presented for the pure Ising lattice gas in one and two dimensions.     

The generalized second law in universes with quantum corrected entropy relations

We apply the generalized second law of thermodynamics to discriminate among quantum corrections (whether logarithmic or power-law) to the entropy of the apparent horizon in spatially Friedmann–Robertson–Walker universes. We use the corresponding modified Friedmann equations along with either Clausius relation or the principle of equipartition of the energy to set limits on the value of a characteristic parameter entering the said corrections.     

Conditional expectations,conditional distributions,anda posteriori ensembles in generalized probability theory

A general probabilistic framework containing the essential mathematical structure of any statistical physical theory is reviewed and enlarged to enable the generalization of some concepts of classical probability theory. In particular, generalized conditional probabilities of effects and conditional distributions of observables are introduced and their interpretation is discussed in terms of successive measurements. The existence of generalized conditional distributions is proved, and the relation to M. Ozawa'sa posteriori states is investigated. Examples concerning classical as well as quantum probability are discussed.     

Subspace ensembles for classification

Ensemble learning constitutes one of the principal current directions in machine learning and data mining. In this paper, we explore subspace ensembles for classification by manipulating different feature subspaces. Commencing with the nature of ensemble efficacy, we probe into the microcosmic meaning of ensemble diversity, and propose to use region partitioning and region weighting to implement effective subspace ensembles. Individual classifiers possessing eminent performance on a partitioned region reflected by high neighborhood accuracies are deemed to contribute largely to this region, and are assigned large weights in determining the labels of instances in this area. A robust algorithm “Sena” that incarnates the mechanism is presented, which is insensitive to the number of nearest neighbors chosen to calculate neighborhood accuracies. The algorithm exhibits improved performance over the well-known ensembles of bagging, AdaBoost and random subspace. The difference of its effectivity with varying base classifiers is also investigated.     

Steepest descent path for the microcanonical ensemble-resolution of an ambiguity

The microcanonical entropy plays an essential role in the equilibrium statistical mechanics of gravitating systems. A peculiar feature of many of these systems is the existence of stable thermodynamic equilibrium configurations with negative heat capacities. Different methods have been developed for calculating the microcanonical entropy involving multivariate integrals of constraints and functional integrations. An apparent ambiguity between an approach due to Hawking and Gibbons, based on an entropy definition involving an inverse Laplace transform of the partition function, which they developed to treat quantum systems with gravity, and a different approach developed by Horwitz and Katz defining the entropy as an equal weight sum over a constant energy surface developed originally to treat Newtonian and classical GR systems is shown here to be spurious, at least at the level of quadratic fluctuations of all variables about the extremal solutions. The two approaches involve distinct contours for different orders of integration, each of which is shown to be the appropriate steepest descent path corresponding to the given order of investigation. Up to quadratic fluctuations both methods yield identical results. However, they represent different perturbation expansions for the gravitational modes of freedom with different radii of convergence. The discussion is made in terms of a particular convenient model, a system of point particles interacting via Newtonian forces, confined to a sphere, but results are quite general.     

Kohn-Sham theory for ground-state ensembles

An electron density distribution n(r) which can be represented by that of a single-determinant ground state of noninteracting electrons in an external potential v(r) is called pure-state v-representable (P-VR). Most physical electronic systems are P-VR. Systems which require a weighted sum of several such determinants to represent their density are called ensemble v-representable (E-VR). This paper develops formal Kohn-Sham equations for E-VR physical systems, using the appropriate coupling constant integration. It also derives local density- and generalized gradient approximations, and conditions and corrections specific to ensembles.     

Robust quantum enhanced phase estimation in a multimode interferometer [corrected

By exploiting the correlation properties of ultracold atoms in a multimode interferometer, we show how quantum enhanced measurement precision can be achieved with strong robustness to particle loss. While the potential for enhanced measurement precision is limited for even moderate loss in two-mode schemes, multimode schemes can be more robust. A ring interferometer for sensing rotational motion with noninteracting fermionic atoms can realize an uncertainty scaling of 1/(N√η) for N particles with a fraction η remaining after loss, which undercuts the shot-noise limit of two-mode interferometers. A second scheme with strongly interacting bosons achieves a comparable measurement precision and improved readout.