From fractional exclusion statistics back to Bose and Fermi distributions

Fractional exclusion statistics (FES) is a generalization of the Bose and Fermi statistics. Typically, systems of interacting particles are described as ideal FES systems and the properties of the FES systems are calculated from the properties of the interacting systems. In this Letter I reverse the process and I show that a FES system may be described in general as a gas of quasiparticles which obey Bose or Fermi distributions; the energies of the newly defined quasiparticles are calculated starting from the FES equations for the equilibrium particle distribution. In the end I use a system in the effective mass approximation as an example to show how the procedure works.     

A path integral method for data assimilation

Described here is a path integral, sampling-based approach for data assimilation, of sequential data and evolutionary models. Since it makes no assumptions on linearity in the dynamics, or on Gaussianity in the statistics, it permits consideration of very general estimation problems. The method can be used for such tasks as computing a smoother solution, parameter estimation, and data/model initialization.Speedup in the Monte Carlo sampling process is essential if the path integral method has any chance of being a viable estimator on moderately large problems. Here a variety of strategies are proposed and compared for their relative ability to improve the sampling efficiency of the resulting estimator. Provided as well are details useful for its implementation and testing.The method is applied to a problem in which standard methods are known to fail, an idealized flow/drifter problem, which has been used as a testbed for assimilation strategies involving Lagrangian data. It is in this kind of context that the method may prove to be a useful assimilation tool in oceanic studies.     

I-theorem for Fermi and Bose gases

The work is concerned with the change of discrimination information during self-organization of Fermi and Bose gases. A new measure of the discrimination information which determines the degree of order of states in the open systems is presented. The I-theorem for a system with known effective Hamiltonian and a system whose Hamiltonian form is not defined is proved.     

Exact Maxwell–Boltzmann,Bose–Einstein and Fermi–Dirac statistics

The exact Maxwell–Boltzmann (MB), Bose–Einstein (BE) and Fermi–Dirac (FD) entropies and probabilistic distributions are derived by the combinatorial method of Boltzmann, without Stirling's approximation. The new entropy measures are explicit functions of the probability and degeneracy of each state, and the total number of entities, N. By analysis of the cost of a “binary decision”, exact BE and FD statistics are shown to have profound consequences for the behaviour of quantum mechanical systems.     

Stochastic path integral formulation of full counting statistics

We derive a stochastic path integral representation of counting statistics in semiclassical systems. The formalism is introduced on the simple case of a single chaotic cavity with two quantum point contacts, and then further generalized to find the propagator for charge distributions with an arbitrary number of counting fields and generalized charges. The counting statistics is given by the saddle-point approximation to the path integral, and fluctuations around the saddle point are suppressed in the semiclassical approximation. We use this approach to derive the current cumulants of a chaotic cavity in the hot-electron regime.     

Isotope effects in water as investigated by neutron diffraction and path integral molecular dynamics

The structures of heavy and light water at 300?K were investigated by using a joint approach in which the method of neutron diffraction with oxygen isotope substitution was complemented by path integral molecular dynamics simulations. The diffraction results, which give intra-molecular O-D and O-H bond distances of 0.985(5) and 0.990(5)??, were found to be in best agreement with those obtained by using the flexible anharmonic TTM3-F water model. Both techniques show a difference of ??0.5% between the O-D and O-H intra-molecular bond lengths, and the results support a competing quantum effects model for water in which its structural and dynamical properties are governed by an offset between intra-molecular and inter-molecular quantum contributions. Further consideration of the O-O correlations is needed in order to improve agreement with experiment.     

Conversion of an atomic Fermi gas to a long-lived molecular Bose gas

We have converted an ultracold Fermi gas of 6Li atoms into an ultracold gas of 6Li2 molecules by adiabatic passage through a Feshbach resonance. Approximately 1.5 x 10(5) molecules in the least-bound, v=38, vibrational level of the X1Sigma(+)(g) singlet state are produced with an efficiency of 50%. The molecules remain confined in an optical trap for times of up to 1 s before we dissociate them by a reverse adiabatic sweep.     

Electron wave packets: Quantum statistics in path integral representation

Fundamental relations between ensemble-averaged kinetic energy, pressure, and virial are obtained for quantum many-particle systems at finite temperatures. A path-integral method is developed for calculating kinetic energy for mixed (quantum-classical) systems. The method eliminates the problem of divergent variance. Path-integral Monte Carlo simulations are performed to validate the method as applied to single-electron and electron-pair wave packets, with exact treatment of exchange symmetry and spin states. Quantitative results are compared with corresponding characteristics calculated for point charges.     

Suppression of molecular decay in ultracold gases without Fermi statistics

We study inelastic processes for ultracold three-body systems in which only one interaction is resonant. We show that at ultracold temperatures three-body recombination in such systems leads mainly to the formation of weakly bound molecules. In addition, and perhaps more importantly, we have found that the decay rates for weakly bound molecules due to collisions with other atoms can be suppressed not only without fermionic statistics but also when bosonic statistics applies. These results indicate that recombination in three-component atomic gases can be used as an efficient mechanism for molecular formation, allowing the achievement of high molecular densities.     

Reformulated spin pseudoclassical dynamics and its path integral quantization

The path integral quantization of the pseudoclassical dynamics, the basic dynamical variables of which are three pairs of the canonical real Grassmann variables, is presented. The exact results (propagators, eigenvalues and eigenfunctions) are obtained for the two simplest but physically acceptable Hamiltonians.     

Quantum dynamics and statistics of a Bose condensate generated by an atomic laser

A self-consistent quantum theory is developed for an atomic laser utilizing cooling of atoms in a trap by the method of stimulated evaporation. The model describes the pumping and extraction of the atomic field from a trap upon its interaction with independent atomic reservoirs. The stimulated collisions between atoms in the trap, which produce a Bose condensate in the lower state of the trap, are considered. The interaction of atoms with a phonon field causes spontaneous transitions between the discrete states of the trap. Calculations performed for the three-and four-level models of the trap showed the possibility of generation of a strongly squeezed sub-Poisson Bose condensate.     

Cost of s-fold decisions in exact Maxwell–Boltzmann,Bose–Einstein and Fermi–Dirac statistics

The exact forms of the degenerate Maxwell–Boltzmann (MB), Bose–Einstein (BE) and Fermi–Dirac (FD) entropy functions, derived by Boltzmann's principle without the Stirling approximation [R.K. Niven, Physics Letters A, 342(4) (2005) 286], are further examined. Firstly, an apparent paradox in quantization effects is resolved using the Laplace–Jaynes interpretation of probability. The energy cost of learning that a system, distributed over s equiprobable states, is in one such state (an “s-fold decision”) is then calculated for each statistic. The analysis confirms that the cost depends on one's knowledge of the number of entities N and (for BE and FD statistics) the degeneracy, extending the findings of Niven (2005).     

Neural network learning dynamics in a path integral framework

Using the local-density approximation, calculating the Hellmann-Feynman forces and applying the direct method, the phonon dispersion relations for the rutile-like structure of crystalline SnO₂ have been derived for the first time. The phonon frequencies at the point agree very well with Raman and infrared data and other phenomenological model calculations. The LO/TO splitting is estimated by calculating phonons from an elongated supercell. The computations under pressure reveal a soft mode of B_1g symmetry which leads to a ferroelastic phase transition. The pressure-dependence of the lattice constants and the Grüneisen parameters of the modes are calculated. Received 18 July 1999     

A riemann integral approach to Feynman's path integral

It is a well known result that the Feynman's path integral (FPI) approach to quantum mechanics is equivalent to Schrödinger's equation when we use as integration measure the Wiener-Lebesgue measure. This results in little practical applicability due to the great algebraic complexibity involved, and the fact is that almost all applications of (FPI) practical calculations — are done using a Riemann measure. In this paper we present an expansion to all orders in time of FPI in a quest for a representation of the latter solely in terms of differentiable trajetories and Riemann measure. We show that this expansion agrees with a similar expansion obtained from Schrödinger's equation only up to first order in a Riemann integral context, although by chance both expansions referred to above agree for the free particle and harmonic oscillator cases. Our results permit, from the mathematical point of view, to estimate the many errors done in practical calculations of the FPI appearing in the literature and, from the physical point of view, our results supports the stochastic approach to the problem.     

Two-species mixture of quantum degenerate Bose and Fermi gases

We have produced a macroscopic quantum system in which a 6Li Fermi sea coexists with a large and stable 23Na Bose-Einstein condensate. This was accomplished using interspecies sympathetic cooling of fermionic 6Li in a thermal bath of bosonic 23Na. The system features rapid thermalization and long lifetimes.     

Thermodynamics of two-parameter quantum group Bose and Fermi gases

The high and low temperature thermodynamical properties of the two-parameter deformed quantum group Bose and Fermi gases with SU _p/q (2) symmetry are studied. Starting with a SU _p/q (2)-invariant bosonic as well as fermionic Hamiltonian, several thermodynamical functions of the system such as the average number of particles, internal energy and equation of state are derived. The effects of two real independent deformation parameters p and q on the properties of the systems are discussed. Particular emphasis is given to a discussion of the Bose-Einstein condensation phenomenon for the two-parameter deformed quantum group Bose gas. The results are also compared with earlier undeformed and one-parameter deformed versions of Bose and Fermi gas models. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Correlation integral as a tool for distinguishing between dynamics and statistics in time series data

We propose a new test for time series data for proper choice of processing technique: dynamical or statistical. It is based upon the normalized slope of the correlation integral (, m) = m⁻¹d(ln C()) /d ln , where m is the embedding dimension. It is shown that when does not tend to 0 on the resolved range of scales as m grows, then there will be serious limitations for dynamical methods even if the data are dynamical by nature. In the latter case it means that the length of time series does not allow to resolve small scales, and on large scales the delay reconstruction for any m mixes true and false neighbours of points and therefore restricts the application of dynamical techniques, such as estimating Lyapunov exponents or predicting time series.