arXiv subjects

Tony C. Scott

Publications and source records attributed to Tony C. Scott.

5 recordsLinked to original sources

arXiv2025

Probability Bracket Notation for Probability Modeling

Following the Dirac Notation in Quantum Mechanics (QM), we propose the Bracket Notation (PBN) by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc. Using the PBN, many formulae, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions, which are easy to expand by inserting a proper P-identity. The time evolution of homogeneous Markov processes can also be formatted in such a way. Our system P-kets are identified with probability vectors, and our system P-bra is comparable to the Doi state function or the Peliti standard bra. In the Heisenberg picture of the PBN, a random variable becomes a stochastic process, and the Chapman-Kolmogorov equations are obtained by inserting a time-dependent P-identity. Also, some QM expressions in the Dirac notation are naturally transformed into probability expressions in PBN by a special Wick rotation. Potential applications show the usefulness of the PBN beyond the constrained domain and range of Hermitian operators on Hilbert Spaces in QM all the way to IT.

math.PR↗

arXiv2020

Resolving the puzzle of sound propagation in liquid helium at low temperatures

Experimental data suggests that, at temperatures below 1 K, the pressure in liquid helium has a cubic dependence on density. Thus the speed of sound scales as a cubic root of pressure. Near a critical pressure point, this speed approaches zero whereby the critical pressure is negative, thus indicating a cavitation instability regime. We demonstrate that to explain this dependence, one has to view liquid helium as a mixture of three quantum Bose liquids: dilute (Gross-Pitaevskii-type) Bose-Einstein condensate, Ginzburg-Sobyanin-type fluid, and logarithmic superfluid. Therefore, the dynamics of such a mixture is described by a quantum wave equation, which contains not only the polynomial (Gross-Pitaevskii and Ginzburg-Sobyanin) nonlinearities with respect to a condensate wavefunction, but also a non-polynomial logarithmic nonlinearity. We derive an equation of state and speed of sound in our model, and show their agreement with experiment.

cond-mat.quant-gas↗

arXiv2015

Trimer superfluid and supersolid on two-dimensional optical lattices

By the photoassociation method, the trimer superfluid phase disappears in the one dimensional state-dependent optical lattice if the ratio of the three-body interaction $W$ to the trimer tunneling $J$is kept at $W/J=12$ [Phys Rev A. {\bf 90}, 033622(2014)]. To search for a trimer superfluid and trimer supersolid, we load the cold atom into two-dimensional lattices, whose coordinate number $z$ and kinetic energy $-zJ$ are respectively larger and lower than those of a one dimensional lattice. Herein, we study the Bose-Hubbard model, which has an additional trimer tunneling term, a three-body interaction and a next-nearest repulsion. The on-site trimer and trimer superfluid exist if the on-site two-body repulsion and three-body repulsion are smaller than some thresholds. With atom-tunneling terms, the phase transitions from trimer superfluid phase to both atom superfluid and atom supersolid phases are first order. With $W/J=12$, in a one dimensional lattice, the trimer superfluid phase does not exist at all. In contrast, the trimer superfluid phase, exists in the lower density regions $0 \textless ρ\textless2$ on the square lattice if $J$ is not very large. The trimer superfluid phase emerges in a wider range $0 \textless ρ\textless3$ in the triangular lattice, or in the cubic lattice ($z=6$). When the three-body interaction is turned on, a trimer supersolid phase emerges due to the classical degeneracy between the quasi trimer solid and the trimer solid being broken by the quantum fluctuation. The phase transitions from the trimer supersolid phase to quasi trimer solid are first order and the phase transition from the trimer supersolid phase to trimer solid is continuous. Our results, obtained by mean-field and quantum Monte Carlo methods, will be helpful in realizing the trimer superfluid and supersolid by cold atom experiments.

cond-mat.quant-gas↗

arXiv2006

General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function

Herein, we present a canonical form for a natural and necessary generalization of the Lambert W function, natural in that it requires minimal mathematical definitions for this generalization, and necessary in that it provides a means of expressing solutions to a number of physical problems of fundamental nature. In particular, this generalization expresses the exact solutions for general-relativistic self-gravitating 2-body and 3-body systems in one spatial and one time dimension. It also expresses the solution to a previously unknown mathematical link between the lineal gravity problem and the Schroedinger equation.

math-ph↗

arXiv2006

New Approach for the Electronic Energies of the Hydrogen Molecular Ion

Herein, we present analytical solutions for the electronic energy eigenvalues of the hydrogen molecular ion H2+, namely the one-electron two-fixed-center problem. These are given for the homonuclear case for the countable infinity of discrete states when the magnetic quantum number m is zero. In this case, these solutions are the roots of a set of two coupled three-term recurrence relations. The eigensolutions are obtained from an application of EXPERIMENTAL MATHEMATICS using Computer Algebra as its principal tool and are vindicated by numerical and algebraic demonstrations. Finally, the mathematical nature of the eigenenergies is identified. The eigenenergies are related to a generalization of the Lambert W function.

physics.chem-ph↗