FRACTIONAL STATISTICS IN ONE DIMENSION: MODELING BY MEANS OF 1/x2 INTERACTION AND STATISTICAL MECHANICS

1994 ◽  
Vol 09 (15) ◽  
pp. 2563-2582 ◽  
Author(s):  
SERGEI B. ISAKOV
Keyword(s):  
Functional Equation ◽  
Statistical Mechanics ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Statistical Distribution ◽  
Harmonic Potential ◽  
One Dimension ◽  
Fractional Statistics ◽  
First Case ◽  
System Of Particles

The equivalence of a quantum system of particles interacting with a two-body inverse square potential to a system of noninteracting particles obeying 1D fractional statistics (1D anyons), stated by Polychronakos for particles on a line, is studied for the cases where the interacting system is placed (i) into a harmonic potential on a line, and (ii) on a ring, with imposing periodic boundary conditions. In the first case, reducibility of the interacting system to the Calogero system is used to explore the statistical distribution for free 1D anyons. On a ring, the thermodynamic limit is discussed in terms of the thermodynamic (asymptotic) Bethe ansatz. Yang and Yang’s integral equation is treated in this case as describing the statistical mechanics of free 1D anyons. It gives a functional equation for the statistical distribution of 1D anyons consistent with the harmonic potential approach. We show that on a ring of a finite circumference, a system of two free 1D anyons is equivalent to a system of two particles interacting with an inverse sine square potential (the Sutherland system). We also discuss the relation to the statistical mechanics of free anyons in 2+1 dimensions.

References and Remarks on Rotating Fluids

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Initial Data ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Incompressible Limit ◽  
Rotating Fluids ◽  
Inviscid Fluids ◽  
First Case ◽  
The Family ◽  
Whole Space

The problem investigated in this part can be seen as a particular case of the study of the asymptotic behavior (when ε tends to 0) of solutions of systems of the type where Δε is a non-negative operator of order 2 possibly depending on ε, and A is a skew-symmetric operator. This framework contains of course a lot of problems including hyperbolic cases when Δε = 0. Let us notice that, formally, any element of the weak closure of the family (uε)ε>0 belongs to the kernel of A. We can distinguish from the beginning two types of problems depending on the nature of the initial data. The first case, known as the well-prepared case, is the case when the initial data belong to the kernel of A. The second case, known as the ill-prepared case, is the general case. In the well-prepared case, let us mention the pioneer paper by S. Klainerman and A. Majda about the incompressible limit for inviscid fluids. A lot of work has been done in this case. In the more specific case of rotating fluids, let us mention the work by T. Beale and A. Bourgeois and T. Colin and P. Fabrie. In the case of ill-prepared data, the nature of the domain plays a crucial role. The first result in this case was established in 1994 in the pioneering work by S. Schochet for periodic boundary conditions. In the context of general hyperbolic problems, he introduced the key concept of limiting system (see the definition on page 125). In the more specific case of viscous rotating fluids, E. Grenier proved in 1997 in Theorem 6.3, page 125, of this book. At this point, it is impossible not to mention the role of the inspiration played by the papers by J.-L. Joly, G. Métivier and J. Rauch (see for instance and). In spite of the fact that the corresponding theorems have been proved afterwards, the case of the whole space, the purpose of Chapter 5 of this book, appears to be simpler because of the dispersion phenomena.

INVESTIGATION OF THE CONSTRAINTS ON TWISTED AND UNTWISTED YANG–MILLS THEORIES IN A SLAB

1991 ◽  
Vol 06 (31) ◽  
pp. 2893-2900
Author(s):  
A. R. LEVI
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
One Dimension ◽  
Yang Mills ◽  
Quantum Constraints

BRST is used to investigate the consistency of the quantum constraints for Yang–Mills theories based on twisted and untwisted SU (N) in a slab with periodic boundary conditions in one dimension.

The Harmonic Solid

2019 ◽  
pp. 331-349
Author(s):  
Robert H. Swendsen
Keyword(s):  
Boundary Conditions ◽  
Specific Heat ◽  
Periodic Boundary ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Normal Modes ◽  
Periodic Boundary Conditions ◽  
One Dimension ◽  
Debye Approximation

In Chapter 26 we return to calculating the contributions to the specific heat of a crystal from the vibrations of the atoms. The vibrations of a model of a solid, for which the interactions are quadratic in form, is investigated. Calculations are restricted to one dimension for simplicity in the derivations of the Fourier modes and the equations of motion. Both pinned and periodic boundary conditions are discussed. The representation of the Hamiltonian in terms of normal modes and the solution in terms of the equations of motion are derived. The Debye approximation is then introduced for three-dimensional systems.

scholarly journals Uniform Asymptotics of Toeplitz Determinants with Fisher–Hartwig Singularities

2021 ◽  
Vol 383 (2) ◽  
pp. 685-730
Author(s):  
B. Fahs
Keyword(s):  
Asymptotic Formula ◽  
Boundary Conditions ◽  
Characteristic Polynomial ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Fixed Number ◽  
One Dimension ◽  
Uniform Asymptotics ◽  
Toeplitz Determinants ◽  
Unitary Ensemble

AbstractWe obtain an asymptotic formula for $$n\times n$$ n × n Toeplitz determinants as $$n\rightarrow \infty $$ n → ∞ , for non-negative symbols with any fixed number of Fisher–Hartwig singularities, which is uniform with respect to the location of the singularities. As an application, we prove a conjecture by Fyodorov and Keating (Philos Trans R Soc A 372: 20120503, 2014) regarding moments of averages of the characteristic polynomial of the Circular Unitary Ensemble. In addition, we obtain an asymptotic formula regarding the momentum of impenetrable bosons in one dimension with periodic boundary conditions.

GENERALIZATION OF STATISTICS FOR SEVERAL SPECIES OF IDENTICAL PARTICLES

1994 ◽  
Vol 08 (05) ◽  
pp. 319-327 ◽  
Author(s):  
SERGEI B. ISAKOV
Keyword(s):  
Statistical Mechanics ◽  
One Dimension ◽  
Fractional Statistics ◽  
Identical Particles ◽  
Statistical Interaction

We present a framework for treating the statistical mechanics of systems including several species of particles with a statistical interaction between particles of different species. The statistical interaction is assumed to occur only between particles with the same momentum. Generalization of fractional statistics in one dimension to the case of several species of particles is discussed.

Molecular Dynamics Using Non-Variational Polarizable Force Fields: Theory, Periodic Boundary Conditions Implementation and Application to the Bond Capacity Model

2019 ◽  
Author(s):  
Pier Paolo Poier ◽  
Louis Lagardere ◽  
Jean-Philip Piquemal ◽  
Frank Jensen
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Force Fields ◽  
Periodic Boundary Conditions ◽  
Computational Effort ◽  
Polarizable Force Fields ◽  
Energy Models ◽  
Capacity Model ◽  
Energy Gradient ◽  
Polarization Models

<div> <div> <div> <p>We extend the framework for polarizable force fields to include the case where the electrostatic multipoles are not determined by a variational minimization of the electrostatic energy. Such models formally require that the polarization response is calculated for all possible geometrical perturbations in order to obtain the energy gradient required for performing molecular dynamics simulations. </p><div> <div> <div> <p>By making use of a Lagrange formalism, however, this computational demanding task can be re- placed by solving a single equation similar to that for determining the electrostatic variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p><div><div><div> </div> </div> </div> <p> </p><div> <div> <div> <p>variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p> </div> </div> </div> </div> </div> </div> </div> </div> </div>

scholarly journals Global classical solutions to the elastodynamic equations with damping

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mengmeng Liu ◽  
Xueyun Lin
Keyword(s):  
Initial Data ◽  
Periodic Boundary ◽  
Energy Estimate ◽  
Periodic Boundary Conditions ◽  
Deformation Tensor ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Weighted Energy ◽  
Global Classical Solutions ◽  
Weighted Energy Estimate

AbstractIn this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry.

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