78,980 research outputs found
Hamilton-Jacobi Approach for Power-Law Potentials
The classical and relativistic Hamilton-Jacobi approach is applied to the
one-dimensional homogeneous potential, , where and
are continuously varying parameters. In the non-relativistic case, the
exact analytical solution is determined in terms of , and the total
energy . It is also shown that the non-linear equation of motion can be
linearized by constructing a hypergeometric differential equation for the
inverse problem . A variable transformation reducing the general problem
to that one of a particle subjected to a linear force is also established. For
any value of , it leads to a simple harmonic oscillator if , an
"anti-oscillator" if , or a free particle if E=0. However, such a
reduction is not possible in the relativistic case. For a bounded relativistic
motion, the first order correction to the period is determined for any value of
. For , it is found that the correction is just twice that one
deduced for the simple harmonic oscillator (), and does not depend on the
specific value of .Comment: 12 pages, Late
Reflection matrices for the vertex model
The graded reflection equation is investigated for the
vertex model. We have found four classes of diagonal
solutions and twelve classes of non-diagonal ones. The number of free
parameters for some solutions depends on the number of bosonic and fermionic
degrees of freedom considered.Comment: 30 page
A Numerical Test of a High-Penetrability Approximation for the One-Dimensional Penetrable-Square-Well Model
The one-dimensional penetrable-square-well fluid is studied using both
analytical tools and specialized Monte Carlo simulations. The model consists of
a penetrable core characterized by a finite repulsive energy combined with a
short-range attractive well. This is a many-body one-dimensional problem,
lacking an exact analytical solution, for which the usual van Hove theorem on
the absence of phase transition does not apply. We determine a
high-penetrability approximation complementing a similar low-penetrability
approximation presented in previous work. This is shown to be equivalent to the
usual Debye-H\"{u}ckel theory for simple charged fluids for which the virial
and energy routes are identical. The internal thermodynamic consistency with
the compressibility route and the validity of the approximation in describing
the radial distribution function is assessed by a comparison against numerical
simulations. The Fisher-Widom line separating the oscillatory and monotonic
large-distance behavior of the radial distribution function is computed within
the high-penetrability approximation and compared with the opposite regime,
thus providing a strong indication of the location of the line in all possible
regimes. The high-penetrability approximation predicts the existence of a
critical point and a spinodal line, but this occurs outside the applicability
domain of the theory. We investigate the possibility of a fluid-fluid
transition by Gibbs ensemble Monte Carlo techniques, not finding any evidence
of such a transition. Additional analytical arguments are given to support this
claim. Finally, we find a clustering transition when Ruelle's stability
criterion is not fulfilled. The consequences of these findings on the
three-dimensional phase diagrams are also discussed.Comment: 17 pages, 12 figures; to be published in JC
- …