Algorithms for the computation of solutions of the Ornstein-Zernike equation
Peplow, A. T. LU
; Beardmore, R. E.
and Bresme, F.
(2006)
In Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
74(4).
- Abstract
We introduce a robust and efficient methodology to solve the Ornstein-Zernike integral equation using the pseudoarc length (PAL) continuation method that reformulates the integral equation in an equivalent but nonstandard form. This enables the computation of solutions in regions where the compressibility experiences large changes or where the existence of multiple solutions and so-called branch points prevents Newton’s method from converging. We illustrate the use of the algorithm with a difficult problem that arises in the numerical solution of integral equations, namely the evaluation of the so-called no-solution line of the Ornstein-Zernike hypernetted chain (HNC) integral equation for the Lennard-Jones potential. We are able to... (More)
We introduce a robust and efficient methodology to solve the Ornstein-Zernike integral equation using the pseudoarc length (PAL) continuation method that reformulates the integral equation in an equivalent but nonstandard form. This enables the computation of solutions in regions where the compressibility experiences large changes or where the existence of multiple solutions and so-called branch points prevents Newton’s method from converging. We illustrate the use of the algorithm with a difficult problem that arises in the numerical solution of integral equations, namely the evaluation of the so-called no-solution line of the Ornstein-Zernike hypernetted chain (HNC) integral equation for the Lennard-Jones potential. We are able to use the PAL algorithm to solve the integral equation along this line and to connect physical and nonphysical solution branches (both isotherms and isochores) where appropriate. We also show that PAL continuation can compute solutions within the no-solution region that cannot be computed when Newton and Picard methods are applied directly to the integral equation. While many solutions that we find are new, some correspond to states with negative compressibility and consequently are not physical.
(Less)
- author
- Peplow, A. T.
LU
; Beardmore, R. E.
and Bresme, F.
- publishing date
- 2006
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
- volume
- 74
- issue
- 4
- article number
- 046705
- publisher
- American Physical Society
- external identifiers
-
- scopus:33750188845
- ISSN
- 1539-3755
- DOI
- 10.1103/PhysRevE.74.046705
- language
- English
- LU publication?
- no
- additional info
- Copyright: Copyright 2008 Elsevier B.V., All rights reserved.
- id
- 731a46c2-5c8c-43a5-8bc5-0a057b8f67d5
- date added to LUP
- 2021-03-08 15:17:17
- date last changed
- 2022-02-01 20:36:57
@article{731a46c2-5c8c-43a5-8bc5-0a057b8f67d5,
abstract = {{<p>We introduce a robust and efficient methodology to solve the Ornstein-Zernike integral equation using the pseudoarc length (PAL) continuation method that reformulates the integral equation in an equivalent but nonstandard form. This enables the computation of solutions in regions where the compressibility experiences large changes or where the existence of multiple solutions and so-called branch points prevents Newton’s method from converging. We illustrate the use of the algorithm with a difficult problem that arises in the numerical solution of integral equations, namely the evaluation of the so-called no-solution line of the Ornstein-Zernike hypernetted chain (HNC) integral equation for the Lennard-Jones potential. We are able to use the PAL algorithm to solve the integral equation along this line and to connect physical and nonphysical solution branches (both isotherms and isochores) where appropriate. We also show that PAL continuation can compute solutions within the no-solution region that cannot be computed when Newton and Picard methods are applied directly to the integral equation. While many solutions that we find are new, some correspond to states with negative compressibility and consequently are not physical.</p>}},
author = {{Peplow, A. T. and Beardmore, R. E. and Bresme, F.}},
issn = {{1539-3755}},
language = {{eng}},
number = {{4}},
publisher = {{American Physical Society}},
series = {{Physical Review E - Statistical, Nonlinear, and Soft Matter Physics}},
title = {{Algorithms for the computation of solutions of the Ornstein-Zernike equation}},
url = {{http://dx.doi.org/10.1103/PhysRevE.74.046705}},
doi = {{10.1103/PhysRevE.74.046705}},
volume = {{74}},
year = {{2006}},
}