LennardJones.Rd
\name{LennardJones}
\alias{LennardJones}
\title{The Lennard-Jones Potential}
\description{
Creates the Lennard-Jones pairwise interaction structure
which can then be fitted to point pattern data.
}
\usage{
LennardJones()
}
\value{
An object of class \code{"interact"}
describing the Lennard-Jones interpoint interaction
structure.
}
\details{
In a pairwise interaction point process with the
Lennard-Jones pair potential,
each pair of points in the point pattern,
a distance \eqn{d} apart,
contributes a factor
\deqn{
\exp \left\{
-
\left(
\frac{\sigma}{d}
\right)^{12}
+ \tau
\left(
\frac{\sigma}{d}
\right)^6
\right\}
}{
exp( - (sigma/d)^12 + tau (sigma/d)^6)
}
to the probability density,
where \eqn{\sigma}{sigma} and \eqn{\tau}{tau} are
positive parameters to be estimated.
See \bold{Examples} for a plot of this expression.
This potential causes very strong inhibition between points at short range,
and attraction between points at medium range.
Roughly speaking, \eqn{\sigma}{sigma} controls the scale of both
types of interaction, and \eqn{\tau}{tau} determines the strength of
attraction.
The potential switches from inhibition to attraction at
\eqn{d=\sigma/\tau^{1/6}}{d=sigma/tau^(1/6)}.
Maximum attraction occurs at distance
\eqn{d = (\frac 2 \tau)^{1/6} \sigma}{d = (2/tau)^(1/6) sigma}
and the maximum achieved is \eqn{\exp(\tau^2/4)}{exp((tau^2)/4)}.
Interaction is negligible for distances
\eqn{d > 2 \sigma \max\{1,\tau^{1/6}\}}{d > 2 * sigma * max(1, tau^(1/6))}.
This potential
is used (in a slightly different parameterisation)
to model interactions between uncharged molecules in statistical physics.
The function \code{\link{ppm}()}, which fits point process models to
point pattern data, requires an argument
of class \code{"interact"} describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the Lennard-Jones pairwise interaction is
yielded by the function \code{LennardJones()}.
See the examples below.
The ``canonical regular parameters'' estimated by \code{\link{ppm}} are
\eqn{\theta_1 = \sigma^{12}}{theta1 = sigma^12}
and
\eqn{\theta_2 = \tau \sigma^6}{theta2 = tau * sigma^6}.
}
\seealso{
\code{\link{ppm}},
\code{\link{pairwise.family}},
\code{\link{ppm.object}}
}
\examples{
X <- rpoispp(100)
ppm(X, ~1, LennardJones(), correction="translate")
# Typically yields very small values for theta_1, theta_2
# so the values of sigma, tau may not be sensible
##########
# How to plot the pair potential function (exponentiated)
plotLJ <- function(sigma, tau) {
dmax <- 2 * sigma * max(1, tau)^(1/6)
d <- seq(dmax * 0.0001, dmax, length=1000)
plot(d, exp(- (sigma/d)^12 + tau * (sigma/d)^6), type="l",
ylab="Lennard-Jones",
main=substitute(list(sigma==s, tau==t),
list(s=sigma,t=tau)))
abline(h=1, lty=2)
}
plotLJ(1,1)
}
\author{Adrian Baddeley
\email{adrian@maths.uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{models}