\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(sigma0=NA)
}
\value{
  An object of class \code{"interact"}
  describing the Lennard-Jones interpoint interaction
  structure.
}
\arguments{
  \item{sigma0}{
    Optional. Initial estimate of the parameter \eqn{\sigma}{sigma}.
    A positive number.
  }
}
\details{
  In a pairwise interaction point process with the
  Lennard-Jones pair potential (Lennard-Jones, 1924)
  each pair of points in the point pattern,
  a distance \eqn{d} apart,
  contributes a factor
  \deqn{
    v(d) = \exp \left\{
    -
    4\epsilon
    \left[
       \left(
          \frac{\sigma}{d}
       \right)^{12}
       -
       \left(
          \frac{\sigma}{d}
       \right)^6
    \right]
    \right\}
  }{
    v(d) = exp( - 4 * epsilon * ((sigma/d)^12 - (sigma/d)^6))
  }
  to the probability density,
  where \eqn{\sigma}{sigma} and \eqn{\epsilon}{epsilon} 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.
  The parameter  \eqn{\sigma}{sigma} is called the
  \emph{characteristic diameter} and controls the scale of interaction.
  The parameter \eqn{\epsilon}{epsilon} is called the \emph{well depth}
  and determines the strength of attraction.
  The potential switches from inhibition to attraction at
  \eqn{d=\sigma}{d=sigma}.
  The maximum value of the pair potential is
  \eqn{\exp(\epsilon)}{exp(epsilon)}
  occuring at distance
  \eqn{d = 2^{1/6} \sigma}{d = 2^(1/6) * sigma}.
  Interaction is usually considered to be negligible for distances
  \eqn{d > 2.5 \sigma \max\{1,\epsilon^{1/6}\}}{d > 2.5 * sigma * max(1, epsilon^(1/6))}.

  This potential is used 
  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.
}
\section{Rescaling}{
  To avoid numerical instability,
  the interpoint distances \code{d} are rescaled
  when fitting the model.
  
  Distances are rescaled by dividing by \code{sigma0}.
  In the formula for \eqn{v(d)} above,
  the interpoint distance \eqn{d} will be replaced by \code{d/sigma0}.

  The rescaling happens automatically by default.
  If the argument \code{sigma0} is missing or \code{NA} (the default),
  then \code{sigma0} is taken to be the minimum
  nearest-neighbour distance in the data point pattern (in the
  call to \code{\link{ppm}}). 
  
  If the argument \code{sigma0} is given, it should be a positive
  number, and it should be a rough estimate of the
  parameter \eqn{\sigma}{sigma}. 
  
  The ``canonical regular parameters'' estimated by \code{\link{ppm}} are
  \eqn{\theta_1 = 4 \epsilon (\sigma/\sigma_0)^{12}}{theta1 = 4 * epsilon * (sigma/sigma0)^12}
  and 
  \eqn{\theta_2 = 4 \epsilon (\sigma/\sigma_0)^6}{theta2 = 4 * epsilon * (sigma/sigma0)^6}.
}
\section{Warnings and Errors}{
  Fitting the Lennard-Jones model is extremely unstable, because
  of the strong dependence between the functions \eqn{d^{-12}}{d^(-12)}
  and \eqn{d^{-6}}{d^(-6)}. The fitting algorithm often fails to
  converge. Try increasing the number of
  iterations of the GLM fitting algorithm, by setting
  \code{gcontrol=list(maxit=1e3)} in the call to \code{\link{ppm}}.
  
  Errors are likely to occur if this model is fitted to a point pattern dataset
  which does not exhibit both short-range inhibition and
  medium-range attraction between points.  The values of the parameters
  \eqn{\sigma}{sigma} and \eqn{\epsilon}{epsilon} may be \code{NA}
  (because the fitted canonical parameters have opposite sign, which
  usually occurs when the pattern is completely random).

  An absence of warnings does not mean that the fitted model is sensible.
  A negative value of \eqn{\epsilon}{epsilon} may be obtained (usually when
  the pattern is strongly clustered); this does not correspond
  to a valid point process model, but the software does not issue a warning.
}
\seealso{
  \code{\link{ppm}},
  \code{\link{pairwise.family}},
  \code{\link{ppm.object}}
}
\examples{
   fit <- ppm(cells ~1, LennardJones(), rbord=0.1)
   fit
   plot(fitin(fit))
}
\references{
  Lennard-Jones, J.E. (1924) On the determination of molecular fields.
  \emph{Proc Royal Soc London A} \bold{106}, 463--477.
}
\author{Adrian Baddeley
  \email{Adrian.Baddeley@uwa.edu.au}
  \url{http://www.maths.uwa.edu.au/~adrian/}
  and Rolf Turner
  \email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{models}
