# LennardJones

##### The Lennard-Jones Potential

Creates the Lennard-Jones pairwise interaction structure which can then be fitted to point pattern data.

##### Usage

`LennardJones()`

##### Details

In a pairwise interaction point process with the Lennard-Jones pair potential, each pair of points in the point pattern, a distance $d$ apart, contributes a factor $$\exp \left{ - \left( \frac{\sigma}{d} \right)^{12} + \tau \left( \frac{\sigma}{d} \right)^6 \right}$$ to the probability density, where $\sigma$ and $\tau$ are positive parameters to be estimated.

See **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, $\sigma$ controls the scale of both
types of interaction, and $\tau$ determines the strength of
attraction.
The potential switches from inhibition to attraction at
$d=\sigma/\tau^{1/6}$.
Maximum attraction occurs at distance
$d = (\frac 2 \tau)^{1/6} \sigma$
and the maximum achieved is $\exp(\tau^2/4)$.
Interaction is negligible for distances
$d > 2 \sigma \max{1,\tau^{1/6}}$.

This potential
is used (in a slightly different parameterisation)
to model interactions between uncharged molecules in statistical physics.
The function `ppm()`

, which fits point process models to
point pattern data, requires an argument
of class `"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 `LennardJones()`

.
See the examples below.
The ``canonical regular parameters'' estimated by `ppm`

are
$\theta_1 = \sigma^{12}$
and
$\theta_2 = \tau \sigma^6$.

##### Value

- An object of class
`"interact"`

describing the Lennard-Jones interpoint interaction structure.

##### See Also

##### 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)
```

*Documentation reproduced from package spatstat, version 1.11-8, License: GPL version 2 or newer*