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
$\theta_2 = \tau \sigma^6$.