LennardJones: The Lennard-Jones Potential

Description

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

Usage

LennardJones()

Arguments

Value

An object of class "interact" describing the Lennard-Jones interpoint interaction structure.

Warnings

Fitting the Lennard-Jones model is extremely unstable, because of the strong dependence between the functions $d^{-12}$ and $d^{-6}$. The fitting algorithm often fails to converge. The fitting algorithm may also fail to converge when the spatial window of the point pattern dataset has dimensions that are much greater than 1 unit or much less than 1 unit, leading to overflow or underflow in the 12th power of the coordinates. To fix this problem, rescale the coordinates so that the window is close to the unit square (see Examples). 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 $\sigma$ and $\epsilon$ may be 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 $\epsilon$ 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.

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 $d$ apart, contributes a factor $$\exp \left{ - 4\epsilon \left[ \left( \frac{\sigma}{d} \right)^{12} - \left( \frac{\sigma}{d} \right)^6 \right] \right}$$ to the probability density, where $\sigma$ and $\epsilon$ 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. The parameter $\sigma$ is called the characteristic diameter and controls the scale of interaction. The parameter $\epsilon$ is called the well depth and determines the strength of attraction. The potential switches from inhibition to attraction at $d=\sigma$. The maximum value of the pair potential is $\exp(\epsilon)$ occuring at distance $d = 2^{1/6} \sigma$. Interaction is usually considered to be negligible for distances $d > 2.5 \sigma \max{1,\epsilon^{1/6}}$.

This potential is used 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 $\theta_1 = 4 \epsilon \sigma^{12}$ and $\theta_2 = 4 \epsilon \sigma^6$.

References

Lennard-Jones, J.E. (1924) On the determination of molecular fields. Proc Royal Soc London A 106, 463--477.

Examples

Run this code

data(demopat)
   demopat
   X <- rescale(unmark(demopat), 5000)
   X
   fit <- ppm(X, ~1, LennardJones(), rbord=0.1)
   fit
   plot(fitin(fit), xlim=c(0,0.01))

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