LennardJones            package:spatstat            R Documentation

_T_h_e _L_e_n_n_a_r_d-_J_o_n_e_s _P_o_t_e_n_t_i_a_l

_D_e_s_c_r_i_p_t_i_o_n:

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

_U_s_a_g_e:

       LennardJones()

_D_e_t_a_i_l_s:

     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( - (sigma/d)^12 + tau (sigma/d)^6)

     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 =
     (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 theta1
     = sigma^12 and  theta2 = tau * sigma^6.

_V_a_l_u_e:

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

_A_u_t_h_o_r(_s):

     Adrian Baddeley adrian@maths.uwa.edu.au <URL:
     http://www.maths.uwa.edu.au/~adrian/> and Rolf Turner
     rolf@math.unb.ca <URL: http://www.math.unb.ca/~rolf>

_S_e_e _A_l_s_o:

     'ppm', 'pairwise.family', 'ppm.object'

_E_x_a_m_p_l_e_s:

        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)

