This applet demonstrates the futility of performing a direct quadrature of a configurational integral.
The applet attempts to sample all of configuration space by moving each atom methodically through the box, as one would do if attempting a multidimensional trapezoid-rule integration. The progress bar shows at any time how far the simulation has gone in completing this sampling
Adjust the number of atoms in the simulation, and the step size of the quadrature process, using the text boxes. Hit the "Slow down" button to get a better picture of what is happening in the quadrature.
The atom in the innermost nested integral (with the most frequently changing position) is colored blue, while the atom in the outermost integral is colored red. The other atom colors are something between blue and red, depending on the location of their integral in the nest. At the outset all atoms are on top of one another in the upper left corner. Bluer atoms are drawn on top of redder ones, so the outermost-integral atom (the reddest) becomes visible only when all other atoms have moved ahead. For integration parameters only slightly larger than the default values, it may be a long wait for the red atom to emerge.
As you increase the parameter values, note that the time to compute the complete multidimensional integral (100% completion) very quickly becomes too long to be bear waiting. For any practical values of these parameters (say 100 atoms and 500 steps/dimension) direct calculation of the complete integral in this manner is absolutely hopeless.
Note:
This applet is fragile. Choosing the integration steps to be nine or greater may cause an error (nothing in the applet will work until you refresh the browser); this breakage may also occur (but not too often) when the number of atoms is changed; just Refresh to fix it.