T - IfB

High dimensional integralConsider n hard spheres of radius R in a 3d3box.Phase space is 3n dimensional: (x(i , y i , z i ), i = 1,…,n.r ( x x ) ( y y ) ( z z)2 2 2ij i j i j i jwith hard sphere constraint( x x ) ( y y ) ( z z ) R2 2 2ijijij2Calculate average distance:1 2r r dx ,.., dx dy ,.., dy dz ,.., dzij ij 1 n 1 n 1 nZ nn ( 1)ij11MC strategy• Choose particle position.• If the excluded volume condition isnot fulfilled then reject.• Once the n particles are placed,calculate the average of r ij over allpairs (i,j).12