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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
Not smooth integralsg(x)ax ineed more precisionb13Calculation of integralsgx ( )1 gx (i)g( x) dx p( x) dx ( b a)px ( ) N px ( )b b Naai1iif x i randomly distributed according to p(x).Good convergence when functiong( x)px ( )smooth.„importancesampling“14
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