Systems with a large number of degrees of freedom are often of interest in physics. Among these are the many atoms in a chunk of condensed matter, the many electrons in an atom, or the infinitely many values of a quantum field at all points in a region of space-time. The description of such systems often involves (or can be reduced to) the evaluation of integrals of very high dimension. For example, the classical partition function for a gas of A atoms at a temperature 1 /(3 interacting through a pair-wise potential v is proportional to the 3A-dimensional integral

= J d3 r*i . . . d3 r& exp - f iY jir n ) i<j

The straightforward evaluation of an integral like this by one of the quadrature formulas discussed in Chapters 1 or 4 is completely out of the question except for the very smallest values of A . To see why, suppose that the quadrature allows each coordinate to take on 10 different values (not a very fine discretization), so that the integrand must be evaluated at 103A points. For a modest value of A = 20 and a very fast computer capable of some 107 evaluations per second, this would take some 1053 seconds, more than 1034 times the age of the universe! Of course, tricks like exploiting the permutation symmetry of the integrand can reduce this estimate considerably, but it still should be clear that direct quadrature is hopeless.