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Additional resources for Thermodynamics Statistical Mechanics Notes
For now, it is su cient to know that g(r) is a measure of the probability that a particle will be located a distance r from a another particle in the system. The gure below shows the function g(r) for the ideal gas and for the liquid argon systems. It can be seen that the radial distribution function for the ideal gas is completely featureless signifying that it is equally likely to nd a particle at any distance r from a given particle. (Since the probability is uniform, g(r) 1=r2 for small r. This is the particularly normalization condition on g(r) that gives rise to uniform probability.
Dr2 drN e; U (r1 ::: rN ) N Z N =Z dr2 drN e; U (r1 ::: rN ) N 2 Thus, if we integrate over all r1 , we nd that 1 V Z dr1 (1) (r1 ) = N V = Thus, (1) actually counts the number of particles likely to be found, on average, in the volume element dr1 at the point r1 . Thus, integrating over the available volume, one nds, not surprisingly, all the particles in the system. 2651: Statistical Mechanics Notes for Lecture 9 I. DISTRIBUTION FUNCTIONS IN CLASSICAL LIQUIDS AND GASES (CONT'D) A. General correlation functions A general correlation function can be de ned in terms of the probability distribution function (n) (r1 ::: rn ) according to g(n) (r ::: r ) = 1 (n) (r ::: r ) 1 n n n 1 Z nN !
H (r1 ; r01 ) (r2 ; r02 )i Z 2 V ( N ; 1) = dr3 drN e; U (r1 ::: rN ) correlation NZN = N (N 2; 1) h (r1 ; r01 ) (r2 ; r02 )ir1 ::: rN 0 0 In general, for homogeneous systems in equilibrium, there are no special points in space, so that g(2) should depend only on the relative position of the particles or the di erence r1 ; r2 . In this case, it proves useful to introduce the change of variables r = r1 ; r 2 R = 12 (r1 + r2 ) r1 = R + 12 r r2 = R ; 21 r Then, we obtain a new function g~(2) , a function of r and R: Z 2 g~(2)(r R) = V (N ; 1) dr3 drN e; U (R+ 21 r R; 12 r r3 ::: rN ) NZN N ( = N ; 1) 2 R + 21 r ; r01 1 R ; 12 r ; r02 r1 ::: rN 0 0 In general, we are only interested in the dependence on r.
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