A Formal Solution of the Equations of Statistical Equilibrium

Cover A Formal Solution of the Equations of Statistical Equilibrium
A Formal Solution of the Equations of Statistical Equilibrium
Bruno Zumino
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For the purpose of identification with known resxilts one ceui show quite easily that the condition (l6) for a can be written as (17) af: nb^(ac)°-^ = 1 n=l where (18) ^1 = ^' \=^|^n(W-^n)^S - 5 are the well-known cluster integrals L-^-' . Therefore the product (19) ac = z is the activity and (I6), or (20) c = zH°[x, z], expresses the relation between density and activity. Equation (l4) can now be written (21) h[x, X] =f H°[x, |X + z] .
Equations (9), (12), (20) and (21) together describe the d
...esired solution. The solution appears expressed in terms of the activity z; if one desires an expression in terms of the density, one must first invert eqviation (20). 5. Expansion to first order in the density As an example, we shall now use the obtained solution to calciilate the distribution function F neglecting terms of second order in the density. From (20) (22) f = H°[x, z] = 1 + z Equation (22) shows that (25) z = c +0(c^) so that, in the following, we will have to retain only terms of first order f G°(x, x^)dx^ + 0(z^).

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