Stochastic Models for Many Body Systems I Infinite Systems in Thermal Equilibr

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Stochastic Models for Many Body Systems I Infinite Systems in Thermal Equilibr
R H Kraichnan
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9) et seq. It is clear that the only change in the expansion for S, ( ^ ) is that the factor '^^^Y^ ^ in rule b) K a pqx o P"-^ 22 above must be replaced by the factor ^ V . We readily find that ■^ '^ n. M;p-s p-s ^9 precisely the same diagrams survive in the present case as for the pre- vious ladder model. We shall illustrate the equivalence by two examples.
Consider first the diagram of Fig. 12c and equip it vith momentum labels as in Fig. 6a. Its contribution contains the factors j-i[(k-s)+(
...s-s')+(s'-k)]. D^^^l (6. 11) according to our rules and to (2. 9). The phase of this expression is identically zero, and consequently the contribution survives when it is summed over the particle labels n and m in accord with rule c) . Next, however, consider the ring diagram Fig. 12d, with momentum labels as in Fig. 7a- • Its contribution contains the factors VVV exp -iq-(d +d +d )L (6. 12) qqq \_ ^ n, i t, m m, nj where q = k-s and we have used the momentum conservation relations. The phase of this expression fluctuates at random as we sum over all values of n, t, and m, so that the contribution does not survive in the limit jj ^ M^ il -^ 00 .

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