Functions Kronecker delta

The time evolution of an arbitrary density pt=o x) may be obtained from equation 4.80 as follows. The Kronecker-delta function, 6 x — xq), evolves into the function 6 x — f xo)) after one step. Rewriting this in the form... 

In order to improve upon the mean-field approximation given in equation 7.112, we must somehow account for possible site-site correlations. Let us go back to the deterministic version of the basic Life rule (equation 7.110). We could take a formal expectation of this equation but we first need a way to compute expectation values of Kronecker delta functions. Schulman and Seiden [schul78] provide a simple means to do precisely that. We state their result without proof... 

The elements of the unit matrix are dy, the Kronecker delta function. 

Kronecker delta-function 5y Deformation at plastic yield... 

Equation (12.42) can be further simplified by noting that since the coordinates are independent, dxi/dxj = Sl, where StJ is the usual Kronecker delta function. (Sij = 1 for i = j Sij = 0 for i j.) After simplification we are left with... 

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