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I have a question of the form, "how did they go from this step to that step?" I am following a paper from R. Kubo, called "The fluctuation-dissipation theorem". I suspect variable substitution and a simple assumption are used, but I don't know how. Here is the basic expression from which everything else follows.

[tex]D=lim_{t\rightarrow\infty}\frac{1}{2t}<\{x(t)-x(0)\}^2>[/tex]

Since we have

where the limits of the integration are 0 to t, we substitute the second equation into the first equation to get

again, the limits of both integrals are from 0 to t. (Here comes the step I don't understand)

[tex]D=lim_{t\rightarrow\infty}\frac{1}{t}\int{dt_{1}}\int{dt'<v(t_{1})v(t_{1}+t')}>[/tex]

where the limits of the outer integral are from 0 to t, and the limits of the inner integral are from 0 to [tex](t-t_{1})[/tex]

It looks like the substitution [tex]t'=t_{2}-t_{1}[/tex] was made. But what happened to the upper limit of integration?

[tex]D=lim_{t\rightarrow\infty}\frac{1}{2t}<\{x(t)-x(0)\}^2>[/tex]

Since we have

[tex]x(t)-x(0)=\int{v(t')dt'}[/tex]

where the limits of the integration are 0 to t, we substitute the second equation into the first equation to get

[tex]D=lim_{t\rightarrow\infty}\frac{1}{2t}\int{\int{dt_{1}dt_{2}<v(t_{1})v(t_{2})}}>[/tex]

again, the limits of both integrals are from 0 to t. (Here comes the step I don't understand)

[tex]D=lim_{t\rightarrow\infty}\frac{1}{t}\int{dt_{1}}\int{dt'<v(t_{1})v(t_{1}+t')}>[/tex]

where the limits of the outer integral are from 0 to t, and the limits of the inner integral are from 0 to [tex](t-t_{1})[/tex]

It looks like the substitution [tex]t'=t_{2}-t_{1}[/tex] was made. But what happened to the upper limit of integration?

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