- #1

Opus_723

- 178

- 3

<F1(t1)F2(t2)> = s

^{2}*d(t1-t2)*D12

Where s is the standard deviation of the Gaussian, little d is the delta function, and big D is the kronecker delta. For concreteness and to keep track of units, say F represents a force.

What is the 4th-order correlation of this same white noise?

<F(t1)F(t2)F(t3)F(t4)> = ?

My first guess, would be to simply let s

^{2}become 3s

^{4}to capture the higher moment in the Gaussian distribution, and add a couple of dirac-deltas and kroneckers to make sure it's only nonzero when all t1,t2,t3,t4 are the same:

<F(t1)F(t2)F(t3)F(t4)> = 3s

^{4}*d(t1-t2)d(t2-t3)d(t3-t4)*D12*D23*D34

But this is, of course, wrong. We would need three dirac deltas to ensure that the result is nonzero only when all times are the same, but this introduces too many units of 1/time. It no longer makes any sense as a 4th-order correlation of forces. Somehow this must be accomplished with a different structure.

So what is the right way to do this? Do we need to specify additional properties of the white noise in order to calculate this?