0

I would like to know if for a linear system, the Green function and the linear response function finally represent the same thing ?

Indeed, if I have an "ouput" A of a system that is caused by an excitation F, I have by definition of the linear response function :

$$ A(t)=\int \chi(t,t')F(t')dt' $$

Because of the linearity of the system, the ouput $A(t)$ is solution of a linear differential equation with a source term.

The green function is how the system react to a dirac.

We can proove that :

$$A(t)=\int dt' G(t,t') F(t') dt'$$

So can we say that susceptibility and green functions are exactly the same thing ?

Then why would it be two different notions ?

StarBucK
  • 689
  • 1
    in a LTI system, the Green function is the convolutive inverse of the impulse reponse – reuns Oct 02 '16 at 20:51
  • I agree but it doesn't really answer my question :/ – StarBucK Oct 02 '16 at 20:52
  • it does. If the system is LTI then $A(t) = \chi \ast F(t)$ and $G(t) \ast A(t) = F(t)$, i.e. $G \ast \chi(t) = \delta(t)$ (the Dirac delta, the neutral element of the convolution) – reuns Oct 02 '16 at 20:53
  • Yes I didn't understand what you meant. So ok, the green function is here the inverse (for the convoluion product) of the linear response function. But do you agree with my post ? Because if you agree, then we would have $\chi=G$ which seems contradictory with what you said. So there is something I still misunderstand. – StarBucK Oct 02 '16 at 20:58
  • 1
    Your formula for $A(t)$ is much simpler when the system is LTI : $A(t) = \int_{-\infty}^\infty \chi(t-t')F(t')dt' = \chi \ast F(t)$. And "The green function is how the system react to a dirac" isn't correct. – reuns Oct 02 '16 at 21:01
  • If I have the differential linear equation like this : $ \Theta A(t) = F(t) $ where $\Theta$ is my differential operator. I have $ \Theta G(t,t_0) = \delta (t-t_0) $. And I have $ A(t)=\int dt' G(t,t')F(t') $ So my Green function $G$ is also the response to a dirac ? – StarBucK Oct 03 '16 at 00:10
  • 1
    I just read on wikipedia green functions : "In mathematics, a Green's function is the impulse response of an inhomogeneous differential equation defined on a domain, with specified initial conditions or boundary conditions". So it seems the green function is the response to a dirac ? But maybe I misunderstood what you wanted to say – StarBucK Oct 03 '16 at 00:12
  • 1
    This is unclear, and wrong. It is the impulse response of the inverse operator $L^{-1}$, where your PDE is $L u = f$, so that $u = L^{-1} f = L^{-1} \delta \ast f = G \ast f$. See the 1st line of https://en.wikipedia.org/wiki/Green%27s_function#Definition_and_uses – reuns Oct 03 '16 at 00:27
  • @reuns please see my question at https://math.stackexchange.com/questions/2432092/greens-function-impulse-response-confusion/2432303#2432303 and consider using your comment as an answer there--it is important and should be archived. Also it seems that you are saying that wiki is wrong. Ie. the Greens function of L is not the impulse response of L as wiki stated. This is the essence of my question. – user45664 Sep 17 '17 at 16:16

0 Answers0