Dear Mr. Parrondo, thank you so much for your exceptionally clear and incredibly helpful lectures! It might be very obvious, but I get lost once during the derivation of the Fokker-Planck equation (around 31:00). I would be very grateful if you could help me out of my confusion! When replacing the average $\langle \dot A (t) angle$ by its integral definition $\int dx ho(x,t) \dot A (t)$, I don't understand why $$ \langle \dot A angle = \int \frac{\partial ho}{\partial t} A $$ holds or how one would get there...
Hia Lana, thanks for your comment!! "A" is introduced as an arbitrary function of x, so it's A(x). Then we define A(t) as A(x(t)). Then, the average is == int dx rho(x,t) A(x). Imagine for instance that A(x)=x, then ==int dx rho(x,t) x. Now, if we differentiate the equation == int dx rho(x,t) A(x) with respect to time, we get: d / dt = int dx [\partial rho(x,t)/\partial t] A(x) The l.h.s. is \langle \dot A angle I hope this will help!
Dear Dr. Parrondo, thank you for the lectures. I have one question. How one should deal with stochastic DE when the stochastic part of the equation is nonlinear in \xi, but some function of \xi?
Hi Roman. Thank you for your comment. I do not know the answer to your question. Just a couple of comments: If the noise appears in a term as g(x) h(xi), where xi is Gaussian white noise and h(.) is a nonlinear function, then this is equivalent to considering a non-gaussian noise. There are theorems that prove that some nongaussian white noises are equivalent to gaussian white noise (see for instance the limit of a dichotomous noise in the book by Horsthemke and Lefebver, Noise Induced Transitions). But I don't know if this is general. In fact, the dichotomous noise cannot be written as a nonlinear function of a gaussian noise. If the noise appears in a more complicated way, like sin(x xi) then I guess things are even more complicated. But I don't know the literature on this topic.
@@juanmrparrondo1375 Thank you for a prompt response. I am reading a couple of articles on supression/introduction of chaos in nonlinear systems by random phase in the drive (e.g., doi:10.1155/2011/53820, doi:10.1016/j.chaos.2004.04.014, both refers Runge-Kutta-Verner method for simulation of SDE). Unfortunately I fail to find anything else but Ito calculus for SDE numerical methods which it its turn deals only with linear \xi(t) at r.h.s. Now I came to some understanding that it seems that for simulation it should be enough to deal with argument of cos(\omega*t+\sigma\xi(t)) as separate differential equation in the SDE system, so there would be one additional equation d\theta/dt=\omega+\sigma*d(\xi(t))/dt (e.g.,10.1006/jsvi.1996.0869). However, I am still confused about usage of white noise and Wiener process as \xi(t). It seems that different papers mean different thing for \xi(t) in cosine argument. So I believe I still have to do some research.
Amazing lecture! Around 45:28 , I believe the second term on the right of the FPE for Stratanovich interpretation must have a positive sign and not negative. Please correct me if I am wrong☺
Thanks for the video
Dear Mr. Parrondo,
thank you so much for your exceptionally clear and incredibly helpful lectures! It might be very obvious, but I get lost once during the derivation of the Fokker-Planck equation (around 31:00). I would be very grateful if you could help me out of my confusion!
When replacing the average $\langle \dot A (t)
angle$ by its integral definition $\int dx
ho(x,t) \dot A (t)$, I don't understand why
$$
\langle \dot A
angle = \int \frac{\partial
ho}{\partial t} A
$$
holds or how one would get there...
Hia Lana, thanks for your comment!! "A" is introduced as an arbitrary function of x, so it's A(x). Then we define A(t) as A(x(t)). Then, the average is == int dx rho(x,t) A(x).
Imagine for instance that A(x)=x, then ==int dx rho(x,t) x.
Now, if we differentiate the equation == int dx rho(x,t) A(x) with respect to time, we get:
d / dt = int dx [\partial rho(x,t)/\partial t] A(x)
The l.h.s. is \langle \dot A
angle
I hope this will help!
@@juanmrparrondo1375 Now it all makes sense, thank you very much!
Dear Dr. Parrondo, thank you for the lectures. I have one question. How one should deal with stochastic DE when the stochastic part of the equation is nonlinear in \xi, but some function of \xi?
Hi Roman. Thank you for your comment. I do not know the answer to your question. Just a couple of comments: If the noise appears in a term as g(x) h(xi), where xi is Gaussian white noise and h(.) is a nonlinear function, then this is equivalent to considering a non-gaussian noise. There are theorems that prove that some nongaussian white noises are equivalent to gaussian white noise (see for instance the limit of a dichotomous noise in the book by Horsthemke and Lefebver, Noise Induced Transitions). But I don't know if this is general. In fact, the dichotomous noise cannot be written as a nonlinear function of a gaussian noise. If the noise appears in a more complicated way, like sin(x xi) then I guess things are even more complicated. But I don't know the literature on this topic.
@@juanmrparrondo1375 Thank you for a prompt response. I am reading a couple of articles on supression/introduction of chaos in nonlinear systems by random phase in the drive (e.g., doi:10.1155/2011/53820, doi:10.1016/j.chaos.2004.04.014, both refers Runge-Kutta-Verner method for simulation of SDE). Unfortunately I fail to find anything else but Ito calculus for SDE numerical methods which it its turn deals only with linear \xi(t) at r.h.s. Now I came to some understanding that it seems that for simulation it should be enough to deal with argument of cos(\omega*t+\sigma\xi(t)) as separate differential equation in the SDE system, so there would be one additional equation d\theta/dt=\omega+\sigma*d(\xi(t))/dt (e.g.,10.1006/jsvi.1996.0869). However, I am still confused about usage of white noise and Wiener process as \xi(t). It seems that different papers mean different thing for \xi(t) in cosine argument. So I believe I still have to do some research.
Amazing lecture! Around 45:28 , I believe the second term on the right of the FPE for Stratanovich interpretation must have a positive sign and not negative. Please correct me if I am wrong☺
Sure, you're right. The error is there from minute 44:20 to 45:30 approx. Thanks for pointing it out!!