y**c
发帖数: 133 | 1 Is there anybody can help me to solve this question:
A sphere of radius=R0 has gas in it;
There has a potential V(R)=V0*exp[-(R-R0)/a].
The atoms are colliding in the sphere, and the potential acts on the particles.
The question is:
How to get the Fourier transform of V(t), i.e.
Integral(from -inf to +inf) [V(t)*V(t+tau)]*exp(-i omega t) d tau
you may consider potential is much smaller than the kinematic energy, if
otherwise too difficult to solve.
Thank you! | w*****g
发帖数: 9 | 2 I am confused. V is a function of R or a function of t? | y**c
发帖数: 133 | 3 Sorry, I did not explain it clearly.
There are some gas in a sphere. For one particle, it is randomly walking,
Brown motation, and at the same time, this particle can see the potential
which is exponentially dropping depend on how close to the wall of the
sphere, exp(-(R0-R)/a).
The potential itself is static, i.e. is NOT of function of t. But the
potential seen by the particles are not static. It is a function of space,
and time. In this question, Fourier transfer of auto-correlation function
【在 w*****g 的大作中提到】
: I am confused. V is a function of R or a function of t?
| w*****g
发帖数: 9 | 4 I am afraid this is a hard problem and you need to be more specific. As far
as I know, for a Markov random walk without the potential, the diffusion
distance is proportional to D*t^2 where D is the diffusion constant. Then
the correlation function is in a exponential form:
exp(-\sqrt{D} \tau). It is easy to get the power spectrum by doing the
Fourier transform. I think it is delta function, but not sure.
For other random processes (e.g., Gaussian) or including the potential, I
don't have a clue.
【在 y**c 的大作中提到】
: Sorry, I did not explain it clearly.
: There are some gas in a sphere. For one particle, it is randomly walking,
: Brown motation, and at the same time, this particle can see the potential
: which is exponentially dropping depend on how close to the wall of the
: sphere, exp(-(R0-R)/a).
: The potential itself is static, i.e. is NOT of function of t. But the
: potential seen by the particles are not static. It is a function of space,
: and time. In this question, Fourier transfer of auto-correlation function
|
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