The fundamental solution to the Diffusion Equation in one dimension is
c(x, t)=1/(4(pi)Dt)^1/2 e^( - (x - xo)^2/4Dt) ,
where D is the Diffusion constant of the diffusing particle, t is the time, x is the position along the x-axis, and xo is the initial position of the diffusing particle.
In the above equation, the center (mean) of the pulse is taken to be stationary at x = xo.
If the pulse were to drift in one direction along the x axis so that its center (mean) moved with velocity v to the right, the moving pulse would be expressed mathematically as
c(x, t)=1/(4(pi)Dt)^1/2 e^( - (x - vt)^2/4Dt) ,
where v is the velocity of the center (mean) of the pulse along the x-axis.
However, if the center (mean) of the pulse were moving randomly, so that it undergoes Brownian Motion, how would c(x, t) be expressed so that it could be evaluated (numerically) for each given (x, t) pair?
Any information related to the answer of this question would be greatly appreciated.
Hi,
I don't have any reference to give you, but, as I understand your question, you have two independent diffusion processes:
* There is a particle (2) diffusing with constant D_2 with respect to a medium.
* There is a medium (1) diffusing with constant D_1 with respect to the desired frame of reference (0).
* Let x_21 be the coordinate of the particle w.r.t. the medium.
* Let x_10 be the coordinate of the medium w.r.t. the frame of reference.
* The probability density function (PDF) for the particle w.r.t. the medium is
c_21(x_21, t)=1/(4(pi)D_2t)^1/2 e^( - (x_21)^2/4D_2t)
* The PDF for the medium w.r.t. the frame of refernce is
c_10(x_10, t)=1/(4(pi)D_1t)^1/2 e^( - (x_10)^2/4D_1t)
* The position of the particle w.r.t. the frame of reference is
x = x_21 + x_10 => x_10 = x - x_21
* If the diffusion processes are independent, we have
c(x, t) = integral c_21(x_21, t) * c_10(x - x_21, t) d x_21,
with the integral taken of the domain of x_21.
I don't know if there is an analytical solution to this convolution, but you can easily find out with Maple/Mathematica/Matlab.
Good luck,
/Stefan
- Badges
- Science topic
- Similar topics
- Elementary Particle Physics
- Particle