
#1
Apr3009, 07:54 AM

P: 1

Hello,
I am trying to (numerically) solve the following reactiondiffusion equation for the probability density of the a pair, [tex]\rho (\vec{r}_1,\vec{r}_2)[/tex]: [tex]\dot{\rho} (\vec{r}_1,\vec{r}_2,t) = D_1 \nabla^2_1 \rho (\vec{r}_1,\vec{r}_2,t) + D_2 \nabla^2_2 \rho (\vec{r}_1,\vec{r}_2,t)  k \left( \left\ \vec{r}_1  \vec{r}_2 \right\ \right) [/tex], where the subscripts refer to the first and second particle, respectively. In 2D and polar coordinates, [tex]r_i[/tex] and [tex]\theta_i [/tex]: [tex] \nabla^2_i = \frac{1}{r_i} \frac{\partial}{\partial r_i} r_i \frac{\partial}{\partial r_i} + \frac{1}{r_i^2} \frac{\partial}{\partial \theta_i} [/tex]. The domain is confined by two concentric spheres: [tex] 0 \leq \left\ \vec{r}_1 \right\ \leq R [/tex] and [tex] \left\ \vec{r}_2 \right\ \geq R [/tex]. The initial condition are spherically symmetric, i.e. only depends on the [tex]r_i[/tex]s. The reaction term is a function of the distance of the two particles, i.e. in 2D [tex] k( \left\ \vec{r}_1  \vec{r}_2 \right\ ) = k( \sqrt{r_1^2 + r_2^2  2 r_1 r_2 \cos ( \theta_1\theta_2)} ) [/tex]. I hoped to get rid of at least 1 coordinate by a variable transformation and separation of variables. However, so far I just could not come up with a separable problem. Do I really have to retain all 4 variables? Any suggestions of how to reduce this problem to something manageable are highly welcome. Eventually I will be interested in 3D and 4D as well. Thank you, Daniel 



#2
May1409, 10:24 AM

P: 330

Just throwing idea.
If the forces between particles 1 and 2 are conservative, try working in the centre of mass frame. In mechanics we use this frame to solve the central force motion and scattering problem. 


Register to reply 
Related Discussions  
Potential between 2 concentric spheres  Advanced Physics Homework  4  
Gauss's Law/Energy Problem with Concentric Spheres  Introductory Physics Homework  8  
Capacitance of a system of 3 concentric spheres  Introductory Physics Homework  0  
Electric Potentials: concentric spheres  Introductory Physics Homework  3  
two concentric spheres  Introductory Physics Homework  8 