How Do You Calculate Steady-State Concentration in a Quadratic Potential Trap?

[klam997]

Homework Statement

N particles diffuse in one dimension in the potential U(x)=ax2, with a > 0. For example, such a potential could be provided by a line-shaped optical tweezer trap. The particles have the diffusion constant D.

Find the steady-state concentration, C0 (x).

Homework Equations

diffusion equation: dc/dt= D d^2(c)/d(x^2)

Fick's law: j = -D dc/dx

Diffusion concentration in 3 dimensions: c(r,t) = N/ [(4*pi*D*t)^3/2] * e^(-r^2/(4Dt)

Nernst-Planck formula and Nernst relation?

The Attempt at a Solution

I wasn't sure if I needed to use the Nernst-Planck formula or the Nernst relation. I know the flux of the system is zero because the particles are trapped within the potential. Therefore, j=0, dc/dx must be zero. At a steady state, I know that dc/dt is zero. I'm not really sure how I can approach this problem maybe except for adding boundaries conditions at the potential. Any thought or help is greatly appreciated!

Thanks in advance!

 

[grey_earl]

Your diffusion equation and the diffusion concentration in 3D you gave are for the case of zero potential. Look in your notes where those formulae are derived from Newton's law and add an external force (which is the gradient of the potential you are given). Then you should arrive at the correct formulae, and yes, d/dt c(t) = 0 is exactly the equilibrium condition you need then.

 

[klam997]

I realized it is derived into the Nernst-Planck equation. Thank you very much!