Fick's Second law from Boltzmann Equation

In summary: F + F.∇2φ)= ∫d3r(∂F/∂t + v.∇F + F.∇2φ)= ∫d3r(∂F/∂t + v.∇F + F.∇2φ)= ∫d3r(∂F/∂t + v.∇F + F.∇2φ)= ∫d3r(∂F/∂t + v.∇F + F.∇2φ)= ∫d3r(∂F/∂t + v.∇F + F.
  • #1
The_Foetus
3
0

Homework Statement


Starting from the kinetic equation for the distribution function F*(t, r, v) of some
labelled particle admixture in a gas, derive the self-diffusion equation

∂n*/∂t = D∇2n*
for the number density n*(t,r) = ∫d3vF*(t,r,v) of the labelled particles. Derive also the expression for the self-diffusion coefficient D.

Homework Equations


Boltzmann Equation ∂F/∂t + v.∇F = (∂F/dt)c

The Attempt at a Solution


Dropping the star notation on n and F
∂n/∂t = ∂/∂t∫Fd3v

= ∫∂F/∂t d3v

=∫((∂F/∂t)c - v.∇F)d3v

(∂F/∂t)c term drops out by definition (

As v is a single vector uniform in space, can write it as ∇φ

Therefore ∂n/∂t = ∫∇φ.∇Fd3v

Stuck from here on, don't know how to find D either.
 
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  • #2


Hello, thank you for your post. I will provide a possible solution to the problem, but please note that there may be other approaches and solutions that are equally valid.

To derive the self-diffusion equation, we start with the Boltzmann equation:

∂F/∂t + v.∇F = (∂F/∂t)c

We can rewrite this equation in terms of the number density n(t,r) as follows:

∂n/∂t + ∫vd3v∇F = (∂n/∂t)c

Next, we use the definition of the number density n as the integral of the distribution function F over all velocities:

n = ∫Fd3v

Therefore, we can rewrite the above equation as:

∂n/∂t + ∫vd3v∇F = ∫∂F/∂td3v

Now, let's consider the left-hand side of the equation. We can use the chain rule to rewrite the integral over velocity as an integral over position and velocity:

∫vd3v∇F = ∫d3r∫vd3v(∇F)

= ∫d3r∫vd3v(∂F/∂t + v.∇F)

= ∫d3r(∂F/∂t + ∫vd3vv.∇F)

= ∫d3r(∂F/∂t + ∇.∫vd3vvF)

= ∫d3r(∂F/∂t + ∇.(vF))

= ∫d3r(∂F/∂t + v.∇F + F.∇v)

= ∫d3r(∂F/∂t + v.∇F + F.(∂v/∂r))

= ∫d3r(∂F/∂t + v.∇F + F.(∇v))

= ∫d3r(∂F/∂t + v.∇F + F.∇(∇φ))

= ∫d3r(∂F/∂t + v.∇F + F.∇2φ)

= ∫d3r(∂F/∂
 

1. What is Fick's Second law from Boltzmann Equation?

Fick's Second law from Boltzmann Equation is a mathematical equation that describes the diffusion of particles in a fluid or gas. It relates the flux of particles to the concentration gradient and the diffusion coefficient.

2. How is Fick's Second law derived from Boltzmann Equation?

Fick's Second law is derived from Boltzmann Equation through the assumption of molecular chaos and the use of statistical mechanics. The equation describes the probability of particles moving from a region of high concentration to a region of low concentration.

3. What is the physical interpretation of Fick's Second law?

The physical interpretation of Fick's Second law is that it describes the rate at which particles move from areas of high concentration to areas of low concentration due to random thermal motion. This process is known as diffusion.

4. What factors affect the diffusion described by Fick's Second law?

The diffusion described by Fick's Second law is affected by the concentration gradient, the diffusion coefficient, and the temperature of the system. A steeper concentration gradient and higher temperature will result in a faster diffusion rate, while a lower diffusion coefficient will slow down the diffusion process.

5. What are the limitations of Fick's Second law from Boltzmann Equation?

Fick's Second law has some limitations, including the assumption of molecular chaos and the neglect of other factors that may affect diffusion, such as convection and chemical reactions. It also only applies to dilute solutions and does not account for non-ideal behaviors or interactions between particles.

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