Entropy of Diffusion: Delta Initial Condition

In summary, the conversation discusses finding the entropy of a diffusion constant as a function of time, and the use of the heat/diffusion kernel and entropy definition to do so. There is also a question about integrating over space to find the entropy equation. The specific scenario being considered is a heat equation without a source term, with an open boundary at infinity and a delta function initial condition.
  • #1
simoncks
29
0

Homework Statement


The problem requires me to find the entropy of a diffusion constant as a function of time (I guess in terms of diffusion coefficient)

Homework Equations


Perhaps Heat / Diffusion kernel
S = k p lnp

The Attempt at a Solution


I assume it was a delta initial condition then apply the kernel. However I need to turn the entropy definition into an integral over space. The kernel times differential volume is the probability finding the particle in that space but the natural log term is tricky.
 
  • #3
In short, I would like to know if there are any entropy equation integrating over space.
 
  • #4
You are asking what entropy S is produced by time t by diffusion through a medium characterized by diffusion coefficient D_{ij}? If so, could you indicate the arrangement of the system at t = 0; is a point-source diffusing?
 
  • #5
It is a heat equation without source term. Open boundary at infinity. Initial condition is a delta function at (x,y,z) = 0.
 

FAQ: Entropy of Diffusion: Delta Initial Condition

1. What is entropy of diffusion?

Entropy of diffusion is a measure of the randomness or disorder in a system as particles move from an area of higher concentration to an area of lower concentration.

2. How is entropy of diffusion measured?

Entropy of diffusion is typically measured using the Boltzmann equation, which takes into account the number of particles in a system, their velocities, and the temperature of the system.

3. What is the relationship between entropy of diffusion and initial conditions?

The initial conditions of a system, such as the concentration gradient and temperature, can affect the entropy of diffusion. For example, a higher concentration gradient will result in a higher entropy of diffusion.

4. How does entropy of diffusion affect the rate of diffusion?

The higher the entropy of diffusion, the faster the rate of diffusion. This is because as the randomness or disorder in a system increases, the particles have more opportunities to move and therefore diffuse at a faster rate.

5. What factors can influence the entropy of diffusion?

Aside from initial conditions, other factors such as pressure, particle size, and the presence of obstacles can also affect the entropy of diffusion. Higher pressure and smaller particle size will result in a higher entropy of diffusion, while obstacles can decrease the entropy of diffusion by limiting the movement of particles.

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