Entropy of Diffusion: Delta Initial Condition

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Homework Help Overview

The problem involves determining the entropy associated with a diffusion process, specifically as a function of time and in relation to the diffusion coefficient. The context includes considerations of initial conditions and the mathematical formulation of entropy in diffusion scenarios.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the need to express entropy as an integral over space, with one participant questioning the integration of entropy equations in this context. Another participant seeks clarification on the initial conditions and system arrangement, particularly regarding the nature of the diffusion source.

Discussion Status

The discussion is active, with participants exploring different aspects of the problem, including the mathematical formulation and initial conditions. Guidance has been offered regarding the setup of the system, but no consensus has been reached on the approach to take.

Contextual Notes

Participants note that the problem involves a heat equation without a source term and that the initial condition is a delta function located at the origin in three-dimensional space.

simoncks
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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.
 
In short, I would like to know if there are any entropy equation integrating over space.
 
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?
 
It is a heat equation without source term. Open boundary at infinity. Initial condition is a delta function at (x,y,z) = 0.
 

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