Entropy of Diffusion: Delta Initial Condition

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The discussion focuses on calculating the entropy of a diffusion process characterized by a delta initial condition. The entropy is expressed in terms of the diffusion coefficient and involves integrating the entropy definition over space. The challenge lies in incorporating the natural logarithm term into the integral. The problem is framed as a heat equation without a source term, with an open boundary condition at infinity. Clarification is sought on the system's arrangement at time t=0, specifically regarding a point-source diffusion scenario.
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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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