Relationship between density and probability in diffusion

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Homework Help Overview

The discussion revolves around the relationship between the density of a substance (ink) and its probability distribution in the context of diffusion, specifically within a water vase. The original poster is uncertain about how to approach the problem and seeks clarification on the connection between density and probability as they relate to diffusion processes.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the nature of the probability being discussed, with some questioning what it means for a particle to be at a precise location. There is an attempt to clarify the concept of probability density in relation to Brownian movement.

Discussion Status

The discussion is ongoing, with participants providing insights and seeking clarification on the definitions and implications of probability in the context of diffusion. Some guidance has been offered regarding the interpretation of probability density functions in relation to Brownian motion.

Contextual Notes

There is a lack of explicit information regarding the setup of the problem, particularly in terms of how the volume is conceptualized in relation to the particles of dye. The original poster indicates that the question is preparatory for deriving and solving the diffusion equation.

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Homework Statement



Consider the diffusion of a drop of ink in a water vase. The density of the ink is ## \rho (\vec{r}, t) ##, and the probability ##P(\vec{r}, t)## obeys the diffusion equation. What is the relationship between ##\rho (\vec{r}, t)## and ##P(\vec{r}, t)##?

Homework Equations

The Attempt at a Solution



I think the probability is proportional to the density, but I am not sure in what way. I don't understand how to approach this problem.
 
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The probability of what?
 
The probability of a particle for being at a position ##\vec{r}## at time ##t##.
 
It's me said:
The probability of a particle for being at a position ##\vec{r}## at time ##t##.
What does it mean to say that there is a particle of dye at a precise point in space? Is there some preamble to the question, telling you to consider the volume as made up of little boxes, each of which contains either one particle of dye or none?
 
Not really, it just says that the probability is that of Brownian movement and we are asked to find that relationship as preparation to derive and solve the diffusion equation.
 
It's me said:
the probability is that of Brownian movement
I still don't think that means anything. How about, it's the probability density function in a Brownian movement?
 
Okay yes that makes sense.
 

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