# Diffusion equation question in 1D?

• magicuniverse
In summary, the solution to the diffusion equation in 1D is given by n'(x,t) = N/sqrt(4piDt) * exp(-x^2/4DT), where n'(x,t) is the concentration of particles at position x at time t, N is the total number of particles, and D is the diffusion coefficient. An expression for the number of particles in a slab of thickness dx at position x is given by integrating the first function between x and x+dx. The probability that a particle is in the slab is equal to P(x)dx, where P(x) is the normalized probability density function. The mean square displacement of a particle is 2Dt. When sketching the solution for
magicuniverse

## Homework Statement

The solution to the diffusion equation in 1D may be written as follows:

n'(x,t) = N/sqrt(4piDt) * exp(-x^2/4DT)

where n'(x,t) is the concentration of the particles at position x at time t, N is the total number of particles and D is the diffusion coefficient.

a) Write down an expression for the number of particles in a slab of thickness dx located at position x.

b) What is the probability that a particle is in the slab?

c) If the particle is in the slab, what distance has it traveled from the origin?

d) Show that the mean square displacement of a particle is 2Dt.

e) Sketch the form of the solution to the diffusion equation for two times t1 and t2 where t2 = 4t1. How will the width of the curves be related at these two times?

f) Consider a 1D random walk where the particle had equal possibilities of moving from left to right. For a journey of 3 steps, a particular sequence of steps might be, for example RRR. Write down all possible sequences for a yourney of 3 steps.

g) What is the total number of journeys possible for a walk of N steps?

h) Write down an expression for the number of journeys in which exactly NL steps are taken to the left if the total number of steps is N. If the random walker starts at the origin, after 4 steps what is the probability that the particle has returned to the origin?

## Homework Equations

At the start N.A. as I go through the question I know that I need standard integrals and the combinations formula but ill raise these when needed.

## The Attempt at a Solution

I am struggling with much of this but can do some of it. I think the most sensible thing to so it to go through it in order.

So I don't know how to integrate the first function between x+dx and x! Any ideas? thanks

magicuniverse said:
So I don't know how to integrate the first function between x+dx and x! Any ideas? thanks

The integral of any function f(x) between x and x+dx is nothing but f(x)dx.

Hope this starts you off.

Note that the integrale of any function f(x) between x and x +dx is simply f(x) dx!

\int_x^{x+dx} f(x') dx' \approx f(x) dx

So = N/sqrt(4piDt) * exp(-x^2/4DT)dx ?

If that's right I don't know how to do the sencond part. I know that since expx^2 term the distirbution is gaussian and that it can be normalised to 1. However what do i get for an answer?

magicuniverse said:
I know that since expx^2 term the distirbution is gaussian and that it can be normalised to 1. However what do i get for an answer?

If after normalization the PDF is P(x), then the reqd probability is again nothing but P(x)dx.

## 1. What is the diffusion equation in 1D?

The diffusion equation in 1D is a mathematical model that describes the process of diffusion in one-dimensional systems. It is a partial differential equation that relates the rate of change of a diffusing substance to its concentration gradient and diffusion coefficient.

## 2. What is the physical meaning of the diffusion equation?

The diffusion equation represents the physical process of particles moving from areas of high concentration to areas of low concentration. This is known as diffusion, and it occurs due to random molecular motion.

## 3. How is the diffusion equation solved in 1D?

The diffusion equation can be solved using various methods, including analytical methods such as separation of variables and numerical methods such as finite difference methods. The appropriate method depends on the boundary and initial conditions of the system.

## 4. What are the applications of the diffusion equation in 1D?

The diffusion equation has many applications in various scientific fields, such as physics, chemistry, biology, and engineering. It is commonly used to model diffusion processes in gases, liquids, and solids, and it has applications in areas such as heat transfer, mass transfer, and chemical reactions.

## 5. How does the diffusion coefficient affect the diffusion process?

The diffusion coefficient is a measure of how easily a substance can diffuse in a given medium. A higher diffusion coefficient means the substance can diffuse more quickly, while a lower diffusion coefficient means it will diffuse more slowly. Therefore, the diffusion coefficient plays a crucial role in determining the rate of diffusion in a system.

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