# Heat transfer and diffusion of particles

# Probabilistic approach of scattering

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

In a cylindrical pipe between and , neutrons are apportioned at with an integer on discrete abscissas with an integer.

Between the moments and , each neutron has the probabilty to disappear.

If it doesn't, it will have the same probability to spring towards one or the other neighbour site, located on its right or left side.

## Question

We will note the probability for a neutron to be located in at .

Express using and .

### Hint

### Solution

A neutron is, at time in point . It was, at time , in point or in point . It can also disappear between and .

Thus, the probability can be written as :

## Question

We will consider the medium as continuous.

Demonstrate that is the solution of such a partial derivative equation :

and express with the information in the statement of the problem.

Of which partial derivative equation is the neutron linear density a solution ?

### Hint

### Solution

We can write (approximation of continuous medium) :

Thus :

Let be the total number of neutrons. The number of neutrons at and at the point is and the number of neutrons located between and is (there are sites) :

Thus :

And, finally, the neutron linear density has to satisfy the same differential equation as :

## Question

Let's suppose now that the material acquires a stationary neutron flux at .

Determine in stationary mode.

### Hint

### Solution

In stationary mode :

The solution is :

With :

This solution has to be symmetrical, so and :

can be determined using the stationary neutron flux at :

Where is the transverse area of the cylindrical pipe.

Finally :