Chapter 4

# Random walk

Take 10 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.

Because of the thermal motion, the particles in a gas or a liquid have broken lines trajectories.

Starting from the initial point O at , a particle makes moves ( to ).

After each collision, it goes towards another direction, independent from the previous one.

## Question

What is the average distance between O and the particle after ( is huge), where is the average time between to following collisions ?

Express with , (free mean path) and (mean square velocity).

### Solution

After moves , the particle is in :

So, in modulus :

And, as the definition of the free mean path states :

Furthermore, for really huge, the random characteristic of scattering makes cosinus equally distributed between and , so :

It remains, with  :

So :

## Question

Numerical application : estimate the time required by a fragrance to diffuse after the opening to a distance ; comment on it.

### Solution

Numericale application : .

Scattering phenomenons are really slow.

In reality, the molecules are heavier than air, so is smaller and is longer, and here is the interest of convection.

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Scattering of peak concentrations
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To test the understanding of the lesson