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I am reading a Gardiner's Stochastic Methods handbook and I am wondering about the meaning of the following (this is the very beginning of the chapter):

$dn = n \phi(\Delta) d \Delta$

This is arrived at like so (looking at the particles suspended in a liquid):

There are $n$ particles suspended in a liquid. We choose time, $\tau$ at which the moves of the single particle are independent of its own moves at the previous time step $\tau$. Moves of all particles are also independent of each other. Introduce quantity $\Delta$, which is the move in the $x$ direction at time $\tau$ (this quantity can be either positive or negative). There is a certain frequency to these moves, so that the number $dn$ of particles that experience a move in the range $[\Delta, \Delta + d\Delta]$ is the equation that I have written initially.

I look at the $\phi(\Delta)$ as the probability that a particle experienced a move of size $\Delta$ then the number of particles that experienced such a move is expected to be $n\phi(\Delta)$ and I am not sure where the $d\Delta$ comes in there. Also, $\Delta$ is already a change in the $x$ direction of the position of the particle, then $d\Delta$ is like a change of the change? So the whole confusion I am experiencing comes from that last $d\Delta$ in that first equation.

Naz
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1 Answers1

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If you chose an integer between 1 and 10, the probability for each is 1/10.

Now chose a real value between 1 and 10, what is the probability for 3? It is zero! You can compute the probability for your number to be between 2.9 and 3.1 though, this has a clear meaning.
So to describe this, you use a probability density. Then the probability for the above interval is $0.2\cdot\phi(3)$. This is an approximation in so far, as the probability density is changing over the interval, to be precise you would have to integrate.

That's what is happening in your formula. $\Delta$ describes a change, but is otherwise a normal variable. To get the probability of a particle to be in an interval $\mathrm d\Delta$ around $\Delta$ you take $\phi(\Delta)\mathrm d\Delta$. Multiplying by $n$ gives the number of particles in that interval.

Ilja
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