An issue with the distribution function

Question

I am reading a book about Boltzmann equation, here is a quotation:

For a gas of $N$ particles, the number of particles having velocities in the $x$ direction between $c_x$ and $c_x + \mathrm dc_x$ is $Nf(c_x)\mathrm dc_x$. The function $f(c_x)$ is the fraction of the particles having velocities in the interval $c_x$ and $c_x + \mathrm dc_x$; in the x-direction.

I am really confused, why the number of particles having velocities in $c_x$ and $c_x+\mathrm dc_x$ is given by $Nf(c_x)\mathrm dc_x$? why do we multiply by $\mathrm dc_x$?

Can you explain please?

Thank you

score 0 · Answer 1 · answered Jun 26 '16 at 07:50

0

This is roughly a first order approximation for the distribution of velocities. Strictly speaking, the fraction of particles with velocities in the interval is $\int_{c_x}^{c_x+dc_x}f(c)dc$. But if $dc_x$ is small and $f$ is roughly continuous, then one can assume that $f$ is constant on the interval to obtain $$\int_{c_x}^{c_x+dc_x}f(c)dc\approx\int_{c_x}^{c_x+dc_x}f(c_x)dc=f(c_x)dc_x.$$ Then multiply by $N$ to turn this fraction into a quantity of particles.

If this seems imprecise, it's because you're reading a book about physics and not math!

answered Jun 26 '16 at 07:50

Funktorality

3,221

But the fraction of particles (if it is the average) then it will be $f(c_x)=\dfrac{\int_{c_x}^{c_x+dc_x}f(c)dc}{\int_{c_x}^{c_x+dc_x}dc}$ – Syntax_ErrorX00 Jun 26 '16 at 07:55
@Syntax_ErrorX00 I'm having trouble reading that equation. Have you studied probability density functions? – Funktorality Jun 26 '16 at 07:57
but the book states that the fraction of particles is $f(c_x)$ not $f(c_x)dc_x$ – Syntax_ErrorX00 Jun 26 '16 at 08:14
They're not explaining their notation very precisely. You can see that $f(c)$ does not give the fraction itself since then it wouldn't depend on the choice of $dc_x$. This is the confusion that arises when authors try to use PDFs but avoid any mention of calculus. – Funktorality Jun 26 '16 at 08:22
what do you mean by PDF? does it mean the PDF file? – Syntax_ErrorX00 Jun 26 '16 at 08:25
Sorry, I mean probability density function, which is what the authors probably intend with $f$. – Funktorality Jun 26 '16 at 08:27
anyway, that was really a great answer, thank you again :) – Syntax_ErrorX00 Jun 26 '16 at 08:29
Let us continue this discussion in chat. – Funktorality Jun 26 '16 at 09:16

An issue with the distribution function

1 Answers1