Integral related to particle diffusion In the context of particle diffusion, I am trying to understand the equations that describe Brownian motion as a macroscopic process.
Assume $N(x,t)$ is a number concentration and $D$ is a diffusion constant, then we can write:
$$\frac{\partial N(x,t)}{\partial t} = D \frac{\partial^2}{\partial x^2} N(x,t)$$
next, we can multiply by the mean square displacement $x^2$ and integrate over $x$ from $-\infty$ to $\infty$:
$$\int x^2 \frac{\partial N(x,t)}{\partial t} dx = \int x^2 D \frac{\partial^2}{\partial x^2} N(x,t) dx $$
Now the script that I'm reading writes that the left-hand side can be written as:
$$\int x^2 \frac{\partial N(x,t)}{\partial t} dx = N_0 \frac{\partial \langle x^2 \rangle}{\partial t} $$
where I assume that $N_0$ is the initial number concentration of particles and $\langle \rangle$ is the mean square displacement.
I do not understand the last equation. Could somebody explain why I can express the integral as the partial derivative wrt to time of the mean square displacement (multiplied by $N_0$).
EDIT: $\langle x^2 \rangle$ is the mean square displacement (contrarily to what I have written earlier)
 A: Since the comment answered your question I'll just go ahead and set out a more generalised version. It's straightforward to simplify things back down to your case. Consider the following continuity equation:
$$ \dot{N}(x,t) = -\nabla\cdot\vec{\Gamma}(x,t) + S(x,t), $$
where $\vec{\Gamma}(x,t)$ is the flux (in your case $\vec{\Gamma}(x,t)= -D\nabla N(x,t)$, you can easily extend this) and $S(x,t)$ is a source term representing net creation/destruction of particles (in your case $S=0$). Multiplying by $x^n$ and integrating:
$$ \int \mathrm{d}x\ x^n\dot{N}(x,t) = -\int \mathrm{d}x\ x^n\nabla\cdot\vec{\Gamma}(x,t) + \int \mathrm{d}x\ x^nS(x,t). $$
If the domain of integration is constant:
$$ \int \mathrm{d}x\ x^n\dot{N}(x,t) = \frac{\mathrm{d}}{\mathrm{d}t} (N(t)\langle x^n \rangle), $$
where $N(t)$ is the total number of particles in the domain of integration. Taking the domain of integration as all space we find:
$$\begin{array}{lrcl}
n=0: & \frac{\mathrm{d}}{\mathrm{d}t} N(t) &=& S(t), \\
n=1: & \frac{\mathrm{d}}{\mathrm{d}t} (N(t)\langle x \rangle) &=& -\int \mathrm{d}x\ x\nabla\cdot\vec{\Gamma}(x,t) + \int \mathrm{d}x\ x S(x,t) = \int \mathrm{d}x\ x S(x,t),
\end{array}$$
etc. The first equation gives conservation of particles. The second can be rearrange using the first to give
$$ \frac{\mathrm{d}\langle x \rangle }{\mathrm{d}t} = \frac{1}{N(t)} \left(\int \mathrm{d}x\ x S(x,t) - S(t) \langle x \rangle \right) = \langle\frac{x S(x,t)}{N(x,t)}\rangle - \frac{S(t) \langle x \rangle }{N(t)}, $$
which is the centre of mass velocity, which is not zero, or even conserved in general, if there is a source.
You can see you get a tower of moment equations, essentially converting a partial differential equation into a sequence of ordinary differential equations. In your case this doesn't really gain you anything, but in more complicated situations (where you have to use, e.g. the Boltzmann equation) this technique is essential for extracting useful information about a system.
