coffe and neutrons
I was trying to describe how coffee particles get smaller as I grind them. The equation looked familiar from neutron transport. Both count particles leaving one state and arriving from other states.
counting fractures
Lets put mass on a grid $x_i=i\Delta x$, with spacing $\Delta x$ and positive integers $i$. For each mass we keep track of
- $N_i(t)$, the average number of particles of mass $x_i$ at time $t$
- $a_i$, how often one of those particles breaks, per unit time
- $b_{ij}$, the average number of pieces of mass $x_i$ made when one parent of mass $x_j$ breaks
In a short time $\Delta t$, a parent at $x_j$ breaks with probability approximately $a_j\Delta t$. So the change $\Delta N_i=N_i(t+\Delta t)-N_i(t)$ is
$$ \frac{\Delta N_i}{\Delta t}\simeq\sum_{j\gt i}a_jb_{ij}N_j-a_iN_i. $$
We gain pieces from larger masses, $j\gt i$. We lose pieces at $x_i$ when they break into smaller ones.
Now make the mass spacing small. Write $N_i\simeq n(x_i,t)\Delta x$ and $b_{ij}\simeq b(x_i\mid x_j)\Delta x$. $n(x,t)\,dx$ counts particles in $[x,x+dx]$. $b(x\mid y)\,dx$ counts the pieces in that interval from a broken parent of mass $y$.
As the time step and mass spacing go to zero, we get
$$ \partial_t n(x,t)=\int_x^\infty a(y)b(x\mid y)n(y,t)\,dy-a(x)n(x,t). $$
The integral starts at $x$ because a parent smaller than $x$ cannot produce a fragment of mass $x$.
the neutron version
Neutrons change energy and direction when they collide with nuclei. For scattering and absorption, without fission, the neutron transport equation with time dependence is
$$ \frac{1}{v(E)}\partial_t\psi+\Omega\cdot\nabla\psi=-\Sigma_t\psi+\int_0^\infty\!\int_{4\pi}\Sigma_s(E',\Omega'\!\to E,\Omega)\psi'\,d\Omega'\,dE'+q. $$
Here $\psi=\psi(\mathbf r,E,\Omega,t)$ is angular flux. It is neutron density per volume, energy and solid angle, multiplied by the speed $v(E)$. Solid angle measures a patch of directions. $\mathbf r$ is position, $E$ is energy and $\Omega$ is the unit vector giving travel direction. Primes mark incoming states, so $\psi'=\psi(\mathbf r,E',\Omega',t)$.
$\Sigma_t$ is the total collision probability per distance. $\Sigma_s$ gives the probability of scattering from $(E',\Omega')$ to $(E,\Omega)$ per distance, outgoing energy and solid angle. The double integral adds arrivals from all incoming energies and directions. $4\pi$ covers the full sphere of directions.
$\Omega\cdot\nabla\psi$ accounts for motion through space. $\nabla$ takes derivatives with respect to position. $q$ counts neutrons supplied by an external source per volume, time, energy and solid angle.
To compare rates per unit time, use neutron density $h=\psi/v(E)$. Then we can line up the terms
| Fracture | Neutrons | |
|---|---|---|
| Change in time | $\partial_t n$ | $\partial_t h$ |
| Loss rate | $a(x)$ | $v(E)\Sigma_t(E)$ |
| Gain kernel | $a(y)b(x\mid y)$ | $v(E')\Sigma_s(E',\Omega'\to E,\Omega)$ |
The fracture equation above only tracks changes in mass. The neutron equation also has motion through space and an external source. Neutrons can gain or lose energy to nuclei. A fragment has to be smaller than its parent.