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.

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

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

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

The integral starts at $x$ because a parent smaller than $x$ cannot produce a fragment of mass $x$.

Neutrons change energy and direction when they collide with nuclei. For scattering and absorption, without fission, the neutron transport equation with time dependence is

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

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.