# The growth rate

The growth rate is the one parameter that decides what kind of packing the
Lubachevsky-Stillinger algorithm produces. This page defines it, explains why it is
defined this way, and gives guidance for choosing it.

## Definition

$$
\Gamma = \frac{1}{v_\mathrm{th}}\,\frac{\mathrm{d}\langle d\rangle}{\mathrm{d}t},
\qquad v_\mathrm{th} = \sqrt{k_BT/m},
$$

where $\langle d\rangle$ is the arithmetic mean diameter of the spheres and
$v_\mathrm{th}$ the thermal speed: the standard deviation of one Cartesian velocity
component. $\Gamma$ is a ratio of two speeds, so it is dimensionless and independent of
the units of length.

**Physical meaning.** While a sphere moving at the thermal speed travels one mean
diameter, the mean diameter grows by the fraction $\Gamma$. Equivalently, during a
compression that starts from points, a sphere moving at $v_\mathrm{th}$ travels
$1/\Gamma$ final mean diameters. At $\Gamma = 0.02$ that is 50 diameters: enough
collisions for the spheres to rearrange locally, too few for them to find a crystal.

The mean speed is $\langle|\mathbf v|\rangle = \sqrt{8/\pi}\,v_\mathrm{th} \approx 1.60\,v_\mathrm{th}$
in 3D and $\sqrt{\pi/2}\,v_\mathrm{th} \approx 1.25\,v_\mathrm{th}$ in 2D. To express
the growth rate relative to the mean speed, divide $\Gamma$ by these factors.

## Why this definition

- **It contains both the size and the speed.** The behaviour of the algorithm depends
  on how far spheres travel, in diameters, while they grow. A growth rate in units of
  1/time (as in the legacy code) only means something together with the particle size
  and the velocity scale; $\Gamma$ combines them.
- **It enters the collision rule directly.** For equal spheres every contact distance
  grows at speed $\Gamma v_\mathrm{th}$, which is the minimum separation speed after a
  collision (see {doc}`algorithm`).
- **It is the same in 2D and 3D.** The normal approach speed of colliding spheres has
  the same distribution in both dimensions when measured in $v_\mathrm{th}$; measured
  in the mean speed it would not.
- **The arithmetic mean diameter is the natural reference.** All spheres have the same
  mass, so their speeds do not depend on their size, and the contact distance of a
  random pair, $a_i + a_j$, grows on average at $\mathrm{d}\langle d\rangle/\mathrm{d}t$.
  Every diameter grows in proportion: sphere $i$ grows at
  $\Gamma\,v_\mathrm{th}\,d_i/\langle d\rangle$.

## Choosing the growth rate

Measured with spheropack: 1000 spheres in a periodic box, packed until jammed
(`density="max"`, default `Jammed()` stop, default collision rule), mean of 5 seeds.
The crystalline fraction uses {func}`spheropack.analysis.crystalline`; "contacts" is
the number of contacts between force-bearing spheres relative to the isostatic number
({func}`spheropack.analysis.isostaticity`). The data are in
`docs/data/growth_rate_sweep.json` and are regenerated by
`benchmarks/sweep_growth_rate.py`.

| $\Gamma$ | equal spheres: $\phi_J$ | crystalline | contacts | 10% polydisperse: $\phi_J$ | collisions per sphere |
|---|---|---|---|---|---|
| 0.3 | 0.637 | 0.2% | 0.956 | 0.639 | 3000 |
| 0.1 | 0.634 | 0.4% | 0.953 | 0.636 | 3300 |
| 0.03 | 0.640 | 0.7% | 0.960 | 0.642 | 3700 |
| 0.01 | 0.644 | 1.2% | 0.975 | 0.647 | 5800 |
| 0.003 | 0.648 | 2.7% | 0.985 | 0.651 | 13600 |
| 0.001 | 0.650 | 4.1% (up to 14%) | 0.990 | 0.652 | 35500 |

The standard deviation of $\phi_J$ between seeds is about 0.001. The run time is
proportional to the number of collisions: about 12 s for the fastest and 2 minutes
for the slowest equal-sphere runs on one core.

What this shows:

- **Jammed packings are dense for every growth rate.** With the default collision
  rule the jamming density of equal spheres only varies between 0.634 and 0.650.
  Slower growth gives denser packings, closer to isostatic: the spheres have more
  time to rearrange before they jam. This agrees with the known dependence of the
  jamming density on the compression rate of frictionless spheres.
- **Crystallisation needs very slow growth.** At $\Gamma = 10^{-3}$ only a few percent
  of 1000 equal spheres are crystalline; nucleation becomes likely only for slower
  growth or larger systems, and the density then rises towards the close packing
  limit $\pi/\sqrt{18} \approx 0.7405$. Polydispersity of 10% suppresses
  crystallisation completely in this range.
- **Loose random packings are not jammed packings.** Packings with a density well
  below 0.64 are obtained with a target density (`density=0.58`), which stops the
  growth before jamming. The legacy collision rule (`collision_rule="legacy"`) jams
  at lower densities when growth is fast (about 0.60 at $\Gamma = 0.16$ with
  `stop=Pressure(1e6)`), because it lets clusters of spheres lock prematurely; use it
  only to reproduce results of the legacy code.

Practical advice:

- The default $\Gamma = 0.02$ gives random close packing at a moderate cost. Use
  0.003 to 0.01 for denser, more nearly isostatic packings, and check the
  crystalline fraction for large systems of equal spheres.
- Equal disks in 2D crystallise easily. For random 2D packings use a mixture, for
  example 50:50 with size ratio 1.4.
- For a target density below jamming a larger growth rate is faster, but the
  structure then deviates more from an equilibrium hard-sphere fluid at that
  density. Grow slowly (0.01 or less) when an equilibrium structure matters.

## Relation to the legacy code

The legacy command-line tools took a growth rate $g$ in units of 1/time, with radii
$r_i(t) = r_i\,g t/2$ and velocity components of unit variance. This corresponds to
$\Gamma = g\,d/2$, with $d$ the final diameter. The settings of the legacy notebooks
($g = 0.32/d$) correspond to $\Gamma = 0.16$.