L3 · Chapter 4

Where the jamming point is, and how pressure grows past it

Sandbox Physics · 2026-10-29

Start from a random arrangement and slowly grow the disks. At some moment they begin to push on each other. That packing fraction is this start’s jamming threshold. Another random start gives a slightly different threshold, but the more disks there are, the tighter the thresholds cluster; large systems nearly all jam at one packing fraction. Past the threshold the pressure grows from zero, in a way set only by the contact law.

How to find one packing’s threshold

The protocol of O’Hern and co-workers: scatter centres at random in a periodic cell, a configuration at infinite temperature, then find a local minimum of the energy. If the energy per disk is below $10^{-16}\,\varepsilon$ , the disks do not overlap and the packing is not jammed; otherwise it is.

They minimised with conjugate gradients; we use FIRE 2.0. Both find the same kind of state: a configuration with every force balanced. If it is not jammed, grow every disk by the same factor; if it is, shrink them; halve the step each time the direction switches, until the step is below one part in a hundred thousand. The packing fraction where the search stops is that start’s threshold.

Source: C. S. O’Hern · J. Guénolé

Thresholds form a distribution that narrows with size

Every random start gives one threshold. Fig. 5 of O’Hern et al. plots the fraction of starts that are jammed against packing fraction; its derivative is the threshold distribution of Fig. 6. In large systems the full width at half maximum falls as in Eq. (3), with $w_0=0.16\pm0.04$ and $\Omega=0.55\pm0.03$ from Fig. 7.

We took 24 random seeds each for 32, 64, 128 and 256 disks and estimated the full width as the sample standard deviation times $2\sqrt{2\ln2}$ , which assumes a Gaussian shape. The widths are 0.0259, 0.0148, 0.0129 and 0.00654, all inside the range Eq. (3) allows with its uncertainties. The mean thresholds are 0.827, 0.835, 0.838 and 0.839, rising with size toward the large-system value near 0.84 in Fig. 5(a).

As the number of disks grows without bound, the width goes to zero: nearly every random start jams at the same packing fraction. That is why the jamming point has a well-defined location.

w=w_0\,N^{-\Omega},\qquad w_0=0.16\pm0.04,\quad \Omega=0.55\pm0.03

Eq. (3) of O’Hern et al., for large systems only. They note that with very few particles the width first grows with size and only follows this power law past its maximum.

Source: C. S. O’Hern

How pressure grows past the threshold

Write $\Delta\phi=\phi-\phi_c$ . Just past the threshold a typical overlap is proportional to $\Delta\phi$ , because the extra area has to be absorbed by overlaps. Chapter 1 gave $f\propto\delta^{\alpha-1}$ . The contact number barely changes near the threshold, and the pressure of Chapter 3 is a sum of force times distance, so $p\propto\Delta\phi^{\alpha-1}$ . Harmonic contacts give a linear rise, Hertzian contacts a 3/2 power.

Table I of O’Hern et al. gives, for 2D bidisperse systems, $\psi=1.01$ and $p_0=0.34$ (harmonic) and $\psi=1.50$ and $p_0=0.27$ (Hertzian). With 128 disks and 8 seeds, fitting $\Delta\phi$ from 0.001 to 0.1, we get $1.039\pm0.021$ with $p_0=0.35$ (harmonic) and $1.52\pm0.02$ with $p_0=0.233$ (Hertzian). The exponents reproduce; the Hertzian prefactor falls below Table I’s uncertainty.

p=p_0\,(\phi-\phi_c)^{\psi},\qquad \psi=\alpha-1

Table I quotes ±0.05 on

p_0

and ±0.03 on

\psi

. The argument above fixes only the exponent; the prefactor has to be computed.

Pressure against compression past each packing’s own threshold, both axes logarithmic. Dots: mean of 8 seeds of 128 disks, recomputed by this course. Dashed lines: the Table I fits of O’Hern et al.,

0.34\,\Delta\phi^{1.01}

for harmonic contacts (cyan) and

0.27\,\Delta\phi^{1.50}

for Hertzian contacts (amber), drawn as a separate reference.

Source: C. S. O’Hern

The contact-number scaling did not reproduce

Table I also gives $Z-Z_c=Z_0\,\Delta\phi^{\zeta}$ , with $\zeta=0.49$ and $Z_0=3.6$ for 2D harmonic disks. At 128 disks we get $\zeta=0.577\pm0.009$ and $Z_0=4.41$ , clearly steeper.

Goodrich, Liu and Nagel explain the trouble with small systems: when $p^{1/2}N$ is of order one, the square-root law for the contact number fails. We fitted only data with $p^{1/2}N\ge3$ and the exponent is still too steep. At 128 disks the exponent did not reproduce, and checking it needs larger systems.

Source: C. S. O’Hern · C. P. Goodrich

Worked example: one percent past the threshold

Question: for 2D harmonic bidisperse disks squeezed to $\Delta\phi=0.01$ , roughly what are the pressure and the mean contact number?

From Table I: $p=0.34\times0.01^{1.01}=3.2\times10^{-3}\,\varepsilon/d^2$ and $Z=3.98+3.6\times0.01^{0.49}=3.98+0.38=4.36$ .

Compare with our numbers. Squeezing 12 seeds each of 64, 128 and 256 disks in periodic cells to $\phi_c+0.01$ gives mean pressures of $(2.62\pm0.22)\times10^{-3}$ , $(2.73\pm0.13)\times10^{-3}$ and $(2.45\pm0.23)\times10^{-3}$ . They sit about 15 % to 25 % below the Table I fit and do not drift systematically with size. Table I’s 2D values come from fits to 1024-disk runs across a range of compressions, so a gap of this size at one point is within reason. Read the Table I prefactor as an estimate good to about 20 %.

Source: C. S. O’Hern

Ruling out false jamming by crystals

Disks of one size readily form a hexagonal lattice and jam at a much higher packing fraction. We give each disk a local six-fold order and call it crystal-like above 0.85. At 0.01 past the threshold, crystal-like disks make up 14.5 % of a bidisperse packing on average and 21.1 % at most; for equal disks they make up 64.6 % on average, and the mean threshold rises to 0.859.

A bidisperse packing with more than 35 % crystal-like disks is flagged as crystallised and left out of the statistics. No benchmark seed crossed that line. The Lab’s same-size preset exists to show this counter-example.

Source: C. S. O’Hern

Try it in the Lab

Compare the harmonic and Hertzian onsets

Predict first: just past the onset, which contact law gives the flatter pressure curve?
Play the default preset and see whether the pressure curve leaves zero as a straight line or bends upward. Then choose the Hertzian preset and compare the two curves near the onset.
Choose the same-size preset, read its onset packing fraction and mean local six-fold order, compare them with the default, and say why the benchmark statistics exclude crystals.

Exercises with explained answers

Going from 0.01 to 0.02 past the threshold, by what factor does the pressure grow for harmonic and for Hertzian contacts?
Show the answer
With the Table I exponents, about $2^{1.01}=2.01$ for harmonic and $2^{1.50}=2.83$ for Hertzian contacts. The prefactor $p_0$ cancels in the ratio.
By Eq. (3), roughly how many disks are needed for the full width of the threshold distribution to fall below 0.001?
Show the answer
$N=(w_0/w)^{1/\Omega}=160^{1/0.55}\approx1.0\times10^{4}$ . Given the uncertainties of $w_0$ and $\Omega$ , the answer could be off by a factor of a few, so “about ten thousand” is all one can say.
In a periodic cell of 32 disks, roughly where will one random start’s threshold fall? Does one such run tell you $\phi_c$ for a large system?
Show the answer
We measured a mean of 0.827 with a standard deviation of 0.011, so a single result falls roughly between 0.81 and 0.84. It sits systematically below the large-system value near 0.84, and its scatter is about as large as that offset. The large-system $\phi_c$ has to come from larger systems.

Sources and what we recomputed

C. S. O’Hern, L. E. Silbert, A. J. Liu and S. R. Nagel, Jamming at zero temperature and zero applied stress: the epitome of disorder, arXiv:cond-mat/0304421v1; Physical Review E 68, 011306 (2003)
Version 1, 17 April 2003. Eq. (1): the finite-range repulsion between soft spheres. Section II: random infinite-temperature starts, conjugate-gradient minimisation in a periodic cell, and the energy test for an unjammed state. Figs. 5 to 7 and Eq. (3): the fraction of jammed states, the distribution of thresholds and its width, which falls as the number of particles to the power −0.55. Table I: pressure and contact-number exponents and prefactors. Fig. 20: a 2D force network. We recomputed the threshold width, the pressure exponent and the contact-number exponent with our own code; the contact-number exponent did not reproduce at 128 disks.
C. P. Goodrich, A. J. Liu and S. R. Nagel, Finite-size scaling at the jamming transition, arXiv:1204.4915v3; Physical Review Letters 109, 095704 (2012)
Version 3, 6 June 2012. Eq. (2): the contact count needed to remove every zero mode except global translations. Eq. (3): one more contact is needed for a positive bulk modulus, so a periodic packing of N disks in two dimensions needs at least 2N − 1 contacts. Eqs. (4) and (5): the corresponding minimum contact number. The finite-size crossover where the usual power laws break down at low pressure. We checked the 2N − 1 count on our own threshold packings.
J. Guénolé, W. G. Nöhring, A. Vaid, F. Houllé, Z. Xie, A. Prakash and E. Bitzek, Assessment and optimization of the fast inertial relaxation engine (FIRE) for energy minimization in atomistic simulations and its implementation in LAMMPS, arXiv:1908.02038v4 (2020)
Version 4, 30 January 2020. Algorithm 2: FIRE 2.0 with a semi-implicit Euler step and the half-step correction after an uphill move. Table 1: the default parameters we use. The relaxation method only; nothing about jamming.