Skip to main content
Sandbox Physics
Active matter · A course with experiments

L3 · Chapter 4

Calculate when a density ripple grows

In the local, slowly varying approximation without positional Brownian diffusion, a uniform state is unstable when the logarithmic fall of swim speed with density is steeper than minus one. This is a long-wavelength linear test. It identifies growth from a small ripple. Determining the eventual coexistence densities requires the nonlinear problem.

Sandbox Physics ·

Write a current before drawing a phase boundary

Let density be the number of particles per area. Let the local swim speed depend only on this density, and keep the orientational persistence time constant. Consider length scales much larger than the persistence length, so headings relax faster than the density pattern changes.

The active part of the current has two contributions: ordinary spreading down a density gradient and accumulation where speed is lower. Substituting a speed that depends on density combines both into one collective diffusion coefficient. The last term is independent positional Brownian diffusion. The current is particle number crossing a unit line per unit time.

J=−D(ρ)∇ρ,D(ρ)=Dt+v(ρ)2τ2[1+ρdln⁡vdρ].\begin{aligned}\mathbf J&=-\mathcal D(\rho)\nabla\rho,\\\mathcal D(\rho)&=D_t+\frac{v(\rho)^2\tau}{2}\left[1+\rho\frac{d\ln v}{d\rho}\right].\end{aligned}

Ask what happens to one weak ripple

Conservation says density changes by minus the divergence of the current. Add a tiny sinusoidal perturbation to a constant mean density. To first order, the collective diffusion coefficient can be evaluated at that mean. A positive coefficient damps the ripple; a negative coefficient amplifies it.

With no positional Brownian diffusion, the sign changes when the bracket becomes negative. With finite positional diffusion, the active negative contribution must first overcome that positive term. At low swim speed, positional Brownian spreading can overwhelm the weakened active feedback.

δρ∝eσtcos⁡(kx),σ=−k2D(ρ0),Dt=0:ρ0dln⁡vdρ∣ρ0<−1\delta\rho\propto e^{\sigma t}\cos(kx),\qquad\sigma=-k^2\mathcal D(\rho_0),\qquad D_t=0:\quad\rho_0\frac{d\ln v}{d\rho}\bigg|_{\rho_0}<-1

Worked example

A solvable slowdown law

Choose an exponential speed law. The density scale in its denominator has the same units as density, so the exponential argument is dimensionless. The logarithmic derivative is simply minus density divided by that scale. With no Brownian contribution, a mean density one and a half times the scale is unstable in the local approximation.

At that density, suppose the active diffusivity before multiplying by the bracket is 0.2 length units squared per time unit. The bracket is minus 0.5. Adding a Brownian diffusivity of 0.03 gives a collective diffusivity of minus 0.07. A wave number of 0.5 inverse length units then has a predicted growth rate of 0.0175 inverse time units. These numbers define a separate analytic example. A nonlocal browser model must be analysed with its own speed law and smoothing kernel.

v(ρ)=v0e−ρ/ρ∗,D(ρ0)=0.03+0.2(1−1.5)=−0.07,σ=0.0175v(\rho)=v_0e^{-\rho/\rho_*},\quad\mathcal D(\rho_0)=0.03+0.2(1-1.5)=-0.07,\quad\sigma=0.0175

The smallest visible ripple depends on the box

A periodic box admits only wavelengths that fit inside it. A box of side length twenty has a smallest nonzero wave number of about 0.314 inverse length units. Doubling the box permits a wave number half as large. Even for a negative collective diffusivity, that longer mode initially grows four times more slowly in this local calculation.

Short wavelengths need the density-smoothing kernel or higher-gradient terms that the local approximation omitted. Those terms regularise the unphysical unlimited growth at high wave number. Record the kernel radius and time step alongside particle number and box length; otherwise a change called a finite-size effect might really be a change of model.

kmin⁡=2πL,ρ0=NL2k_{\min}=\frac{2\pi}{L},\qquad \rho_0=\frac{N}{L^2}

Compare records at fixed mean density

Increase box length by a factor of two and particle number by a factor of four. Keep the local interaction length, time step and all speed-law parameters fixed. Repeat for several independent seeds. Save the time series so that the growth process remains available.

For each record, count particles in equally sized physical cells and normalise by cell area. Compare density histograms, correlation lengths and the growth of selected Fourier amplitudes over the same physical time interval. State how a cluster is defined before reporting its largest fraction. A distance threshold chosen after looking at the picture can manufacture an apparent change.

The continuum criterion tests the slow, weak-ripple limit. Finite particle records test a specified stochastic implementation. Agreement is strongest when both the scale separation and numerical convergence are visible.

An experiment to think through

Predict, change, explain

  1. Predict the sign of the growth rate for a chosen local speed law before running it.
  2. Compare exported records at constant mean density, using the same density-smoothing length and several seeds.
  3. Separate a linear instability, a finite cluster event and macroscopic coexistence in your conclusion.
Watch swimmers gather

These Labs generate synthetic particle or continuum records. Their displayed model notes state the implemented dynamics and units.

Try explaining it yourself

1. For the exponential speed law, what happens exactly at the density scale if positional diffusion is nonzero?

Read an explained answer

The active bracket vanishes there, leaving positive positional diffusion. The weak ripple decays. The instability threshold shifts to a higher density.

2. Does the linear instability condition give the final dense-phase density?

Read an explained answer

No. It determines whether a small perturbation grows around a chosen uniform density. Coexistence requires the nonlinear constitutive law, interfaces and an appropriate selection calculation.

Sources to read alongside

  1. Cates & Tailleur, Motility-Induced Phase Separation, arXiv:1406.3533v1 (2014); Annual Review of Condensed Matter Physics 6 (2015)

    Sections 2–4: orientational persistence, accumulation in slow regions and the local density-instability criterion. The course derives its own worked examples from these stated assumptions.

  2. Fily & Marchetti, Athermal Phase Separation of Self-Propelled Particles with No Alignment, arXiv:1201.4847v2, Physical Review Letters 108, 235702 (2012)

    Primary numerical study of repulsive self-propelled disks without alignment. Its microscopic force law differs from the browser’s density-dependent motility model.

  3. Stenhammar et al., A continuum theory of phase separation kinetics for active Brownian particles, arXiv:1307.4373v2, Physical Review Letters 111, 145702 (2013)

    Source for how leading density-gradient terms extend a local motility description. The course’s weak-ripple example is a separate local-limit calculation.

FeedbackSomething broken, confusing, or missing on this page?Tell us
What kind of feedback?

Sent with this page’s address, its Lab ID, the site language and your device type. Please leave out personal details. Feedback is kept for up to 12 months.Privacy policy