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Sandbox Physics
Active matter · A course with experiments

L4 · Chapter 8

Read active turbulence through a spectrum and a material path

A spectrum tells us which length scales carry rotational motion; a tracer map tells us how fluid parcels separate over time. Use both viewpoints. This reading pins Giomi’s arXiv:1409.1555v2 and Serra and colleagues’ 2023 Nature Physics paper, and reproduces one theoretical spectrum shape from the former.

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Find the model before reading the curve

Open Giomi’s version 2, dated 3 August 2015. Equation (1) specifies a coupled nematic–fluid model; Section I.2 introduces the active length from elastic and active stresses. Its square-root scaling estimates where those stresses balance. It is a scale relation, not a universal coefficient for a biological sample.

Figure 5C–D shows numerical enstrophy and energy spectra at three activities. The following mean-field calculation describes their scaling. The mean-field construction treats vortices as uncorrelated circular objects with an exponentially distributed area. Our reproduction evaluates the normalised theoretical enstrophy curve in equations (9) and (14). The reproduction contains the calculated curve and its independent integral check.

ℓa∼K/∣α∣\ell_{\rm a}\sim\sqrt{K/|\alpha|}

Recalculate the spectrum by two routes

Enstrophy measures squared vorticity. Its spectrum distributes that quantity over wave number. Scale wave number by the characteristic vortex radius and divide out the normalisation constant. The resulting dimensionless curve can be computed from modified Bessel functions, or from an independent integral over vortex radii.

The plot below is calculated from that theoretical expression. The independent quadrature checks the special-function evaluation. Small wave numbers have a slope approaching positive one on logarithmic axes; large wave numbers approach minus two. The horizontal scale is dimensionless. Neither axis is an experimentally calibrated time or length.

Ω(κ)C=κe−κ2/2 ⁣[I0 ⁣(κ22)−I1 ⁣(κ22)]=4κ∫0∞t3e−t2J1(κt)2 dt.\begin{aligned}\frac{\Omega(\kappa)}{C}&=\kappa e^{-\kappa^2/2}\!\left[I_0\!\left(\frac{\kappa^2}{2}\right)-I_1\!\left(\frac{\kappa^2}{2}\right)\right]\\&=\frac4\kappa\int_0^\infty t^3e^{-t^2}J_1(\kappa t)^2\,dt.\end{aligned}
Recalculated dimensionless mean-field enstrophy curve; the low-wave-number slope approaches one and the high-wave-number slope approaches minus two.
Original calculation of the normalised Giomi mean-field curve, equations (9) and (14), with independent quadrature. This is a theoretical curve associated with Figure 5, with no copied or digitised simulation points.

Worked example

Use a simple flow to understand the tracer diagnostic

A flow map assigns each starting point its position after a chosen time interval. Its derivative describes how an initially small patch deforms. Multiply that derivative by its transpose in the correct order to form the right Cauchy–Green tensor. Its largest eigenvalue gives the square of the strongest local stretching factor.

For a planar area-preserving extension, one coordinate grows exponentially and the other shrinks at the same rate. The largest finite-time Lyapunov exponent equals the extension rate. This worked example has no defects and no active stress: it checks the kinematic diagnostic independently of the physics that generated the velocity.

u=(ax,−ay),FT(x0,y0)=(eaTx0,e−aTy0),C=(∇FT)T∇FT,ΛT=ln⁡λmax⁡(C)∣T∣=∣a∣.\begin{gathered}\mathbf u=(a x,-a y),\qquad F_T(x_0,y_0)=(e^{aT}x_0,e^{-aT}y_0),\\C=(\nabla F_T)^{\mathsf T}\nabla F_T,\qquad\Lambda_T=\frac{\ln\sqrt{\lambda_{\max}(C)}}{|T|}=|a|.\end{gathered}

Read the 2023 figures in material coordinates

Open Serra and colleagues, DOI 10.1038/s41567-023-02062-y, published 25 May 2023. Figure 1 distinguishes initial positions from final positions and sketches stretching and folding. Equations (1)–(3) define the flow map and finite-time stretching. Figure 2 applies the analysis to measured microtubule–kinesin velocities, using a 100-second interval for the illustrated forward and backward maps.

A forward map starts at initial positions and quantifies separation; a backward map starts at final positions and diagnoses convergence over the same interval. Compare labels before comparing ridges. The linked browser model supplies synthetic Q fields, velocities and tracers for a separate numerical experiment.

Choose the next reproducible comparison

To test the theoretical spectrum, keep its vortex assumptions and compare the two numerical evaluations. To test a continuum solver, use the manufactured sinusoidal force-to-flow example and refine the grid. To investigate defect control, compare matched trajectories and the chosen flux with and without anchoring. These checks answer different questions and belong beside the result they support.

For the next paper, write down the version, the figure and axis units, what was measured or simulated, and which parameters are specified. Choose a comparison you can calculate independently. Save the seed, initial field, numerical resolution and observation interval before interpreting a new pattern.

An experiment to think through

Predict, change, explain

  1. Predict how changing activity magnitude changes the active-length estimate while elasticity stays fixed.
  2. Compare the theoretical spectrum plot with its limiting slopes, then use the flow Lab to follow a defect and nearby tracers.
  3. State which part of your conclusion follows from a theoretical curve, which from synthetic trajectories and which belongs to the paper’s experimental evidence.
Follow defects and fluid together Find half-turn defects

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

Try explaining it yourself

1. Activity magnitude increases by a factor of four at fixed elasticity. What happens to the active-length estimate?

Read an explained answer

It decreases by a factor of two. This is the scaling estimate; a frictional model also has a screening length and need not preserve a one-length description.

2. Can matching the normalised theoretical spectrum certify that a Q-field solver reproduces the full 2015 turbulent simulation?

Read an explained answer

No. The curve calculation checks a mean-field expression under its own vortex assumptions. Full dynamical reproduction would require matching equations, coefficients, box, resolution, sampling and the paper’s numerical observables.

3. What is the finite-time stretching exponent of rigid planar rotation?

Read an explained answer

Zero. The flow-map derivative is an orthogonal rotation, so the Cauchy–Green tensor is the identity and all stretching factors are one.

Sources to read alongside

  1. Giomi, Geometry and topology of turbulence in active nematics, arXiv:1409.1555v2 (2015), Physical Review X 5, 031003

    Equation (1), Section I.2 and Figure 2 provide the continuum context and active-length scaling. Our screened-flow benchmark is a manufactured solution, separate from the paper’s turbulent simulations.

  2. Serra, Lemma, Giomi, Dogic & Mahadevan, Defect-mediated dynamics of coherent structures in active nematics, Nature Physics 19, 1355–1361 (2023)

    Figure 1 and equations (1)–(3) introduce the flow map and finite-time stretching; Figure 2 uses experimental microtubule–kinesin velocities. This is a guided reading, with no copied images or imported experimental dataset.

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