Generalised Fick’s laws

For clarity, we first revise the case of constant diffusion D~ in one spatial dimension, z. Then, Fick’s first law relates gradients in concentration φ(z, t) and an external drift v_d to the total flux,

Together with the continuity equation, ∂_tφ = −∂_zJ_z, this gives Fick’s second law,

which is also known as the Fokker–Planck equation. This is equivalent to motion described by the following Langevin equation, dzdt=vd+2D~η(t), where η is Gaussian white noise as defined below Eq. (³⁹).

Rather than being constant in space, the diffusivity of tracers near an active carpet depends continuously on position. Such stochastic processes can often be described by an effective Smoluchowski equation⁷⁸, rather than standard Langevin methods which make no reference to individual collisions. Here, we follow this approach as a foundation for the generalised Fick’s laws that describe the diffusion of a tracer particle as a function of distance from an active carpet. Since we only have gradients in the vertical direction, we use the short-hand notations D(z) = D_zz for the vertical diffusivity and v(z)=⟨vz2⟩=Vzz for the mean vertical speed.

With this spatial dependence, the question arises whether the second term in Eq. (⁵⁵) should be interpreted as D∂_zφ or ∂_z(Dφ), or something in between. This question is not well posed, because it depends on the physical processes in question. In other words, generalising this expression for macroscopic quantities requires partial knowledge of the microscopic mechanism for diffusion. In particular, information is needed about the spatial dependence of the memory time τ(z), and of the mean vertical speed, v(z). The diffusivity can then be written as the combination of these ingredients, D(z) = v²(z)τ(z), as shown in Eq. (⁵³). Indeed, either a larger average speed or a longer time between reorientations can give rise to a larger diffusivity. Note that for Ornstein–Uhlenbeck forcing the time scale τ does not depend on position, but this need not be true in general. For example, for rapidly moving actuators the smallest decorrelation time is τ ~ z/V. On the other hand, for slowly moving actuators the rotational diffusion constant D_r sets the smallest memory time.

Using this information about the microscopic interactions, a ‘telegraph model’⁷⁸ can be constructed that describes the space-dependent diffusion process. Inspired by this model, we consider two particle populations, with population densities φ_up(z, t) and φ_down(z, t), respectively, that either move up or down along z with mean speed v(z) due to the active carpet fluctuations. The mean speed is the same for the two populations at any given z since we consider a uniform carpet without net drift, ⟨v⟩=0, as shown in Eq. (³²). The particles switch directions at a mean rate of 12τ−1, set by the memory time of the active carpet. The total particle density is then given by φ(z, t) = φ_up + φ_down, and the diffusive flux of particles is J_diff(z, t) = v(φ_up − φ_down). Subsequently, using continuity of particle flow and conservation of particle number, the up and down populations evolve according to

In each equation, the first term describes spatial gradients in the moving populations, while the last two terms describe the switching between particles moving up and down, and vice versa. By adding and subtracting, the equations (⁵⁷) can be rewritten in terms of the total density and diffusive flux, giving

Taking the time derivative of the first expression and combining with the second expression yields

Then, we assume that the high-frequency behaviour of particle movements can be neglected, so the second time derivative on the left-hand side vanishes,

Integrating this expression, we can solve for the diffusive flux J_diff. The constant of integration is set equal to zero to ensure that J_diff vanishes when the variance of the active fluctuations v² are zero. If the particles are sedimenting with a constant drift velocity v_d, there is an additional flux J_drift = v_dφ, so the total flux is J_z = J_diff + J_drift. Then, combining all this information gives the first generalised Fick’s law in the vertical direction,

We find the other components of the flux by repeating the telegraph model analysis in the x and y directions, which gives

Since the variance tensor only has diagonal components, we can write the generalised flux in three dimensions as

which is written in the main text as Eq. (⁶). Note that the last term only has a vertical component for systems that obey translational invariance along the horizontal directions, when the variance tensor only depends on z. Finally, using the continuity equation we obtain the second generalised Fick’s law,

where repeated indices are summed over.