= Linear-gradient isopycnic focusing
{title2=$z_*=(\rho_b-\rho_0)/G$}
Why does an <isopycnic centrifugation> band return after displacement? Choose a dilute <mathematical model> with outward coordinate $z$, constant local <centrifugal acceleration> $a>0$, fixed displaced <volume> $V>0$, <linear friction coefficient> $f>0$, and an imposed liquid <density> $\rho_f(z)=\rho_0+Gz$ with $G>0$. Treat the species' <buoyant density> $\rho_b$ as constant in this medium. The <sedimentation coefficient> force balance gives the <ordinary differential equation>
$$
\frac{dz}{dt}=\frac{Va}{f}\bigl(\rho_b-\rho_f(z)\bigr)=-k(z-z_*),\qquad k=\frac{VaG}{f},\qquad z_*=\frac{\rho_b-\rho_0}{G}.
$$
Its <exponential function> solution is $z(t)=z_*+(z(0)-z_*)e^{-kt}$. The sign of the <velocity> reverses across $z_*$, proving <stable equilibrium> when $z_*$ lies inside the liquid. Changing $V/f$ changes the approach time $1/k$, while $z_*$ depends on <buoyant density> and the imposed <density gradient>. A real <centrifuge rotor> has radially varying <centrifugal acceleration>; the constant-$a$ approximation describes a short radial interval. Here the <density gradient> is held fixed, rather than modeling its formation.
For an original <dimensionless> example, let $u=z/L$ and $\tau=t/T$, where $L$ is column length and $T$ a chosen time scale. Write $q=z_*/L$ and $\lambda=kT$, so $u(\tau)=q+(u(0)-q)e^{-\lambda\tau}$. Species A and B have the same $q=0.45$ but different friction-dependent rates: $(\lambda_A,u_A(0))=(0.4,0.1)$ and $(\lambda_B,u_B(0))=(1.6,0.95)$. A moves outward and B inward, both toward $u=0.45$. Species C has $(q_C,\lambda_C,u_C(0))=(0.8,0.8,0.1)$ and approaches a separate band at $u=0.8$. At $\tau=4$, their positions are approximately $0.3793$, $0.4508$ and $0.7715$: a finite-time photograph would still show A short of its limiting position.
\Image[/isopycnic-bands-and-restoring-trajectories.png]
{title=<Isopycnic centrifugation>: different approach rates can lead to the same <buoyant density> band}
{description=Original schematic tubes and deterministic trajectories for the stipulated <linear-gradient isopycnic focusing> model. A and B share a matching <density> position; C has another. Tube shading represents an imposed increasing <density gradient>. Band thickness is schematic; <diffusion> is omitted from the plotted trajectories. The final tube shows the mathematical limit while rotation continues, not a measured result or a stopped tube.}
An equilibrium band has finite width because <diffusion> opposes focusing. In the same local model, let $c(z)$ be <concentration> and let the constant <diffusion coefficient> be $D>0$. Combining the restoring drift with <Fick's laws>, zero net <flux> requires
$$
J=-k(z-z_*)c-D\frac{dc}{dz}=0,\qquad c(z)=c(z_*)\exp\!\left[-\frac{k(z-z_*)^2}{2D}\right].
$$
Dividing by $c$ and integrating proves the <Gaussian function> profile, whose <standard deviation> is $\sigma_z=\sqrt{D/k}$. This local result assumes that the band is narrow relative to its distance from the boundaries and that $D$, $k$ and the <density gradient> vary negligibly across it. Equal <buoyant densities> therefore imply coincident centers in this model, not identical widths. A band outside the available <density> range has no such interior restoring point.
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