Brownian vortexes

Mechanical equilibrium at zero temperature does not necessarily imply thermodynamic equilibrium at finite temperature for a particle confined by a static but nonconservative force field. Instead, the diffusing particle can enter into a steady state characterized by toroidal circulation in the probability flux, which we call a Brownian vortex. The circulatory bias in the particle’s thermally driven trajectory is not simply a deterministic response to the solenoidal component of the force but rather reflects interplay between advection and diffusion in which thermal fluctuations extract work from the nonconservative force field. As an example of this previously unrecognized class of stochastic heat engines, we consider a colloidal sphere diffusing in a conventional optical tweezer. We demonstrate both theoretically and experimentally that nonconservative optical forces bias the particle’s fluctuations into toroidal vortexes whose circulation can reverse direction with temperature or laser power.

Figure 1

(Color online) Schematic representation of a colloidal sphere undergoing toroidal circulation in an optical tweezer. (a) The Rayleigh limit, $a<\lambda$. (b) Larger spheres, $a>\lambda$.

Figure 2

(Color online) (a) Streamlines of the computed optical force calculated as tangent curves to $\mathit{F}\left(\mathit{r}\right)$ in the $\left(r,z\right)$ plane. This representation indicates the force’s direction but not its magnitude. Background images show $\rho \left(\mathit{r}\right)$ at $T=21°\text{C}$ and a laser power of 0.3 W. (b) Streamlines of the irrotational component of the force, $-\nabla U$. (c) Streamlines of the solenoidal component of the force, $\nabla \times \mathit{A}$.

Figure 3

Streamlines of the simulated probability flux for a sphere diffusing in the optical force field of Fig. 2. (a) Forward circulation at $P=0.7\text{W}$. (b) Counter-rotating vortexes at $P=0.5\text{W}$. (c) Complete flux reversal at $P=0.3\text{W}$. Images show the normalized vorticity of $\mathit{F}\left(\mathit{r}\right)$.

Figure 4

(Color online) Streamlines of the measured circulation of optically trapped silica spheres. (a) $k_{\perp }=6.56\text{pN}/\mu \text{m}$: single positive toroidal roll. (b) $k_{\perp }=2.43\text{pN}/\mu \text{m}$: concentric counter-rotating rolls. (c) $k_{\perp }=2.27\text{pN}/\mu \text{m}$: flux reversal. Single retrograde roll. Background images show measurements of $\rho \left(\mathit{r}\right)$ for each set of conditions.