Sorting by Resetting

Abstract

A novel paradigm for sorting is introduced, based upon resetting. Using simple examples, we demonstrate that sorting is achieved by resetting the velocity component(s) or orientation of the particles, rather than position. The objects to be sorted are microparticles, modeled as suspended and spatially extended Brownian particles. This sorting-by-resetting scheme illustrates that stochastic resetting can create non-equilibrium conditions which enable tasks forbidden at thermodynamic equilibrium.

Figures (5)

• Figure 1: Illustration of the three different scenarios. (i) Tracer particles of various shapes with fixed orientation in an ideal gas. (ii) Chiral and other colloidal particles suspended in an aqueous solution. (iii) Underdamped spherical Brownian particles of different mass (due to different sizes or densities) in a one-dimensional asymmetric potential landscape.
• Figure 2: (a) Relaxation of the lowest order moments for the rectangular shape object shown in the inset ($\varepsilon = \sqrt{1/20} \approx 0.22$), in response to (re-)setting $v_x=0$ and $v_y=0$ at time $t=0$. The inset shows the relaxation of the second moments. Solid curves: theoretical predictions according to Eqs. \ref{['eq:moments']}. Dots: Simulation results, obtained as an average over $2 \times 10^7$ independent realizations (per data point). (b)-(c) Average displacement velocity of the same triangle as a function of the resetting period $\tau$. Solid lines: theoretical predictions according to Eq. \ref{['eq:tracer:V']}. Dots with standard error bars: simulation results obtained as an average over 500 realizations (per data point). Parameters: $S=1$, $k_{\mathrm{B}} T=1$, $\rho=1$; the time unit is the mean free time between particle-object collisions, i.e. the time values can also be read as the average number of collisions.
• Figure 3: Separation of four different kinds of tracer particles (see legend): shown are the positions of 50 particles per tracer species after a time $5\times10^{7}$; all particles started at the origin at time 0. Upper panel: resetting period $\tau = 10$. Lower panel: resetting period $\tau = 30$. Parameters: $S=1$ (for all species), $k_{\mathrm{B}} T=1$, $\rho=1$; the time unit is the mean free time between particle-object collisions.
• Figure 4: (a) Separation of four differently shaped colloids (see legend). The dots are the positions of 100 colloids per species after time 90000. All particles start at the origin at time 0 and experience the same resetting protocol: orientation is reset along the $x$ axis (this is the orientation shown in the legends) in periodic intervals $\tau = 0.1$. The colloidal objects are assembled from beads of unit diameter as illustrated in the legend. Parameters: thermal energy $k_{\mathrm{B}} T = 4$, viscosity of the solution $\nu = 1$ (this quantity enters the calculation of the friction tensors, see Appendix). When choosing water at room temperature as a solution for the colloidal particles, these parameter values correspond to units of seconds for time scales and of micrometers for length scales. (b) The average displacement velocity (given in Eq. \ref{['eq:colloidal:V']}) for the shown particle as a function of the reset period $\tau$.
• Figure 5: Separation of spherical particles in a spatially asymmetric potential. (a) Average velocity as a function of $\tau$ for four different particle species, combining two different particle densities $\rho$ with two different particle radii $R$ (see legend). The dots are numerical data obtained from averaging over 20000 independent realizations per data point, lines are a guide to the eye, error bars (not shown) are about the symbol size; the spatial component is given in units of the period length $L$. (b) Histograms of the distribution of 500 particles per species after running the resetting protocol with $\tau=0.7$ (dashed line in (a)) for a total time of $5\times10^7$. Other parameters: $U_0=8$, $L=0.5$, $k_{\mathrm{B}} T=4$; the particle mass is $m=(4\pi/3) \rho R^3$, and the friction coefficient $\gamma=6\pi\nu R$ with the viscosity $\nu=10$. Translating these dimensionless parameters into dimensional quantities, one obtains length scales of $\mu$m, time scales of ms, $k_{\mathrm{B}} T$=4 corresponds to the thermal energy at room temperature, $\rho=1$ corresponds to the density of water, and $\nu=10$ corresponds to 1% of the viscosity of water.