Engineering ratchet-based particle separation via extended shortcuts to isothermality

Microscopic particle separation plays a vital role in various scientific and industrial domains. Conventional separation methods relying on external forces or physical barriers inherently exhibit limitations in terms of efficiency, selectivity, and adaptability across diverse particle types. To overcome these limitations, researchers are constantly exploring new separation approaches, among which ratchet-based separation is a noteworthy method. However, in contrast to the extensive numerical studies and experimental investigations on ratchet separation, its theoretical exploration appears weak, particularly lacking in the analysis of energy consumption involved in the separation processes. The latter is of significant importance for achieving energetically efficient separation. In this paper, we propose a nonequilibrium thermodynamic approach, extending the concept of shortcuts to isothermality, to realize controllable separation of overdamped Brownian particles with low energy cost. By utilizing a designed ratchet potential with temporal period $\tau$, we find in the slow-driving regime that the average particle velocity ${\overline{v}}_s\propto (1-D/D^{*}){\tau }^{-1}$, indicating that particles with different diffusion coefficients $D$ can be guided to move in distinct directions with a preset $D^{*}$. It is revealed that an inevitable portion of the energy cost in separation depends on the driving dynamics of the ratchet, with an achievable lower bound $W_{\mathrm{ex}}^{\left(\min \right)}\propto {\mathcal{L}}^2\left|{\overline{v}}_s\right|$. Here, $\mathcal{L}$ is the thermodynamic length of the driving loop in the parametric space. With a sawtooth potential, we numerically test the theoretical findings and illustrate the optimal separation protocol associated with $W_{\mathrm{ex}}^{\left(\min \right)}$. Finally, for practical considerations, we compare our approach with the conventional ratchets in terms of separation velocity and energy consumption. The scalability of the current framework for separating various particles in two-dimensional space is also demonstrated. This paper bridges the gap between thermodynamic process control and particle separation, paving the way for further thermodynamic optimization in ratchet-based particle separation.