PHYSICAL REVIEW E 89, 052131 (2014) Inertial effects in adiabatically driven flashing ratchets Viktor M. Rozenbaum, 1, 2, 3 , * Yurii A. Makhnovskii, 1, 4 Irina V. Shapochkina, 5 Sheh-Yi Sheu, 6 , † Dah-Yen Yang, 1 , ‡ and Sheng Hsien Lin 1, 2 1 Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 106, Taiwan 2 Department of Applied Chemistry, National Chiao Tung University, 1001 Ta Hsuen Road, Hsinchu, Taiwan 3 Chuiko Institute of Surface Chemistry, National Academy of Sciences of Ukraine, Generala Naumova Street 17, Kiev 03164, Ukraine 4 Topchiev Institute of Petrochemical Synthesis, Russian Academy of Sciences, Leninsky Prospect 29, 119991 Moscow, Russia 5 Department of Physics, Belarusian State University, Prospekt Nezavisimosti 4, 220050 Minsk, Belarus 6 Department of Life Sciences and Institute of Genome Sciences, Institute of Biomedical Informatics, National Yang-Ming University, Taipei 112, Taiwan (Received 19 December 2013; published 19 May 2014) We study analytically the effect of a small inertial correction on the properties of adiabatically driven flashing ratchets. Parrondo’s lemma [J. M. R. Parrondo, Phys. Rev. E 57, 7297 (1998)] is generalized to include the inertial term so as to establish the symmetry conditions allowing directed motion (other than in the overdamped massless case) and to obtain a high-temperature expansion of the motion velocity for arbitrary potential profiles. The inertial correction is thus shown to enhance the ratchet effect at all temperatures for sawtooth potentials and at high temperatures for simple potentials described by the first two harmonics. With the special choice of potentials represented by at least the first three harmonics, the correction gives rise to the motion reversal in the high-temperature region. In the low-temperature region, inertia weakens the ratchet effect, with the exception of the on-off model, where diffusion is important. The directed motion adiabatically driven by potential sign fluctuations, though forbidden in the overdamped limit, becomes possible due to purely inertial effects in neither symmetric nor antisymmetric potentials, i.e., not for commonly used sawtooth and two-sinusoid profiles. DOI: 10.1103/PhysRevE.89.052131 PACS number(s): 05.40.−a, 05.60.Cd, 82.20.−w, 87.16.Nn I. INTRODUCTION Nonequilibrium fluctuations can cause a net particle drift along periodic structures lacking reflection symmetry, even without any macroscopic bias force. This phenomenon is known as the ratchet effect [1–3]. Predictive model systems for such nonequilibrium transport, called ratchets or Brownian motors, have been discussed in various contexts, especially in relation to studies of the physics of protein motors [4,5] and nanoscale artificial engines [3,6]. Systems of this kind are usually driven out of equilibrium either by pulsing the potential (flashing ratchets) [7–9] or by rocking it back and forth (rocking ratchets) [10,11]. Most of works on the ratchet effect consider Brownian motion in the overdamped regime, where friction dominates inertia, so the finiteness of the mass is neglected. In many cases, such as in biological applications, this approximation can be well justified on physical grounds [2,12]. However, finite inertia effects are important in many experimental situations and have attracted some recent attention [3,13–20]. For example, particle masses are of significance in experiments on microscopic particle separation [3]. In thermal ratchets, the decay of the massive particle current at short noise-correlation times behaves in a more complicated manner than previously reported in the overdamped limit and the direction of transport may depend on the particle mass [14]. Rocking ratchets are well studied in the overdamped limit [10–12,21,22] and also in the opposite limit, where inertial effects are strong and * vik-roz@mail.ru † sysheu@ym.edu.tw ‡ dyyang@pub.iams.sinica.edu.tw can lead to motion reversal as well as to chaotic dynamics in the absence of noise (inertia ratchets) [13,15,16]. With the appropriate noise level, the inertia ratchets provide efficient mass-sensitive scenarios for particle separation [3,17,18]. As shown in Ref. [19], a small inertia correction always decreases the particle mobility in a stationary periodic potential, but enhances the rocking ratchet effect at high temperatures. The inertial effects in the flashing mechanism have been less frequently investigated, but their significance can be seen from the following observation: Directed motion induced by shifting dichotomic fluctuations of a symmetric potential is impossible in the overdamped regime [23,24] but occurs in the underdamped regime, as numerical simulations have shown [20]. The goal of the present paper is to study analytically the influence of finite inertia in the particle dynamics on the flashing ratchet mechanism. In order to make the problem analytically tractable, we invoke two simplifying assumptions: (i) The inertia corrections are small and (ii) the potential variation is adiabatic. These simplifications allow us to stress essential physical points made in this paper. While the overdamped Brownian motion in a potential is governed by the Smoluchowski equation [25], containing no information about the particle mass and velocity, the rigorous treatment of a Brownian particle with finite mass involves time evolution of the particle probability in the phase space, which satisfies the much more complicated Klein-Kramers equation [26,27], analytically solvable only in a few special cases [28]. So to handle inertial effects in the particle dynamics, one usually invokes approximate approaches. The most well known and commonly used is the high-friction expansion of the Klein-Kramers equation [28–31]. In this way, the Klein- Kramers equation can be reduced to a Smoluchowski-like 1539-3755/2014/89(5)/052131(9) 052131-1 ©2014 American Physical Society