Negative friction memory induces persistent motion
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THE EUROPEAN PHYSICAL JOURNAL E
Regular Article
Negative friction memory induces persistent motion Bernhard G. Mitterwallnera , Laura Lavacchi, and Roland R. Netz Fachbereich Physik, Freie Universit¨ at Berlin, 14195 Berlin, Germany Received 25 May 2020 / Received in final form 6 September 2020 / Accepted 30 September 2020 Published online: 23 October 2020 c The Author(s) 2020. This article is published with open access at Springerlink.com Abstract. We investigate the mean-square displacement (MSD) for random motion governed by the generalized Langevin equation for memory functions that contain two different time scales: In the first model, the memory kernel consists of a delta peak and a single-exponential and in the second model of the sum of two exponentials. In particular, we investigate the scenario where the long-time exponential kernel contribution is negative. The competition between positive and negative friction memory contributions produces an enhanced transient persistent regime in the MSD, which is relevant for biological motility and active matter systems.
1 Introduction If the dynamics of a diffusing particle is coupled to other degrees of freedom, memory effects occur, and the particle dynamics becomes non-Markovian [1,2]. Examples include the diffusion of a tracer bead in viscoleastic [3–6] and heterogeneous [7, 8] media, polymer dynamics [9–13] and dynamics in rough energy landscapes [14–16]. Furthermore, many systems far from equilibrium, such as self-propelled particles [17–19] or passive tracer particles in active media [20–22], exhibit non-Markovian dynamics. The random motion of a diffusing particle is commonly characterized by the mean-square displacement (MSD) 2 (1) CMSD (t) = (x(t) − x(0)) . Diffusive properties may be classified in terms of the timedependent MSD exponent α(t): d ln CMSD (t) . α(t) = d ln t
0
(2)
Normal diffusion (Brownian motion) yields an exponent α = 1. Processes for which α = 1 are broadly referred to as anomalous diffusion, or more specifically, subdiffusion for α < 1 and superdiffusion for α > 1. Much effort has been directed towards modeling anomalous diffusion especially for cases in which the exponent stays anomalous in the long-time limit, for recent reviews see [23, 24]. In many instances, however, the dynamics becomes effectively Markovian at long time scales, which leads to normal diffusion, i.e. α = 1, for t → ∞. The MSD of an underdamped particle (or a reaction coordinate with a a
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non-vanishing effective mass) is ballistic, i.e. persistent, at short times (α = 2). The crossover between the persistent and the diffusive regimes generally depends on details of the non-Markovian dynamics, which are relevant at short and intermediate time scales [25]. The generalized Langevin equation (GLE) offers a convenient framework for the description of non-Markovian dynamics by introducing a memory term in the equation of motion. It has been successfully applied to passive microrheology [4–6], the modeling of (bio-)molecular systems [14, 26–30]
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