Dynamics of a Particle Moving in a Two Dimensional Lorentz Lattice Gas
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Dynamics of a Particle Moving in a Two Dimensional Lorentz Lattice Gas Pranay Bimal Sampat1
· Sameer Kumar1 · Shradha Mishra1
Received: 5 February 2020 / Accepted: 1 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study the dynamics of a particle moving in a square two-dimensional Lorentz lattice-gas. The underlying lattice-gas is occupied by two kinds of rotators, “right-rotator (R)” and “leftrotator (L)” and some of the sites are empty viz. vacancy “V”.The density of R and L are the same and density of V is one of the key parameters of our model. The rotators deterministically rotate the direction of a particle’s velocity to the right or left and vacancies leave it unchanged. We characterise the dynamics of particle motion for different densities of vacancies. Since the system is deterministic, the particle forms a closed trajectory asymptotically. The probability of the particle being in a closed or open trajectory at time t is a function of the density of vacancies. The motion of the particle is uniform throughout in a fully occupied lattice. However, it is divided in two distinct phases in partially vacant lattices: The first phase of the motion, which is the focus of this study, is characterised by anomalous diffusion and a power-law decay of the probability of being in an open trajectory. The second phase of the motion is characterised by subdiffusive motion and an exponential decay of the probability of being in an open trajectory. For lattices with a non-zero density of vacancies, the first phase of motion lasts for a longer period of time as the density of vacancies increases. Keywords Lorentz lattice gas · Diffusion · Anomalous diffusion · Dynamics · Random walk · Deterministic walk
Communicated by Abhishek Dhar.
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Pranay Bimal Sampat [email protected] Sameer Kumar [email protected] Shradha Mishra [email protected]
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Department of Physics, Indian Institute of Technology (BHU), Varanasi, U.P 221005, India
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P. B. Sampat et al.
1 Introduction Most native species, like microscopic particles, encounter random obstacles in the environment in which they move, i.e. they move in an inhomogeneous environment [1–3]. These obstacles or the environment directly influence the motion of the microscopic particles and can be modeled by a pattern of barriers/obstacles in the medium, which control the motion of a particle moving in it. Depending on the characteristics of the environment, motion can be of different kinds, e.g. anomalous diffusion or confined motion [4,5]. Previous studies have addressed the action of the particle(s) in an inhomogeneous environment and its effect on particle dynamics and steady state [6,7]. The Lorentz lattice gas (LLG) [8–10] has turned out to be a useful way to model different types of inhomogeneous environment. Several physical phenomena have been studied using the Lorentz lattice gas [11–17], in one and two dimensions. In [8], the authors have studied a fixed obstacle model, where the properties
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