Conductivity of two-dimensional systems of interacting Brownian particles within the effective potential description

Abstract

We investigate the dynamic properties of two-dimensional systems composed of N interacting Brownian particles subject to a periodic potential. By employing the Langevin dynamic simulations and assuming a repulsive Yukawa potential type, we have computed the effective potential V eff. Our results show that for the incommensurate concentration c=2/3, the barrier height of V eff increases in one direction and decreases in the other compared with the that for the non-interacting particles case. This behaviour explains quantitatively the aspect of the diffusion process which becomes unidirectional. So, the two-dimensional diffusion process can be treated in one-dimensional picture. On the other hand, we have computed the frequency-dependent conductivity σ( ω), of the two-dimensional system with Frenkel–Kontorova pair interaction in the intermediate friction regime, by using the continued fraction expansion method (CFEM). For the determination of the remainder of the development of the continued fraction we need to know an analytical expression of the DC conductivity σ(0), by other means. We are then led to derive an expression of σ(0), in the intermediate friction regime. Moreover, this new expression allows us to recover the two known results computed in the low and high friction limits. Its behaviour is compared with that obtained by Langevin dynamics. We have found a very good agreement between the two results. By varying the density of mobile ions we discuss commensurability effects. For commensurate cases, the resonance frequency of the conductivity is determined only by the external potential.