Coupled Brownian motors: Anomalous hysteresis and zero-bias negative conductance

– We introduce a model of interacting Brownian particles in a symmetric, periodic potential that undergoes a noise-induced non-equilibrium phase transition. The associated spontaneous symmetry breaking entails a ratchet-like transport mechanism. In response to an external force we identify several novel features; among the most prominent being a zero-bias negative conductance and a prima facie counterintuitive, anomalous hysteresis. The subject of this letter lies at the borderline of three topics of current interest. The first —Brownian motors— deals with converting unbiased non-equilibrium fluctuations into useful work, mostly by exploiting the asymmetry of some underlying “ratchet” potential [1]. Proposed as a new paradigm for directed transport in cellular structures and technological applications, their collective behavior, which is our focus here, is clearly of paramount importance [2]. Second, coupled phase oscillators are presently under intense discussion as simple models for the ubiquituous synchronization phenomena in complex biological [3] and physico-chemical systems [4-6]. We will demonstrate here the possibility of spontaneous rotations due to the mere presence of noise and concomitant, quite unexpected response to external forces. Finally, noise-induced phase transitions have been introduced as a spontaneous symmetry breaking caused by non-equilibrium fluctuations in otherwise completely symmetric and monostable systems [7]. This mechanism presents another key ingredient of the present investigation. Model. – Our starting point is a set of N coupled stochastic equations of motion ẋi = −U ′ i(xi, t) + √ 2Tξi(t)−N −1 ∑N j=1 K(xi − xj) (1) in properly scaled units. In doing so we have neglected inertia effects, whereas an important role is played by thermal fluctuations, modeled by the temperature T and the independent δ-correlated Gaussian noises ξi(t). The last term in (1) is assumed to derive from an interaction c © EDP Sciences 546 EUROPHYSICS LETTERS potential, implying K(−x) = −K(x) (actio=reactio). The potentials Ui(x, t) consist of a static part V (x), a fluctuating part W (x), driven by non-thermal noise ηi(t) with strength Q, and possibly an additional bias or “load force” F : Ui(x, t) = V (x) +W (x) √ 2Qηi(t)− F x . (2) The potentials V (x) and W (x), as well as the interactionK(x) are supposed to be periodic with period L, but unlike in common “ratchets” [1], we furthermore assume symmetric potentials with V (−x) = V (x) and W (−x) = W (x). Our standard example will be K(x) = K0 sinx , V (x) = W (x) = − cosx−A cos 2x (3) with K0 > 0 and L = 2π, corresponding to a “flashing” [1] or “pulsating” potential in (2). Such potential fluctuations may either be externally imposed (in applications), or mimic internal degrees of freedom far from thermal equilibrium (in complex biological systems). For simplicity, we focus on Gaussian white noises ηi(t) of zero mean and unit strength, independent of each other and all the ξi(t), and of negligible correlation time (Stratonovich interpretation [8]). Models of the form (1,2) arise in the context of molecular motors [2], active rotator systems [4], charge density waves [5], planar XY spin models [6], and many others [3]. While for molecular motors [2] a later extension to asymmetric potentials is of prominent interest, in all other contexts [3-6], however, the symmetric case is of foremost interest. The global coupling in (1) and the periodicity of K(x) are chosen to make the problem analytically tractable. We will first discuss in detail this idealized model and afterwards address numerical generalizations for nearest-neighbor coupling (of physical relevance, e.g., in XY spin models), non-periodic K(x), and asymmetric potentials. The main collective features of (1,2) are captured by the particle density ρ(x, t) := ∑ j δ(xj(t) − x)/N . Two of them are of particular interest, namely the particle current 〈ẋ〉 := ∂t ∫ xρ(x, t) dx and the reduced density P (x, t) := ∑ n ρ(x+nL, t) in the thermodynamic limit N → ∞. Being an intensive quantity, P (x, t) becomes independent of the specific realization of the noises ξi(t) and ηi(t) when N → ∞ [9]. Equation (1) then yields for the reduced density P (x, t) the non-linear Fokker-Planck equation [8,9] ∂tP (x, t) = ∂x {−f(x, t) + g(x)∂xg(x)}P (x, t) , (4) f(x, t) := −V (x) + F − ∫ L/2 −L/2 K(x− y)P (y, t) dy , g(x) := [T +QW (x)] (5) with ∫ L/2 −L/2 P (x, t)dx = 1 and P (x+ L, t) = P (x, t). For the particle current 〈ẋ〉 one obtains