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Register a new userTime-shift invariance determines the functional shape of the current in dissipative rocking ratchets
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Added by
Renato Alvarez-Nodarse
Publication date
2013
fields
Physics
and research's language is
English
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Ratchets are devices able to rectify an otherwise oscillatory behavior by exploiting an asymmetry of the system. In rocking ratchets the asymmetry is induced through a proper choice of external forces and modulations of nonlinear symmetric potentials. The ratchet currents thus obtained in systems as different as semiconductors, Josephson junctions, optical lattices, or ferrofluids, show a set of universal features. A satisfactory explanation for them has challenged theorist for decades, and so far we still lack a general theory of this phenomenon. Here we provide such a theory by exploring ---through functional analysis--- the constraints that the simple assumption of time-shift invariance of the ratchet current imposes on its dependence on the external drivings. Because the derivation is based on so general a principle, the resulting expression is valid irrespective of the details and the nature of the physical systems to which it is applied, and of whether they are classical, quantum, or stochastic. The theory also explains deviations observed from universality under special conditions, and allows to make predictions of phenomena not yet observed in any experiment or simulation.
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Equations describing the evolution of particles, solitons, or localized structures, driven by a zero-average, periodic, external force, and invariant under time reversal and a half-period time shift, exhibit a ratchet current when the driving force breaks these symmetries. The bi-harmonic force $f(t)=epsilon_1cos(q omega t+phi_1)+epsilon_2cos(pomega t+phi_2)$ does it for almost any choice of $phi_{1}$ and $phi_{2}$, provided $p$ and $q$ are two co-prime integers such that $p+q$ is odd. It has been widely observed, in experiments in Josephson-junctions, photonic crystals, etc., as well as in simulations, that the ratchet current induced by this force has the shape $vproptoepsilon_1^pepsilon_2^qcos(p phi_{1} - q phi_{2} + theta_0)$ for small amplitudes, where $theta_0$ depends on the damping ($theta_0=pi/2$ if there is no damping, and $theta_0=0$ for overdamped systems). We rigorously prove that this precise shape can be obtained solely from the broken symmetries of the system and is independent of the details of the equation describing the system.
We extended our formulation of causal dissipative hydrodynamics [T. Koide textit{et al.}, Phys. Rev. textbf{C75}, 034909 (2007)] to be applicable to the ultra-relativistic regime by considering the extensiveness of irreversible currents. The new equation has a non-linear term which suppresses the effect of viscosity. We found that such a term is necessary to guarantee the positive definiteness of the inertia term and stabilize numerical calculations in ultra-relativistic initial conditions. Because of the suppression of the viscosity, the behavior of the fluid is more close to that of the ideal fluid. Our result is essentially same as that from the extended irreversible thermodynamics, but is different from the Israel-Stewart theory. A possible origin of the difference is discussed.
We study a general model of granular Brownian ratchet consisting of an asymmetric object moving on a line and surrounded by a two-dimensional granular gas, which in turn is coupled to an external random driving force. We discuss the two resulting Boltzmann equations describing the gas and the object in the dilute limit and obtain a closed system for the first few moments of the system velocity distributions. Predictions for the net ratchet drift, the variance of its velocity fluctuations and the transition rates in the Markovian limit, are compared to numerical simulations and a fair agreement is observed.
Motivated by recent work [D. Cubero et al., Phys. Rev. E 82, 041116 (2010)], we examine the mechanisms which determine current reversals in rocking ratchets as observed by varying the frequency of the drive. We found that a class of these current reversals in the frequency domain are precisely determined by dissipation-induced symmetry breaking. Our experimental and theoretical work thus extends and generalizes the previously identified relationship between dynamical and symmetry-breaking mechanisms in the generation of current reversals.
We investigate the properties of time reversibility of a soliton gas, originating from a dispersive regularization of a shock wave, as it propagates in a strongly disordered environment. An original approach combining information measures and spin glass theory shows that time reversal focusing occurs for different replicas of the disorder in forward and backward propagation, provided the disorder varies on a length scale much shorter than the width of the soliton constituents. The analysis is performed by starting from a new class of reflectionless potentials, which describe the most general form of an expanding soliton gas of the defocusing nonlinear Schroedinger equation.
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