Anderson localization as a parametric instability of the linear kicked oscillator

We rigorously analyse the correspondence between the one-dimensional standard Anderson model and a related classical system, the `kicked oscillator' with noisy frequency. We show that the Anderson localization corresponds to a parametric instability of the oscillator, with the localization length determined by an increment of the exponential growth of the energy. Analytical expression for a weak disorder is obtained, which is valid both inside the energy band and at the band edge.Comment: 7 pages, Revtex, no figures, submitted to Phys. Rev.

The Gibbs Paradox Revisited

The Gibbs paradox has frequently been interpreted as a sign that particles of the same kind are fundamentally indistinguishable; and that quantum mechanics, with its identical fermions and bosons, is indispensable for making sense of this. In this article we shall argue, on the contrary, that analysis of the paradox supports the idea that classical particles are always distinguishable. Perhaps surprisingly, this analysis extends to quantum mechanics: even according to quantum mechanics there can be distinguishable particles of the same kind. Our most important general conclusion will accordingly be that the universally accepted notion that quantum particles of the same kind are necessarily indistinguishable rests on a confusion about how particles are represented in quantum theory.Comment: to appear in Proceedings of "The Philosophy of Science in a European Perspective 2009

1D quantum models with correlated disorder vs. classical oscillators with coloured noise

We perform an analytical study of the correspondence between a classical oscillator with frequency perturbed by a coloured noise and the one-dimensional Anderson-type model with correlated diagonal disorder. It is rigorously shown that localisation of electronic states in the quantum model corresponds to exponential divergence of nearby trajectories of the classical random oscillator. We discuss the relation between the localisation length for the quantum model and the rate of energy growth for the stochastic oscillator. Finally, we examine the problem of electron transmission through a finite disordered barrier by considering the evolution of the classical oscillator.Comment: 23 pages, LaTeX fil

Phenomenological approach to non-linear Langevin equations

In this paper we address the problem of consistently construct Langevin equations to describe fluctuations in non-linear systems. Detailed balance severely restricts the choice of the random force, but we prove that this property together with the macroscopic knowledge of the system is not enough to determine all the properties of the random force. If the cause of the fluctuations is weakly coupled to the fluctuating variable, then the statistical properties of the random force can be completely specified. For variables odd under time-reversal, microscopic reversibility and weak coupling impose symmetry relations on the variable-dependent Onsager coefficients. We then analyze the fluctuations in two cases: Brownian motion in position space and an asymmetric diode, for which the analysis based in the master equation approach is known. We find that, to the order of validity of the Langevin equation proposed here, the phenomenological theory is in agreement with the results predicted by more microscopic models.Comment: LaTex file, 2 figures available upon request, to appear in Phys.Rev.

Perturbation theory for a stochastic process with Ornstein-Uhlenbeck noise

The Ornstein-Uhlenbeck process may be used to generate a noise signal with a finite correlation time. If a one-dimensional stochastic process is driven by such a noise source, it may be analysed by solving a Fokker-Planck equation in two dimensions. In the case of motion in the vicinity of an attractive fixed point, it is shown how the solution of this equation can be developed as a power series. The coefficients are determined exactly by using algebraic properties of a system of annihilation and creation operators.Comment: 7 pages, 0 figure

Nonlinear Dynamics in Distributed Systems

We build on a previous statistical model for distributed systems and formulate it in a way that the deterministic and stochastic processes within the system are clearly separable. We show how internal fluctuations can be analysed in a systematic way using Van Kanpen's expansion method for Markov processes. We present some results for both stationary and time-dependent states. Our approach allows the effect of fluctuations to be explored, particularly in finite systems where such processes assume increasing importance.Comment: Two parts: 8 pages LaTeX file and 5 (uuencoded) figures in Postscript forma

The Fokker-Planck equation, and stationary densities

The most general local Markovian stochastic model is investigated, for which it is known that the evolution equation is the Fokker-Planck equation. Special cases are investigated where uncorrelated initial states remain uncorrelated. Finally, stochastic one-dimensional fields with local interactions are studied that have kink-solutions.Comment: 10 page

Continuous time dynamics of the Thermal Minority Game

We study the continuous time dynamics of the Thermal Minority Game. We find that the dynamical equations of the model reduce to a set of stochastic differential equations for an interacting disordered system with non-trivial random diffusion. This is the simplest microscopic description which accounts for all the features of the system. Within this framework, we study the phase structure of the model and find that its macroscopic properties strongly depend on the initial conditions.Comment: 4 pages, 4 figure

Pattern fluctuations in transitional plane Couette flow

In wide enough systems, plane Couette flow, the flow established between two parallel plates translating in opposite directions, displays alternatively turbulent and laminar oblique bands in a given range of Reynolds numbers R. We show that in periodic domains that contain a few bands, for given values of R and size, the orientation and the wavelength of this pattern can fluctuate in time. A procedure is defined to detect well-oriented episodes and to determine the statistics of their lifetimes. The latter turn out to be distributed according to exponentially decreasing laws. This statistics is interpreted in terms of an activated process described by a Langevin equation whose deterministic part is a standard Landau model for two interacting complex amplitudes whereas the noise arises from the turbulent background.Comment: 13 pages, 11 figures. Accepted for publication in Journal of statistical physic

On the inertia of heat

Does heat have inertia? This question is at the core of a long-standing controversy on Eckart's dissipative relativistic hydrodynamics. Here I show that the troublesome inertial term in Eckart's heat flux arises only if one insists on defining thermal diffusivity as a spacetime constant. I argue that this is the most natural definition, and that all confusion disappears if one considers instead the space-dependent comoving diffusivity, in line with the fact that, in the presence of gravity, space is an inhomogeneous medium.Comment: 3 page