Introduction to the physics of Brownian motors

A detailed introduction to directed transport in Brownian motors occurring in spatially periodic systems far from equilibrium is presented. We elucidate the prominent physical concepts and novel phenomena with a representative dissipative Brownian motor dynamics. Its main ingredient is a thermal noise with time-dependent temperature modulations that drive the system out of thermal equilibrium in a spatially asymmetric (ratchet-) potential. Yet, this asymmetric setup does not exhibit a concomitant obvious bias into one or the other direction of motion. Symmetry conditions for the appearance (or not) of directed current, its reversal upon variation of certain parameters, and various other generic features and applications are discussed. In addition, we provide a systematic classification scheme for Brownian motor models and review historical landmark contributions to the field. Received: 9 November 2001 / Accepted: 14 January 2002 / Published online: 22 April 2002     

Exclusive charge exchange reaction pD↦n(pp) within the Bethe-Salpeter formalism

The exclusive charge exchange reaction pD↦n(pp) at intermediate and high energies is studied within the Bethe-Salpeter formalism. The final-state interaction in the detected pp pair at nearly zero excitation energy is described by the ¹ S ₀ component of the Bethe-Salpeter amplitude. Results of numerical calculations of polarization observables and differential cross-section persuade that, as in the non-relativistic case, this reaction i) can be utilized as a “relativistic deuteron polarimeter” and ii) delivers further information about the elementary nucleon-nucleon charge exchange amplitude. Received: 26 November 2002 / Accepted: 30 January 2003 / Published online: 29 April 2003     

Numerical study of chaotic oscillations in the electron beam with virtual cathode in the external non-uniform magnetic fields

In this Letter the results of theoretical investigations of the chaotic microwave oscillator based on the electron beam with a virtual cathode are presented. Nonlinear non-stationary processes in these electron systems are studied by means of numerical analysis of 2.5D model. It was discovered that the non-uniform external magnetic field value controls the dynamical regime of oscillations in the virtual cathode oscillator. The processes of the chaotization of output microwave radiation are described and interpreted from the point of view of the formation and interaction of electron structures (bunches) in the electron beams. The numerical results have shown that the investigated electron system with virtual cathode could be considered as a promising controlled source of wideband chaotic oscillations in the microwave range.     

Optical properties of inhomogeneous composite materials

Using the condition that the forward scattering amplitude S(0) = 0, we extend the treatment of composite media to optical frequencies and to a model system of metal island spheres suspended in a dielectric. Our results are in good agreement with experimental data from cermets of Au in SiO₂ for volume fractions from 10 to 80% Au in SiO₂.     

The Möbius function of separable and decomposable permutations

We give a recursive formula for the Möbius function of an interval [σ,π] in the poset of permutations ordered by pattern containment in the case where π is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1,2,…,k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Möbius function in the case where σ and π are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142.We also show that the Möbius function in the poset of separable permutations admits a combinatorial interpretation in terms of normal embeddings among permutations. A consequence of this interpretation is that the Möbius function of an interval [σ,π] of separable permutations is bounded by the number of occurrences of σ as a pattern in π. Another consequence is that for any separable permutation π the Möbius function of (1,π) is either 0, 1 or −1.     

Occupation games on graphs in which the second player takes almost all vertices

Given a connected graph G=(V,E), two players take turns occupying vertices v∈V by placing black and white tokens so that the current vertex sets B,W⊆V are disjoint, B∩W=0?, and the corresponding induced subgraphs G[B] and G[W] are connected any time. A player must pass whenever (s)he has no legal move. (Obviously, after this, the opponent will take all remaining vertices, since G is assumed connected.) The game is over when all vertices are taken, V=B^∗∪W^∗. Then, Black and White get b=|B^∗|/|V| and w=|W^∗|/|V|, respectively. Thus, the occupation game is one-sum, b+w=1, and we could easily reduce it to a zero-sum game by simply shifting the payoffs, b^′=b−1/2,w^′=w−1/2. Let us also notice that b≥0 and w≥0; moreover, b>0 and w>0 whenever |V|>1.[Let us remark that the so-called Chinese rules define similar payoffs for the classic game of GO, yet, the legal moves are defined in GO differently.]Like in GO, we assume that Black begins. It is easy to construct graphs in which Black can take almost all vertices, more precisely, for each ε>0 there is a graph G for which b>1−ε. In this paper we show that, somewhat surprisingly, there are also graphs in which White can take almost all vertices.     

On the convergence rate of grad-div stabilized Taylor-Hood to Scott-Vogelius solutions for incompressible flow problems

It was recently proven in Case et al. (2010) [2] that, under mild restrictions, grad-div stabilized Taylor-Hood solutions of Navier-Stokes problems converge to the Scott-Vogelius solution of that same problem. However, even though the analytical rate was only shown to be (where γ is the stabilization parameter), the computational results suggest the rate may be improvable to γ−1. We prove herein the analytical rate is indeed γ−1, and extend the result to other incompressible flow problems including Leray-α and MHD. Numerical results are given that verify the theory.     

Supply function equilibria for uniform price auction in oligopolistic markets

We consider game-theoretic models related to the supply function auction for electricity markets. We determine the set of supply function equilibria (SFE), introduced by Klemperer and Mayer (Econometrica 57:1243–1277, 1989), for a symmetric oligopoly with linear demand, fixed marginal cost and capacity constraint. This set depends on the maximum random shock of the demand function. We also study the best response dynamics and show that in general it does not converge to any SFE. We find out sufficient conditions for the convergence and conclude on the optimal parameters of the auction.     

Bose-Einstein and colour interference in W-pair decays

We study effects on the W mass measurements at LEP2 from non-perturbative interference effects in the fully hadronic decay channel. Based on a model for Bose-Einstein interference, which is in agreement with LEP1 data, we argue that there are no Bose-Einstein correlations between bosons coming from the different W's. For small reconnection probabilities we rule out the possible experimental signal of colour interference at LEP2, suggested in [1]. The conclusions from this paper are that the theoretical uncertainties in the W mass determination should be smaller than the experimental statistical error. Received: 3 November 1997 / Published online: 10 March 1998     

Accuracy of optical frequency comb generators and optical frequency interval divider chains

We compared two methods for measuring large optical frequency differences: an optical frequency comb generator, which creates a large number of sidebands from a single-mode laser through electro-optic modulation, and an optical frequency interval divider chain, which divides a frequency gap successively by two until it becomes accessible to a radio-frequency counter. By locking two diode lasers to two modulation sidebands of a comb generator, ~1 THz apart, and measuring this interval with a chain of four phase-locked interval dividers, we demonstrate for the first time to our knowledge the accuracy of both techniques within an experimental limit of 6.8 x 10(-15).