Recurrence and Polya number of general one-dimensional random walks

The recurrence properties of random walks can be characterized by P\'{o}lya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we consider recurrence properties for a general 1D random walk on a line, in which at each time step the walker can move to the left or right with probabilities $l$ and $r$, or remain at the same position with probability $o$ ($l+r+o=1$). We calculate P\'{o}lya number $P$ of this model and find a simple expression for $P$ as, $P=1-\Delta$, where $\Delta$ is the absolute difference of $l$ and $r$ ($\Delta=|l-r|$). We prove this rigorous expression by the method of creative telescoping, and our result suggests that the walk is recurrent if and only if the left-moving probability $l$ equals to the right-moving probability $r$.Comment: 3 page short pape

Characterizing flows with an instrumented particle measuring Lagrangian accelerations

We present in this article a novel Lagrangian measurement technique: an instrumented particle which continuously transmits the force/acceleration acting on it as it is advected in a flow. We develop signal processing methods to extract information on the flow from the acceleration signal transmitted by the particle. Notably, we are able to characterize the force acting on the particle and to identify the presence of a permanent large-scale vortex structure. Our technique provides a fast, robust and efficient tool to characterize flows, and it is particularly suited to obtain Lagrangian statistics along long trajectories or in cases where optical measurement techniques are not or hardly applicable.Comment: submitted to New Journal of Physic

Renyi Entropy of the XY Spin Chain

We consider the one-dimensional XY quantum spin chain in a transverse magnetic field. We are interested in the Renyi entropy of a block of L neighboring spins at zero temperature on an infinite lattice. The Renyi entropy is essentially the trace of some power $\alpha$ of the density matrix of the block. We calculate the asymptotic for $L \to \infty$ analytically in terms of Klein's elliptic $\lambda$ - function. We study the limiting entropy as a function of its parameter $\alpha$. We show that up to the trivial addition terms and multiplicative factors, and after a proper re-scaling, the Renyi entropy is an automorphic function with respect to a certain subgroup of the modular group; moreover, the subgroup depends on whether the magnetic field is above or below its critical value. Using this fact, we derive the transformation properties of the Renyi entropy under the map $\alpha \to \alpha^{-1}$ and show that the entropy becomes an elementary function of the magnetic field and the anisotropy when $\alpha$ is a integer power of 2, this includes the purity $tr \rho^2$. We also analyze the behavior of the entropy as $\alpha \to 0$ and $\infty$ and at the critical magnetic field and in the isotropic limit [XX model].Comment: 28 Pages, 1 Figur

Entropy as a function of Geometric Phase

We give a closed-form solution of von Neumann entropy as a function of geometric phase modulated by visibility and average distinguishability in Hilbert spaces of two and three dimensions. We show that the same type of dependence also exists in higher dimensions. We also outline a method for measuring both the entropy and the phase experimentally using a simple Mach-Zehnder type interferometer which explains physically why the two concepts are related.Comment: 19 pages, 7 figure

Quantum Adiabatic Markovian Master Equations

We develop from first principles Markovian master equations suited for studying the time evolution of a system evolving adiabatically while coupled weakly to a thermal bath. We derive two sets of equations in the adiabatic limit, one using the rotating wave (secular) approximation that results in a master equation in Lindblad form, the other without the rotating wave approximation but not in Lindblad form. The two equations make markedly different predictions depending on whether or not the Lamb shift is included. Our analysis keeps track of the various time- and energy-scales associated with the various approximations we make, and thus allows for a systematic inclusion of higher order corrections, in particular beyond the adiabatic limit. We use our formalism to study the evolution of an Ising spin chain in a transverse field and coupled to a thermal bosonic bath, for which we identify four distinct evolution phases. While we do not expect this to be a generic feature, in one of these phases dissipation acts to increase the fidelity of the system state relative to the adiabatic ground state.Comment: 31 pages, 9 figures. v2: Generalized Markov approximation bound. Included a section on thermal equilibration. v3: Added text that appears in NJP version. Generalized Lindblad ME to include degenerate subspaces. v3. Corrections made to Appendix E and F. We thank Kabuki Takada and Hidetoshi Nishimori for pointing out the errors. v4: Corrected a typo in Eqt. B

Computing welfare losses from data under imperfect competition with heterogeneous goods

We study the percentage of welfare losses (PWL) yielded by imperfect competition under product differentiation. When demand is linear, if prices, outputs, costs and the number of firms can be observed, PWL is arbitrary in both Cournot and Bertrand equilibria. If in addition, the elasticity of demand (resp. cross elasticity of demand) is known, we can calculate PWL in Cournot (resp. Bertrand) equilibrium. When demand is isoelastic and there are many firms, PWL can be computed from prices, outputs, costs and the number of .rms. In all these cases we find that price-marginal cost margins and demand elasticities may influence PWL in a counterintuitive way. We also provide conditions under which PWL increases or decreases with concentration

Abundance of unknots in various models of polymer loops

A veritable zoo of different knots is seen in the ensemble of looped polymer chains, whether created computationally or observed in vitro. At short loop lengths, the spectrum of knots is dominated by the trivial knot (unknot). The fractional abundance of this topological state in the ensemble of all conformations of the loop of $N$ segments follows a decaying exponential form, $\sim \exp (-N/N_0)$, where $N_0$ marks the crossover from a mostly unknotted (ie topologically simple) to a mostly knotted (ie topologically complex) ensemble. In the present work we use computational simulation to look closer into the variation of $N_0$ for a variety of polymer models. Among models examined, $N_0$ is smallest (about 240) for the model with all segments of the same length, it is somewhat larger (305) for Gaussian distributed segments, and can be very large (up to many thousands) when the segment length distribution has a fat power law tail.Comment: 13 pages, 6 color figure

TESELA: a new Virtual Observatory tool to determine blank fields for astronomical observations

The observation of blank fields, regions of the sky devoid of stars down to a given threshold magnitude, constitutes one of the typical important calibration procedures required for the proper reduction of astronomical data obtained in imaging mode. This work describes a method, based on the use of the Delaunay triangulation on the surface of a sphere, that allows the easy generation of blank fields catalogues. In addition to that, a new tool named TESELA, accessible through the WEB, has been created to facilitate the user to retrieve, and visualise using the VO-tool Aladin, the blank fields available near a given position in the sky.Comment: Accepted for publication in Monthly Notices of the Royal Astronomical Society. 11 pages, 10 figures. Related Web tool accessible at http://sdc.cab.inta-csic.es/tesel

A photonic basis for deriving nonlinear optical response

Nonlinear optics is generally first presented as an extension of conventional optics. Typically the subject is introduced with reference to a classical oscillatory electric polarization, accommodating correction terms that become significant at high intensities. The material parameters that quantify the extent of the nonlinear response are cast as coefficients in a power series - nonlinear optical susceptibilities signifying a propensity to generate optical harmonics, for example. Taking the subject to a deeper level requires a more detailed knowledge of the structure and properties of each nonlinear susceptibility tensor, the latter differing in form according to the process under investigation. Typically, the derivations involve intricate development based on time-dependent perturbation theory, assisted by recourse to a set of Feynman diagrams. This paper presents a more direct route to the required results, based on photonic rather than semiclassical principles, and offers a significantly clearer perspective on the photophysics underlying nonlinear optical response. The method, here illustrated by specific application to harmonic generation and down-conversion processes, is simple, intuitive and readily amenable for processes of arbitrary photonic order. © 2009 IOP Publishing Ltd