Characterizing and Quantifying Frustration in Quantum Many-Body Systems

We present a general scheme for the study of frustration in quantum systems. We introduce a universal measure of frustration for arbitrary quantum systems and we relate it to a class of entanglement monotones via an exact inequality. If all the (pure) ground states of a given Hamiltonian saturate the inequality, then the system is said to be inequality saturating. We introduce sufficient conditions for a quantum spin system to be inequality saturating and confirm them with extensive numerical tests. These conditions provide a generalization to the quantum domain of the Toulouse criteria for classical frustration-free systems. The models satisfying these conditions can be reasonably identified as geometrically unfrustrated and subject to frustration of purely quantum origin. Our results therefore establish a unified framework for studying the intertwining of geometric and quantum contributions to frustration.Comment: 8 pages, 1 figur

Fractional calculus and continuous-time finance II: the waiting-time distribution

We complement the theory of tick-by-tick dynamics of financial markets based on a Continuous-Time Random Walk (CTRW) model recently proposed by Scalas et al., and we point out its consistency with the behaviour observed in the waiting-time distribution for BUND future prices traded at LIFFE, London.Comment: Revised version, 17 pages, 4 figures. Physica A, Vol. 287, No 3-4, 468--481 (2000). Proceedings of the International Workshop on "Economic Dynamics from the Physics Point of View", Bad-Honnef (Germany), 27-30 March 200

Anomalous diffusion and stretched exponentials in heterogeneous glass-forming liquids: Low-temperature behavior

We propose a model of a heterogeneous glass forming liquid and compute the low-temperature behavior of a tagged molecule moving within it. This model exhibits stretched-exponential decay of the wavenumber-dependent, self intermediate scattering function in the limit of long times. At temperatures close to the glass transition, where the heterogeneities are much larger in extent than the molecular spacing, the time dependence of the scattering function crosses over from stretched-exponential decay with an index $b=1/2$ at large wave numbers to normal, diffusive behavior with $b = 1$ at small wavenumbers. There is a clear separation between early-stage, cage-breaking $\beta$ relaxation and late-stage $\alpha$ relaxation. The spatial representation of the scattering function exhibits an anomalously broad exponential (non-Gaussian) tail for sufficiently large values of the molecular displacement at all finite times.Comment: 9 pages, 6 figure

Modeling of waiting times and price changes in currency exchange data

A theory which describes the share price evolution at financial markets as a continuous-time random walk has been generalized in order to take into account the dependence of waiting times t on price returns x. A joint probability density function (pdf) which uses the concept of a L\'{e}vy stable distribution is worked out. The theory is fitted to high-frequency US$/Japanese Yen exchange rate and low-frequency 19th century Irish stock data. The theory has been fitted both to price return and to waiting time data and the adherence to data, in terms of the chi-squared test statistic, has been improved when compared to the old theory.Comment: 22 pages, 5 postscript figures, LaTeX2e using elsart.cl

Superdiffusion in Decoupled Continuous Time Random Walks

Continuous time random walk models with decoupled waiting time density are studied. When the spatial one jump probability density belongs to the Levy distribution type and the total time transition is exponential a generalized superdiffusive regime is established. This is verified by showing that the square width of the probability distribution (appropriately defined)grows as $t^{2/\gamma}$ with $0<\gamma\leq2$ when $t\to \infty$. An important connection of our results and those of Tsallis' nonextensive statistics is shown. The normalized q-expectation value of $x^2$ calculated with the corresponding probability distribution behaves exactly as $t^{2/\gamma}$ in the asymptotic limit.Comment: 9 pages (.tex file), 1 Postscript figures, uses revtex.st

Forcing anomalous scaling on demographic fluctuations

We discuss the conditions under which a population of anomalously diffusing individuals can be characterized by demographic fluctuations that are anomalously scaling themselves. Two examples are provided in the case of individuals migrating by Gaussian diffusion, and by a sequence of L\'evy flights.Comment: 5 pages 2 figure

Theory of Single File Diffusion in a Force Field

The dynamics of hard-core interacting Brownian particles in an external potential field is studied in one dimension. Using the Jepsen line we find a very general and simple formula relating the motion of the tagged center particle, with the classical, time dependent single particle reflection ${\cal R}$ and transmission ${\cal T}$ coefficients. Our formula describes rich physical behaviors both in equilibrium and the approach to equilibrium of this many body problem.Comment: 4 Phys. Rev. page

Relativistic Weierstrass random walks

The Weierstrass random walk is a paradigmatic Markov chain giving rise to a L\'evy-type superdiffusive behavior. It is well known that Special Relativity prevents the arbitrarily high velocities necessary to establish a superdiffusive behavior in any process occurring in Minkowski spacetime, implying, in particular, that any relativistic Markov chain describing spacetime phenomena must be essentially Gaussian. Here, we introduce a simple relativistic extension of the Weierstrass random walk and show that there must exist a transition time $t_c$ delimiting two qualitative distinct dynamical regimes: the (non-relativistic) superdiffusive L\'evy flights, for $t < t_c$, and the usual (relativistic) Gaussian diffusion, for $t>t_c$. Implications of this crossover between different diffusion regimes are discussed for some explicit examples. The study of such an explicit and simple Markov chain can shed some light on several results obtained in much more involved contexts.Comment: 5 pages, final version to appear in PR

Continuous time random walk and parametric subordination in fractional diffusion

The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW)is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk in space, both walks happening in discrete operational time. In the continuum limit we obtain a generally non-Markovian diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. By what we call parametric subordination, applied to a combination of a Markov process with a positively oriented L\'evy process, we generate and display sample paths for some special cases.Comment: 28 pages, 18 figures. Workshop 'In Search of a Theory of Complexity'. Denton, Texas, August 200

New perspectives on the Ising model

The Ising model, in presence of an external magnetic field, is isomorphic to a model of localized interacting particles satisfying the Fermi statistics. By using this isomorphism, we construct a general solution of the Ising model which holds for any dimensionality of the system. The Hamiltonian of the model is solved in terms of a complete finite set of eigenoperators and eigenvalues. The Green's function and the correlation functions of the fermionic model are exactly known and are expressed in terms of a finite small number of parameters that have to be self-consistently determined. By using the equation of the motion method, we derive a set of equations which connect different spin correlation functions. The scheme that emerges is that it is possible to describe the Ising model from a unified point of view where all the properties are connected to a small number of local parameters, and where the critical behavior is controlled by the energy scales fixed by the eigenvalues of the Hamiltonian. By using algebra and symmetry considerations, we calculate the self-consistent parameters for the one-dimensional case. All the properties of the system are calculated and obviously agree with the exact results reported in the literature.Comment: 19 RevTeX pages, 9 panels, to be published in Eur. Phys. J.