arXiv subjects

Petr Jizba

Publications and source records attributed to Petr Jizba.

At least 55 records · Page 3Linked to original sources

arXiv2016

Comment on "Generalized Shannon--Khinchin axioms and uniqueness theorem for pseudo-additive entropies" [Physica A 411 (2014) 138]

Recently in [Physica A 411 (2014) 138] Ilić and Stanković have suggested that there may be problem for the class of hybrid entropies introduced in [P.~Jizba and T.~Arimitsu, Physica A 340 (2004) 110]. In this Comment we point out that the problem can be traced down to the $q$-additive entropic chain rule and to a peculiar behavior of the DeFinetti--Kolmogorov relation for escort distributions. However, despite this, one can still safely use the proposed hybrid entropies in most of the statistical-thermodynamics considerations.

math-ph↗

arXiv2016

Tsallis thermostatics as a statistical physics of random chains

In this paper we point out that the generalized statistics of Tsallis-Havrda-Charvát can be conveniently used as a conceptual framework for statistical treatment of random chains. In particular, we use the path-integral approach to show that the ensuing partition function can be identified with the partition function of a fluctuating oriented random loop of arbitrary length and shape in a background scalar potential. To put some meat on the bare bones, we illustrate this with two statistical systems; Schultz-Zimm polymer and relativistic particle. Further salient issues such as the $PSL(2;R)$ transformation properties of Tsallis' inverse-temperature parameter and a grandcanonical ensemble of fluctuating random loops related to the Tsallis-Havrda-Charvát statistics are also briefly discussed.

cond-mat.stat-mech↗

arXiv2016

Techniques for multifractal spectrum estimation in financial time series

Multifractal analysis is one of the important approaches that enables us to measure the complexity of various data via the scaling properties. We compare the most common techniques used for multifractal exponents estimation from both theoretical and practical point of view. Particularly, we discuss the methods based on estimation of Rényi entropy, which provide a powerful tool especially in presence of heavy-tailed data. To put some flesh on bare bones, all methods are compared on various real financial datasets, including daily and high-frequency data.

q-fin.ST↗

arXiv2016

One-parameter class of uncertainty relations based on entropy power

We use the concept of entropy power to derive a new one-parameter class of information-theoretic uncertainty relations for pairs of conjugate observables in an infinite-dimensional Hilbert space. This class constitutes an infinite tower of higher-order statistics uncertainty relations, which allows one in principle to determine the shape of the underlying information-distribution function by measuring the relevant entropy powers. We illustrate the capability of the new class by discussing two examples: superpositions of vacuum and squeezed states and the Cauchy-type heavy-tailed wave function.

quant-ph↗

arXiv2016

Effective action approach to dynamical generation of fermion mixing

In this paper we discuss a mechanism for the dynamical generation of flavor mixing, in the framework of the Nambu--Jona Lasinio model. Our approach is illustrated both with the conventional operatorial formalism and with functional integral and ensuing one-loop effective action. The results obtained are briefly discussed.

hep-th↗

arXiv2016

Multi-state stochastic hotchpotch model gives rise to the observed mesoscopic behaviour in the non-stirred Belousov--Zhabotinsky reaction

Mesoscopic dynamics of self-organized structures is the most important aspect in the description of complex living systems. The Belousov--Zhabotinsky (B--Z) reaction is in this respect a convenient testbed because it represents a prototype of chemical self-organization with a rich variety of emergent wave-spiral patterns. Using a multi-state stochastic hotchpotch model, we show here that the mesoscopic behaviour of the non-stirred B--Z reaction is both qualitatively and quantitatively susceptible to the description in terms of stochastic multilevel cellular automata. This further implies that the mesoscopic dynamics of the non-stirred B--Z reaction results from a delicate interplay between a) a maximal number of available states within the elementary time lag (i.e. a minimal time interval needed for demise of a final state) and b) an imprecision or uncertainty in the definition of state. If either the number of time lags is largely different from 7 or the maximal number of available states is smaller than 20, the physicochemical conditions are inappropriate for a formation of the wave-spiral patterns. Furthermore, a white noise seems to be key for the formation of circular structures (target patterns) which could not be as yet systematically explained in existing models.

nlin.CG↗

arXiv2015

Local-time representation of path integrals

We derive a local-time path-integral representation for a generic one-dimensional time-independent system. In particular, we show how to rephrase the matrix elements of the Bloch density matrix as a path integral over x-dependent local-time profiles. The latter quantify the time that the sample paths x(t) in the Feynman path integral spend in the vicinity of an arbitrary point x. Generalization of the local-time representation that includes arbitrary functionals of the local time is also provided. We argue that the results obtained represent a powerful alternative to the traditional Feynman-Kac formula, particularly in the high and low temperature regimes. To illustrate this point, we apply our local-time representation to analyze the asymptotic behavior of the Bloch density matrix at low temperatures. Further salient issues, such as connections with the Sturm-Liouville theory and the Rayleigh-Ritz variational principle are also discussed.

math-ph↗

arXiv2015

On q-non-extensive statistics with non-Tsallisian entropy

We combine an axiomatics of Rényi with the $q$--deformed version of Khinchin axioms to obtain a measure of information (i.e., entropy) which accounts both for systems with embedded self-similarity and non-extensivity. We show that the entropy thus obtained is uniquely solved in terms of a one-parameter family of information measures. The ensuing maximal-entropy distribution is phrased in terms of a special function known as the Lambert W--function. We analyze the corresponding "high" and "low-temperature" asymptotics and reveal a non-trivial structure of the parameter space.

math-ph↗

arXiv2015

Inflationary cosmology from quantum Conformal Gravity

We analyze the functional integral for quantum Conformal Gravity and show that with the help of a Hubbard-Stratonovich transformation, the action can be broken into a local quadratic-curvature theory coupled to a scalar field. A one-loop effective action calculation reveals that strong fluctuations of the metric field are capable of spontaneously generating a dimensionally transmuted parameter which in the weak-field sector of the broken phase induces a Starobinsky-type f(R)-model with a gravi-cosmological constant. A resulting non-trivial relation between Starobinsky'sparameter and the cosmological constant is highlighted and implications for cosmic inflation are briefly discussed and compared with recent PLANCK and BICEP2 data.

hep-th↗

arXiv2014

On the Role of Information Theoretic Uncertainty Relations in Quantum Theory

Uncertainty relations based on information theory for both discrete and continuous distribution functions are briefly reviewed. We extend these results to account for (differential) Rényi entropy and its related entropy power. This allows us to find a new class of information-theoretic uncertainty relations (ITURs). The potency of such uncertainty relations in quantum mechanics is illustrated with a simple two-energy-level model where they outperform both the usual Robertson-Schrödinger uncertainty relation and Kraus-Maassen Shannon entropy based uncertainty relation. In the continuous case the ensuing entropy power uncertainty relations are discussed in the context of heavy tailed wave functions and Schrödinger cat states. Again, improvement over both the Robertson-Schrödinger uncertainty principle and Shannon ITUR is demonstrated in these cases. Further salient issues such as the proof of a generalized entropy power inequality and a geometric picture of information-theoretic uncertainty relations are also discussed.

quant-ph↗

arXiv2014

Multifractal Diffusion Entropy Analysis: Optimal Bin Width of Probability Histograms

In the framework of Multifractal Diffusion Entropy Analysis we propose a method for choosing an optimal bin-width in histograms generated from underlying probability distributions of interest. The method presented uses techniques of Rényi's entropy and the mean squared error analysis to discuss the conditions under which the error in the multifractal spectrum estimation is minimal. We illustrate the utility of our approach by focusing on a scaling behavior of financial time series. In particular, we analyze the S&P500 stock index as sampled at a daily rate in the time period 1950-2013. In order to demonstrate a strength of the method proposed we compare the multifractal $δ$-spectrum for various bin-widths and show the robustness of the method, especially for large values of $q$. For such values, other methods in use, e.g., those based on moment estimation, tend to fail for heavy-tailed data or data with long correlations. Connection between the $δ$-spectrum and Rényi's $q$ parameter is also discussed and elucidated on a simple example of multiscale time series.

q-fin.ST↗

arXiv2014

Path-integral approach to the Wigner-Kirkwood expansion

We study the high-temperature behavior of quantum-mechanical path integrals. Starting from the Feynman-Kac formula, we derive a new functional representation of the Wigner-Kirkwood perturbation expansion for quantum Boltzmann densities. As shown by its applications to different potentials, the presented expansion turns out to be quite efficient in generating analytic form of the higher-order expansion coefficients. To put some flesh on the bare bones we apply the expansion to obtain basic thermodynamic functions of the one-dimensional anharmonic oscillator. Further salient issues, such as generalization to the Bloch density matrix and comparison with the more customary world-line formulation are discussed.

cond-mat.stat-mech↗

arXiv2013

Special Relativity induced by Granular Space

We show that the special relativistic dynamics when combined with quantum mechanics and the concept of superstatistics can be interpreted as arising from two interlocked non-relativistic stochastic processes that operate at different energy scales. This interpretation leads to Feynman amplitudes that are in the Euclidean regime identical to transition probability of a Brownian particle propagating through a granular space. Some kind of spacetime granularity could be held responsible for the emergence at larger scales of various symmetries. For illustration we consider also the dynamics and the propagator of a spinless relativistic particle. Implications for doubly special relativity, quantum field theory, quantum gravity and cosmology are discussed.

hep-th↗

arXiv2012

The emergence of Special and Doubly Special Relativity

Building on our previous work [Phys.Rev.D82,085016(2010)], we show in this paper how a Brownian motion on a short scale can originate a relativistic motion on scales that are larger than particle's Compton wavelength. This can be described in terms of polycrystalline vacuum. Viewed in this way, special relativity is not a primitive concept, but rather it statistically emerges when a coarse graining average over distances of order, or longer than the Compton wavelength is taken. By analyzing the robustness of such a special relativity under small variations in the polycrystalline grain-size distribution we naturally arrive at the notion of doubly-special relativistic dynamics. In this way, a previously unsuspected, common statistical origin of the two frameworks is brought to light. Salient issues such as the role of gauge fixing in emergent relativity, generalized commutation relations, Hausdorff dimensions of representative path-integral trajectories and a connection with Feynman chessboard model are also discussed.

hep-th↗

arXiv2012

Nambu-Goldstone dynamics and generalized coherent-state functional integrals

The present paper gives a new method of attack on the Nambu-Goldstone dynamics in spontaneously broken theories. Since the target space of the Nambu-Goldstone fields is a group coset space, their effective quantum dynamics can be naturally phrased in terms of generalized coherent-state functional integrals. As an explicit example of this line of reasoning we construct a low-energy effective Lagrangian for the Heisenberg ferromagnet in broken phase. The leading field configuration in the WKB approximation leads to the Landau-Lifshitz equation for quantum ferromagnet. The corresponding linearized equations allow to identify the Nambu-Goldstone boson with ferromagnetic magnon.

hep-th↗

arXiv2012

Renyi's information transfer between financial time series

In this paper, we quantify the statistical coherence between financial time series by means of the Renyi entropy. With the help of Campbell's coding theorem we show that the Renyi entropy selectively emphasizes only certain sectors of the underlying empirical distribution while strongly suppressing others. This accentuation is controlled with Renyi's parameter q. To tackle the issue of the information flow between time series we formulate the concept of Renyi's transfer entropy as a measure of information that is transferred only between certain parts of underlying distributions. This is particularly pertinent in financial time series where the knowledge of marginal events such as spikes or sudden jumps is of a crucial importance. We apply the Renyian information flow to stock market time series from 11 world stock indices as sampled at a daily rate in the time period 02.01.1990 - 31.12.2009. Corresponding heat maps and net information flows are represented graphically. A detailed discussion of the transfer entropy between the DAX and S&P500 indices based on minute tick data gathered in the period from 02.04.2008 to 11.09.2009 is also provided. Our analysis shows that the bivariate information flow between world markets is strongly asymmetric with a distinct information surplus flowing from the Asia-Pacific region to both European and US markets. An important yet less dramatic excess of information also flows from Europe to the US. This is particularly clearly seen from a careful analysis of Renyi information flow between the DAX and S&P500 indices.

q-fin.ST↗

arXiv2010

Superstatistics approach to path integral for a relativistic particle

Superstatistics permits the calculation of the Feynman propagator of a relativistic particle in a novel way from a superstatistical average over non-relativistic single-particle paths. We illustrate this for the Klein-Gordon particle in the Feshbach-Villars representation, and for the Dirac particle in the Schroedinger-Dirac representation. As a byproduct we recover the worldline representation of Klein-Gordon and Dirac propagators, and discuss the role of the smearing distributions in fixing the reparametrization freedom. The emergent relativity picture that follows from our approach together with a novel representation of the Lorentz group for the Feshbach-Villars particle are also discussed.

hep-th↗

arXiv2010

Revisiting the gauge principle: enforcing constants of motion as constraints

In this paper we examine an alternative formulation of the gauge principle in which the emphasis is shifted from the symmetry transformations to their generators. We show that the gauge principle can be entirely reformulated in terms of promoting constants of motion - which generate rigid symmetries - to constraints - which generate gauge symmetries. In our exposition we first explain the basic philosophy on mechanical systems, and then with the help of De Donder--Weyl formalism we extend our scenario also to a field-theoretical setting. To put some flesh on bare bones we demonstrate our method in numerous examples, including the massive relativistic particle, the Nambu--Goto string and relativistic field theory.

hep-th↗