Markovian nature, completeness, regularity and correlation properties of Generalized Poisson-Kac processes

We analyze some basic issues associated with Generalized Poisson-Kac (GPK) stochastic processes, starting from the extended notion of the Markovian condition. The extended Markovian nature of GPK processes is established, and the implications of this property derived: the associated adjoint formalism for GPK processes is developed essentially in an analogous way as for the Fokker-Planck operator associated with Langevin equations driven by Wiener processes. Subsequently, the regularity of trajectories is addressed: the occurrence of fractality in the realizations of GPK is a long-term emergent property, and its implication in thermodynamics is discussed. The concept of completeness in the stochastic description of GPK is also introduced. Finally, some observations on the role of correlation properties of noise sources and their influence on the dynamic properties of transport phenomena are addressed, using a Wiener model for comparison

Quantum Einstein Gravity

We give a pedagogical introduction to the basic ideas and concepts of the Asymptotic Safety program in Quantum Einstein Gravity. Using the continuum approach based upon the effective average action, we summarize the state of the art of the field with a particular focus on the evidence supporting the existence of the non-trivial renormalization group fixed point at the heart of the construction. As an application, the multifractal structure of the emerging space-times is discussed in detail. In particular, we compare the continuum prediction for their spectral dimension with Monte Carlo data from the Causal Dynamical Triangulation approach.Comment: 87 pages, 13 figures, review article prepared for the New Journal of Physics focus issue on Quantum Einstein Gravit

From Knowledge, Knowability and the Search for Objective Randomness to a New Vision of Complexity

Herein we consider various concepts of entropy as measures of the complexity of phenomena and in so doing encounter a fundamental problem in physics that affects how we understand the nature of reality. In essence the difficulty has to do with our understanding of randomness, irreversibility and unpredictability using physical theory, and these in turn undermine our certainty regarding what we can and what we cannot know about complex phenomena in general. The sources of complexity examined herein appear to be channels for the amplification of naturally occurring randomness in the physical world. Our analysis suggests that when the conditions for the renormalization group apply, this spontaneous randomness, which is not a reflection of our limited knowledge, but a genuine property of nature, does not realize the conventional thermodynamic state, and a new condition, intermediate between the dynamic and the thermodynamic state, emerges. We argue that with this vision of complexity, life, which with ordinary statistical mechanics seems to be foreign to physics, becomes a natural consequence of dynamical processes.Comment: Phylosophica

Quantification of Beat-To-Beat Variability of Action Potential Durations in Langendorff-Perfused Mouse Hearts

Background: Beat-to-beat variability in action potential duration (APD) is an intrinsic property of cardiac tissue and is altered in pro-arrhythmic states. However, it has never been examined in mice.Methods: Left atrial or ventricular monophasic action potentials (MAPs) were recorded from Langendorff-perfused mouse hearts during regular 8 Hz pacing. Time-domain, frequency-domain and non-linear analyses were used to quantify APD variability.Results: Mean atrial APD (90% repolarization) was 23.5 ± 6.3 ms and standard deviation (SD) was 0.9 ± 0.5 ms (n = 6 hearts). Coefficient of variation (CoV) was 4.0 ± 1.9% and root mean square (RMS) of successive differences in APDs was 0.3 ± 0.2 ms. The peaks for low- and high-frequency were 0.7 ± 0.5 and 2.7 ± 0.9 Hz, respectively, with percentage powers of 39.0 ± 20.5 and 59.3 ± 22.9%. Poincaré plots of APDn+1 against APDn revealed ellipsoid shapes. The ratio of the SD along the line-of-identity (SD2) to the SD perpendicular to the line-of-identity (SD1) was 8.28 ± 4.78. Approximate and sample entropy were 0.57 ± 0.12 and 0.57 ± 0.15, respectively. Detrended fluctuation analysis revealed short- and long-term fluctuation slopes of 1.80 ± 0.15 and 0.85 ± 0.29, respectively. When compared to atrial APDs, ventricular APDs were longer (ANOVA, P < 0.05), showed lower mean SD and CoV but similar RMS of successive differences in APDs and showed lower SD2 (P < 0.05). No difference in the remaining parameters was observed.Conclusion: Beat-to-beat variability in APD is observed in mouse hearts during regular pacing. Atrial MAPs showed greater degree of variability than ventricular MAPs. Non-linear techniques offer further insights on short-term and long-term variability and signal complexity

Average dielectric function and spectral representation of composites with fractal structure

The thesis is organized in three parts, each of them in the form of an independent paper. In Part I, titled A Spectral Theory for Two-Component Porous Media , the spectral representation formalism is applied to successfully explain the observed static (frequency-independent) and dynamic (frequency-dependent) dielectric properties of rock-and-brine systems. In addition, the theory shows scaling properties which were formulated earlier for discrete systems, namely, random resistor networks. A new critical behavior in these porous systems is discovered, whose critical exponent can be related to the transport critical exponents. The exponents describing power law properties of the average dielectric function do not seem to be universal for the continuum porous systems;In Part II, titled Critical Behavior in a New Random Self-Similar System , we model a continuum porous mixture, in which spherical holes (in any arbitrary euclidean dimension) of one medium are first punched out in the second medium, and, subsequently and iteratively, in the resultant medium of the previous punching. A new iterative Effective Medium Theory (EMT) is developed to represent the average dielectric properties of such a system. The spectral representation formalism is applied to extract geometrical properties, as well as three critical exponents of the dielectric function--conductivity exponent t, take off exponent 1, and touch down exponent u--for systems with two to six euclidean dimensions. Their values are computed through numerical iterations. Also, a new feature in the real part of the dielectric function of the composite is found that suggests a value of zero for the capacitative critical exponent s. Remarkably, some of the properties and exponents are quite different from, and opposite to, those observed in the computer-simulated discrete systems, and, possibly, are signatures of the continuum geometry;In Part III, titled Exact Critical Exponents of the Dielectric Function for a New Random Self-Similar System , we theoretically predict the existence of critical behavior for the above iterative system, and derive exact and general analytical values for the critical exponents. Then, as examples, both EMT and Maxwell-Garnett Theory are used to obtain theory-dependent expressions for the critical exponents. The computational results of Part II are found to be quite close to the EMT-based theoretical results derived in this part

Symmetry and Mesoscopic Physics

Symmetry is one of the most important notions in natural science; it lies at the heart of fundamental laws of nature and serves as an important tool for understanding the properties of complex systems, both classical and quantum. Another trend, which has in recent years undergone intensive development, is mesoscopic physics. This branch of physics also combines classical and quantum ideas and methods. Two main directions can be distinguished in mesoscopic physics. One is the study of finite quantum systems of mesoscopic sizes. Such systems, which are between the atomic and macroscopic scales, exhibit a variety of novel phenomena and find numerous applications in creating modern electronic and spintronic devices. At the same time, the behavior of large systems can be influenced by mesoscopic effects, which provides another direction within the framework of mesoscopic physics. The aim of the present book is to emphasize the phenomena that lie at the crossroads between the concept of symmetry and mesoscopic physics

Quantification of beat-to-beat variability of action potential durations in Langendorff-perfused mouse hearts

Beat-to-beat variability in action potential duration (APD) is an intrinsic property of cardiac tissue and is altered in pro-arrhythmic states. However, it has never been examined in mice. Methods: Left atrial or ventricular monophasic action potentials (MAPs) were recorded from Langendorff-perfused mouse hearts during regular 8 Hz pacing. Time-domain, frequency-domain and non-linear analyses were used to quantify APD variability. Results: Mean atrial APD (90% repolarization) was 26.6±3.8 ms and standard deviation (SD) was 1.4±0.8 ms (n=6 hearts). Coefficient of variation (CoV) was 5.9±3.2% and root mean square (RMS) of successive differences in APDs was 0.5±0.2 ms. The peaks for low- and high-frequency were 0.8±0.6 and 2.7±1.1 Hz, respectively, with percentage powers of 37.8±14.5 and 61.5±15.1%. Poincaré plots of APDn+1 against APDn revealed ellipsoid shapes. The ratio of the SD along the line-of-identity (SD2) to the SD perpendicular to the line-of-identity (SD1) was 5.96±1.44. Approximate and sample entropy were 0.45±0.05 and 0.54±0.11, respectively. Detrended fluctuation analysis revealed short- and long-term fluctuation slopes of 1.73±0.17 and 0.68±0.20, respectively. When compared to atrial APDs, ventricular APDs were longer (ANOVA, P<0.05) and showed lower mean SD, CoV and RMS of successive differences in APDs, lower LF power, high HF power and lower SD1 and SD2 (P<0.05) but no difference in the remaining parameters. Conclusion: Beat-to-beat variability in APD is observed in mouse hearts during regular pacing. Atrial MAPs showed greater degree of variability than ventricular MAPs. Non-linear techniques offer further insights on short-term and long-term variability and signal complexity