Use and validity of principles of extremum of entropy production in the study of complex systems

It is shown how both the principles of extremum of entropy production, which are often used in the study of complex systems, follow from the maximization of overall system conductivities, under appropriate constraints. In this way, the maximum rate of entropy production (MEP) occurs when all the forces in the system are kept constant. On the other hand, the minimum rate of entropy production (mEP) occurs when all the currents that cross the system are kept constant. A brief discussion on the validity of the application of the mEP and MEP principles in several cases, and in particular to the Earth’s climate is also presented.     

Towards a physics of Internet traffic in a geographic network

A set of equations from a biased random walk are shown to describe the time-based Gaussian distributions of Internet traffic relative to the Earth’s time zones. The Internet is an example of a more general physical problem dealing with motion near the speed of light relative to different time frames of reference. The second order differential equation (DE) takes the form of ‘time diffusion’ near the speed of light or alternatively considered as a complex variable with real time and imaginary longitudinal components. Congestion waves are generated by peak global traffic from different time zones following the Earth’s revolution. The DE is divided into space and time operators for discussion and each component solution, including constants, is illustrated using data from a global network compiled by the Stanford Linear Accelerator Centre (SLAC). Indices of global and regional phase congestion for the monitoring sites are calculated from standardised regressions from the Earth’s rotation. There is also a J-curve limit to transferring information by the Internet and this is expressed as an inequality underpinned by the speed of light with examples from US and European traffic. The research returns to an often little known theme of Isaac Newton’s: mixing physics with geography. In our case, the equations define trajectories of information packets travelling near the speed of light, navigating within networks and between longitudes, relative to the Earth’s rotation.     

Variability of scaling parameters in non-conservative systems: Geophysical aspect

Newly obtained data on the critical dynamics of the drifting/fracturing sea ice in the Arctic Ocean were analyzed and compared with published data on the dynamic processes in the Earth’s crust. Substantial similarities were found in the scaling behavior of both geophysical systems; the most important of them is the b-value space/time variability including a pre-failure drop of this parameter. The b-value pattern of the ensemble of drifting ice floes is an analogue of the b-value distribution over areas and depths in tectonic structures. A common feature of the pre-failure state in the cryosphere and in the Earth’s crust is the increased degree of the energy conservation that manifests itself both in the sea ice consolidation prior to basin-wide ice pack fragmentations, and in involving harder geological formations in the fracture process before earthquakes, respectively. From the viewpoint of the conservative SOC concept, this relation between the system’s conservativity and the occurrence of large-scale fracture events means that the expected scale level of failure is determined by the system’s closeness to “true” SOC behavior.     

A class of asymmetric pathway distributions and an entropy interpretation

Asymmetric distributions are widely used in probability modeling and statistical analysis. Recently, various asymmetric distributions are being developed by many researchers for modeling various data sets in real life contexts. In the present paper, we introduce a new class of q-type asymmetric distributions which include q-analogues of asymmetric Laplace, exponential power, Weibull etc. and corresponding standard distributions as special cases. Also we show that this pathway model can be obtained by optimizing Mathai’s generalized entropy with more general setup, which is a generalization of various entropy measures due to Shannon and others.     

Spectral Ranking of Causal Influence in Complex Systems

Similar to natural complex systems, such as the Earth’s climate or a living cell, semiconductor lithography systems are characterized by nonlinear dynamics across more than a dozen orders of magnitude in space and time. Thousands of sensors measure relevant process variables at appropriate sampling rates, to provide time series as primary sources for system diagnostics. However, high-dimensionality, non-linearity and non-stationarity of the data are major challenges to efficiently, yet accurately, diagnose rare or new system issues by merely using model-based approaches. To reliably narrow down the causal search space, we validate a ranking algorithm that applies transfer entropy for bivariate interaction analysis of a system’s multivariate time series to obtain a weighted directed graph, and graph eigenvector centrality to identify the system’s most important sources of original information or causal influence. The results suggest that this approach robustly identifies the true drivers or causes of a complex system’s deviant behavior, even when its reconstructed information transfer network includes redundant edges.     

The rank-size scaling law and entropy-maximizing principle

The rank-size regularity known as Zipf’s law is one of the scaling laws and is frequently observed in the natural living world and social institutions. Many scientists have tried to derive the rank-size scaling relation through entropy-maximizing methods, but they have not been entirely successful. By introducing a pivotal constraint condition, I present here a set of new derivations based on the self-similar hierarchy of cities. First, I derive a pair of exponent laws by postulating local entropy maximizing. From the two exponential laws follows a general hierarchical scaling law, which implies the general form of Zipf’s law. Second, I derive a special hierarchical scaling law with the exponent equal to 1 by postulating global entropy maximizing, and this implies the pure form of Zipf’s law. The rank-size scaling law has proven to be one of the special cases of the hierarchical scaling law, and the derivation suggests a certain scaling range with the first or the last data point as an outlier. The entropy maximization of social systems differs from the notion of entropy increase in thermodynamics. For urban systems, entropy maximizing suggests the greatest equilibrium between equity for parts/individuals and efficiency of the whole.     

Shakespeare and other English Renaissance authors as characterized by Information Theory complexity quantifiers

We introduce novel Information Theory quantifiers in a computational linguistic study that involves a large corpus of English Renaissance literature. The 185 texts studied (136 plays and 49 poems in total), with first editions that range from 1580 to 1640, form a representative set of its period. Our data set includes 30 texts unquestionably attributed to Shakespeare; in addition we also included A Lover’s Complaint, a poem which generally appears in Shakespeare collected editions but whose authorship is currently in dispute. Our statistical complexity quantifiers combine the power of Jensen-Shannon’s divergence with the entropy variations as computed from a probability distribution function of the observed word use frequencies. Our results show, among other things, that for a given entropy poems display higher complexity than plays, that Shakespeare’s work falls into two distinct clusters in entropy, and that his work is remarkable for its homogeneity and for its closeness to overall means.     

Resonant phenomena in extended chaotic systems subject to external noise: The Lorenz’96 model case

We have investigated the effects of noise on an extended chaotic system. The chosen model is the Lorenz’96, a type of “toy” model used for climate studies. Through the analysis of the system’s time evolution and its time and space correlations, we have obtained numerical evidence for two distinct stochastic resonance-like behaviors. Such behaviors are seen when both the usual and a generalized signal-to-noise ratio functions are depicted as a function of the external noise intensity, or of the system size. The underlying mechanisms seem to be associated with a noise-induced chaos reduction. The possible relevance of these and other findings for an optimal climate prediction are discussed.     

Laplace transform analysis of a multiplicative asset transfer model

We analyze a simple asset transfer model in which the transfer amount is a fixed fraction f of the giver’s wealth. The model is analyzed in a new way by Laplace transforming the master equation, solving it analytically and numerically for the steady-state distribution, and exploring the solutions for various values of f∈(0,1). The Laplace transform analysis is superior to agent-based simulations as it does not depend on the number of agents, enabling us to study entropy and inequality in regimes that are costly to address with simulations. We demonstrate that Boltzmann entropy is not a suitable (e.g. non-monotonic) measure of disorder in a multiplicative asset transfer system and suggest an asymmetric stochastic process that is equivalent to the asset transfer model.     

Keyword extraction by entropy difference between the intrinsic and extrinsic mode

This paper proposes a new metric to evaluate and rank the relevance of words in a text. The method uses the Shannon’s entropy difference between the intrinsic and extrinsic mode, which refers to the fact that relevant words significantly reflect the author’s writing intention, i.e., their occurrences are modulated by the author’s purpose, while the irrelevant words are distributed randomly in the text. By using The Origin of Species by Charles Darwin as a representative text sample, the performance of our detector is demonstrated and compared to previous proposals. Since a reference text “corpus” is all of an author’s writings, books, papers, etc. his collected works is not needed. Our approach is especially suitable for single documents of which there is no a priori information available.     

Anisotropic Bianchi type V perfect fluid cosmological models in Lyra’s geometry

The law of variation for mean Hubble’s parameter with average scale factor, in an anisotropic Bianchi type V cosmological space–time, is discussed within the frame work of Lyra’s manifold. The variation of Hubble’s parameter, which gives a constant value of deceleration parameter, generates two types of solutions for the average scale factor; one is the power-law and the other one is of exponential form. Using these two forms, new classes of exact solutions of the field equations have been found for a Bianchi type V space–time filled with perfect fluid in Lyra’s geometry by considering a time-dependent displacement field. The physical and kinematical behaviors of the singular and non-singular models of the universe are examined. Exact expressions for look-back time, luminosity distance and event horizon versus redshift are also derived and their significance are discussed in detail. It has been observed that the solutions are compatible with the results of recent observations.     

The world according to Rényi: thermodynamics of multifractal systems

We discuss basic statistical properties of systems with multifractal structure. This is possible by extending the notion of the usual Gibbs-Shannon entropy into more general framework—Rényi’s information entropy. We address the renormalization issue for Rényi’s entropy on (multi)fractal sets and consequently show how Rényi’s parameter is connected with multifractal singularity spectrum. The maximal entropy approach then provides a passage between Rényi’s information entropy and thermodynamics of multifractals. Important issues such as Rényi’s entropy versus Tsallis-Havrda-Charvat entropy and PDF reconstruction theorem are also studied. Finally, some further speculations on a possible relevance of our approach to cosmology are discussed.     

Interior penalty DG methods for Maxwell’s equations in dispersive media

In this paper, we develop a fully-discrete interior penalty discontinuous Galerkin method for solving the time-dependent Maxwell’s equations in dispersive media. The model is described by a vector integral–differential equation. Our scheme is proved to be unconditionally stable and achieve optimal error estimates in both L² norm and energy norm. The scheme is implemented and numerical results supporting our analysis are presented.     

Steepest entropy increase is justified by information theory. The relations between Ziegler’s principle, Onsager’s formalism and Prigogine’s principle

Evidence for Ziegler’s principle of maximum entropy production has been accumulated from different fields such as climatic studies, crystal growth, dynamics of ecosystems and cellular metabolism. However, Ziegler’s principle is still seen with scepticism by the scientific community. The reasons for this scepticism are the absence of an accepted theoretical justification as well as the fact that Ziegler’s principle formulation seems to contradict Prigogine’s principle of minimum entropy production. In this work we aim to provide a theoretical justification for Ziegler’s principle based on information theory, which is at the basis of Gibbs’ formalism for statistical physics. Similar approaches have previously been attempted, however we believe that the justification provided here is simpler and relies in less questionable hypotheses. Once Ziegler’s principle has been formulated as a consequence of information theory, its relations with Onsager’s formulation and Prigogine’s principle are explored.     

Boltzmann,Darwin and Directionality theory

Boltzmann’s statistical thermodynamics is a mathematical theory which relates the macroscopic properties of aggregates of interacting molecules with the laws of their interaction. The theory is based on the concept thermodynamic entropy, a statistical measure of the extent to which energy is spread throughout macroscopic matter. Macroscopic evolution of material aggregates is quantitatively explained in terms of the principle: Thermodynamic entropy increases as the composition of the aggregate changes under molecular collision.     

Rényi’s information transfer between financial time series

In this paper, we quantify the statistical coherence between financial time series by means of the Rényi entropy. With the help of Campbell’s coding theorem, we show that the Rényi entropy selectively emphasizes only certain sectors of the underlying empirical distribution while strongly suppressing others. This accentuation is controlled with Rényi’s parameter q$q$. To tackle the issue of the information flow between time series, we formulate the concept of Rényi’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 Rényian 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 02.04.2008–11.09.2009 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 Rényi information flow between the DAX and S&P500 indices.     

Econophysical visualization of Adam Smith’s invisible hand

Consider a complex system whose macrostate is statistically observable, but yet whose operating mechanism is an unknown black-box. In this paper we address the problem of inferring, from the system’s macrostate statistics, the system’s intrinsic force yielding the observed statistics. The inference is established via two diametrically opposite approaches which result in the very same intrinsic force: a top-down approach based on the notion of entropy, and a bottom-up approach based on the notion of Langevin dynamics. The general results established are applied to the problem of visualizing the intrinsic socioeconomic force–Adam Smith’s invisible hand–shaping the distribution of wealth in human societies. Our analysis yields quantitative econophysical representations of figurative socioeconomic forces, quantitative definitions of “poor” and “rich”, and a quantitative characterization of the “poor-get-poorer” and the “rich-get-richer” phenomena.     

Bifurcation phenomena in a convection problem with temperature dependent viscosity at low aspect ratio

In this paper we analyse convective solutions of a two dimensional fluid layer in which viscosity depends exponentially on temperature. This problem takes in features of mantle convection, since large viscosity variations are to be expected in the Earth’s interior. These solutions are compared with solutions obtained at constant viscosity. Special attention is paid to the influence of the aspect ratio in the solutions presented. The analysis is assisted by bifurcation techniques such as branch continuation, which has proven to be a useful, systematic method for gaining insight into the possible stationary solutions satisfied by the basic equations. One feature presented by the fluid with non constant viscosity is the presence of pitchfork and saddle-node subcritical bifurcations and the presence of convective solutions below the linear critical threshold. The analysis also provides limits of existence of stationary solutions and draws the boundaries for time dependent convection.     

Chapman-Enskog-maximum entropy method on time-dependent neutron transport equation

The time-dependent neutron transport equation in semi and infinite medium with linear anisotropic and Rayleigh scattering is proposed. The problem is solved by means of the flux-limited, Chapman-Enskog-maximum entropy for obtaining the solution of the time-dependent neutron transport. The solution gives the neutron distribution density function which is used to compute numerically the radiant energy density E(x,t), net flux F(x,t) and reflectivity R_f. The behaviour of the approximate flux-limited maximum entropy neutron density function are compared with those found by other theories. Numerical calculations for the radiant energy, net flux and reflectivity of the proposed medium are calculated at different time and space.     

Time-dependent radiative transfer using Chapman-Enskog maximum entropy method

The time-dependent problems of radiative transfer involve a coupling between radiation and material energy fields and are nonlinear because of proposed temperature dependence of the medium characteristics in semi-infinite medium with Rayleigh anisotropic scattering. By means of the limited flux, Chapman-Enskog and maximum entropy technique the time-dependent radiative transfer equation has been solved explicitly. The maximum entropy method is used to solve the resulting differential equation for radiative energy density. The calculations are carried out for temperature (normalized dimensionless) Θ(x,τ), radiative energy density and net flux with Rayleigh and anisotropic scattering for different space at different times.