A statistical physics view of financial fluctuations: Evidence for scaling and universality

The unique scaling behavior of financial time series have attracted the research interest of physicists. Variables such as stock returns, share volume, and number of trades have been found to display distributions that are consistent with a power-law tail. We present an overview of recent research joining practitioners of economic theory and statistical physics to try to understand better some puzzles regarding economic fluctuations. One of these puzzles is how to describe outliers, i.e. phenomena that lie outside of patterns of statistical regularity. We review recent research, which suggests that such outliers may not in fact exist and that the same laws seem to govern outliers as well as day-to-day fluctuations.     

Correlated randomness and switching phenomena

One challenge of biology, medicine, and economics is that the systems treated by these serious scientific disciplines have no perfect metronome in time and no perfect spatial architecture—crystalline or otherwise. Nonetheless, as if by magic, out of nothing but randomness one finds remarkably fine-tuned processes in time and remarkably fine-tuned structures in space. Further, many of these processes and structures have the remarkable feature of “switching” from one behavior to another as if by magic. The past century has, philosophically, been concerned with placing aside the human tendency to see the universe as a fine-tuned machine. Here we will address the challenge of uncovering how, through randomness (albeit, as we shall see, strongly correlated randomness), one can arrive at some of the many spatial and temporal patterns in biology, medicine, and economics and even begin to characterize the switching phenomena that enables a system to pass from one state to another. Inspired by principles developed by A. Nihat Berker and scores of other statistical physicists in recent years, we discuss some applications of correlated randomness to understand switching phenomena in various fields. Specifically, we present evidence from experiments and from computer simulations supporting the hypothesis that water’s anomalies are related to a switching point (which is not unlike the “tipping point” immortalized by Malcolm Gladwell), and that the bubbles in economic phenomena that occur on all scales are not “outliers” (another Gladwell immortalization). Though more speculative, we support the idea of disease as arising from some kind of yet-to-be-understood complex switching phenomenon, by discussing data on selected examples, including heart disease and Alzheimer disease.     

Correlated randomness: Some examples of exotic statistical physics

One challenge of biology, medicine, and economics is that the systems treated by these sciences have no perfect metronome in time and no perfect spatial architecture-crystalline or otherwise. Nonetheless, as if by magic, out of nothing but randomness one finds remarkably fine-tuned processes in time and remarkably fine-tuned structures in space. To understand this ‘miracle’, one might consider placing aside the human tendency to see the universe as a machine. Instead, one might address the challenge of uncovering how, through randomness (albeit, as we shall see, strongly correlated randomness), one can arrive at many spatial and temporal patterns in biology, medicine, and economics. Inspired by principles developed by statistical physics over the past 50 years-scale invariance and universality-we review some recent applications of correlated randomness to fields that might startle Boltzmann if he were alive today.     

Real-time measurements,rare events and photon economics

Rogue events otherwise known as outliers and black swans are singular, rare, events that carry dramatic impact. They appear in seemingly unconnected systems in the form of oceanic rogue waves, stock market crashes, evolution, and communication systems. Attempts to understand the underlying dynamics of such complex systems that lead to spectacular and often cataclysmic outcomes have been frustrated by the scarcity of events, resulting in insufficient statistical data, and by the inability to perform experiments under controlled conditions. Extreme rare events also occur in ultrafast physical sciences where it is possible to collect large data sets, even for rare events, in a short time period. The knowledge gained from observing rare events in ultrafast systems may provide valuable insight into extreme value phenomena that occur over a much slower timescale and that have a closer connection with human experience. One solution is a real-time ultrafast instrument that is capable of capturing singular and randomly occurring non-repetitive events. The time stretch technology developed during the past 13 years is providing a powerful tool box for reaching this goal. This paper reviews this technology and discusses its use in capturing rogue events in electronic signals, spectroscopy, and imaging. We show an example in nonlinear optics where it was possible to capture rare and random solitons whose unusual statistical distribution resemble those observed in financial markets. The ability to observe the true spectrum of each event in real time has led to important insight in understanding the underlying process, which in turn has made it possible to control soliton generation leading to improvement in the coherence of supercontinuum light. We also show a new class of fast imagers which are being considered for early detection of cancer because of their potential ability to detect rare diseased cells (so called rogue cells) in a large population of healthy cells.     

Crises and Collective Socio-Economic Phenomena: Simple Models and Challenges

Financial and economic history is strewn with bubbles and crashes, booms and busts, crises and upheavals of all sorts. Understanding the origin of these events is arguably one of the most important problems in economic theory. In this paper, we review recent efforts to include heterogeneities and interactions in models of decision. We argue that the so-called Random Field Ising model (rfim) provides a unifying framework to account for many collective socio-economic phenomena that lead to sudden ruptures and crises. We discuss different models that can capture potentially destabilizing self-referential feedback loops, induced either by herding, i.e. reference to peers, or trending, i.e. reference to the past, and that account for some of the phenomenology missing in the standard models. We discuss some empirically testable predictions of these models, for example robust signatures of rfim-like herding effects, or the logarithmic decay of spatial correlations of voting patterns. One of the most striking result, inspired by statistical physics methods, is that Adam Smith’s invisible hand can fail badly at solving simple coordination problems. We also insist on the issue of time-scales, that can be extremely long in some cases, and prevent socially optimal equilibria from being reached. As a theoretical challenge, the study of so-called “detailed-balance” violating decision rules is needed to decide whether conclusions based on current models (that all assume detailed-balance) are indeed robust and generic.     

Subordination to periodic processes and synchronization

We study the subordination to a process that is periodic in the natural time scale, and equivalent to a clock with N states. The rationale for this investigation is given by a set of many interacting clocks with N states. The natural time scale representation corresponds to the dynamics of an individual clock with no interaction with the other clocks of this set. We argue that the cooperation among the clocks of this set has the effect of generating a global clock, whose times of sojourn in each of its N states are described by a distribution density with an inverse power law form and power index μ<2. This is equivalent to extending the widely used subordination method from fluctuation-dissipation processes to periodic processes, thereby raising the question of whether special conditions exist of perfect synchronization, signaled by regular oscillations, and especially by oscillations with no damping. We study first the case of a Poisson subordination function. We show that in spite of the random nature of the subordination method the procedure has the effect of creating damped oscillations, whose damping vanishes in the limiting case of N?1, thereby suggesting a condition of perfect synchronization in this limit. The Bateman’s mathematical arguments [H. Bateman, Higher Transcendental Functions, vol. III, Robert K Krieger, Publishing Company, Inc. Krim.Fr. Drive Malabar, FL; Copyright 1953 by McGraw-Hill Book Company Inc.] indicate that the condition of perfect synchronization is possible also in the non-Poisson case, with μ<2, although it may lie beyond the range of computer simulation. To make the theoretical predictions accessible to numerical simulation, we use a subordination function whose survival probability is a Mittag-Leffler exponential function. This method prevents us from directly establishing the macroscopic coherence emerging from μ=2, which generates a perfect form of 1/f noise. However, it affords indirect evidence that perfect synchronization signaled by undamped regular oscillations may be produced in this case. Furthermore, we explore a condition characterized by an excellent agreement between theory and numerical simulation, where the long-time region relaxation, with a perfect inverse power law decay, emerging from the subordination to ordinary fluctuation-dissipation processes, is replaced by exponentially damped regular oscillations.     

Extremal black hole entropy from conformal string sigma model

We present a string-theory derivation of the semiclassical entropy of extremal dyonic black holes in the approach based on the conformal sigma model (NS-NS embedding of the classical solution). We demonstrate (resolving some puzzles existing in previous related discussions) that the degeneracy responsible for the entropy is due to string oscillations in four transverse dimensions ‘intrinsic’ to the black hole: four non-compact directions of the D = 5 black hole case and three non-compact and one compact (responsible for embedding of magnetic charges) dimensions in the D = 4 black hole case. Oscillations in other compact internal dimensions give subleading contributions to statistical entropy in the limit when all charges are large. The dominant term in the statistical entropy is thus universal (i.e. is the same in type II and heterotic string theory) and agrees with the Bekenstein-Hawking expression.     

Nonlocality for graph states

The possibility of preparing two-photon entangled states encoding three or more qubits in each photon leads to the following problem: If n quabits were distributed between two parties, which quantum pure states and qubit distributions would allow all-versus-nothing (or Greenberger-Horne-Zeilinger-like) proofs of Bell’s theorem using only single-qubit measurements? We show a necessary and sufficient condition for the existence of these proofs and provide all existing proofs up to n = 7 qubits. On the other hand, the possibility of preparing n-photon n-qubit graph states leads to the following problem: If n qubits were distributed between n parties, which would be the optimal Bell inequalities? We show all optimal n-party Bell inequalities for the perfect correlations of any graph state of n < 6 qubits. Optimal means that the ratio between the quantum violation and the bound for local hidden-variable theories is the maximum over all possible combinations of perfect correlations. This implies that the required detection efficiencies for loophole-free Bell tests are minimal.     

Exploring an opinion network for taste prediction: An empirical study

We develop a simple statistical method to find affinity relations in a large opinion network which is represented by a very sparse matrix. These relations allow us to predict missing matrix elements. We test our method on the Eachmovie data of thousands of movies and viewers. We found that significant prediction precision can be achieved and it is rather stable. There is an intrinsic limit to further improve the prediction precision by collecting more data, implying perfect prediction can never obtain via statistical means.     

Observational constraints on the time-dependence of dark energy

One of the most important questions nowadays in physics concerns the nature of the so-called dark energy. It is also a consensus among cosmologists that such a question will not be answered on the basis only of observational data. However, it is possible to diminish the range of possibilities for this dark component by comparing different dark energy scenarios and finding which models can be ruled out by current observations. In this paper, by assuming three distinct parametrizations for the low-redshift evolution of the dark energy equation of state (EOS), we consider the possibility of discriminating between evolving dark energy and ΛCDM models from a joint analysis involving the most recent radio sources gravitational lensing sample, namely, the Cosmic all Sky Survey (CLASS) statistical data and the recently published gold SNe Ia sample. It is shown that this particular combination of observational data restricts considerably the dark energy parameter space, which enables possible distinctions between time-dependent and constant EOSs.     

Gamma-ray bursts in the Swift-Fermi era: Confronting data with theory

With the prompt slewing capability of the X-ray and UV-optical telescopes onboard the Swift mission and with the gamma-ray large area telescope onboard the Fermi mission, gamma-ray bursts (GRBs) are now accessible in a full time window and in all electromagnetic wavelengths for the events. Many observational breakthroughs have been made in recent years. I present here a brief review of some observational breakthroughs with the two missions, focusing on how these breakthroughs have revolutionized our understanding of the nature of this phenomenon and puzzles as well as challenges of confronting the conventional models with data.     

Demonstrating mesoscopic superpositions in double-well Bose-Einstein condensates

The availability of Bose-Einstein condensates as mesoscopic or macroscopic quantum objects has aroused new interest in the possibility of making and detecting coherent superpositions involving many atoms. We consider the important problem of distinguishing whether a coherent superposition or a statistical mixture is generated by a given experimental procedure, using the specific example of a double-well condensate. In this system, such a superposition state can be generated by using a Feshbach resonance to tune the inter-atomic interactions. We find that unambiguously distinguishing even a perfect ‘NOON’ state from a statistical mixture using standard detection methods will present experimental difficulties.     

Switching Phenomena in a System with No Switches

It is widely believed that switching phenomena require switches, but this is actually not true. For an intriguing variety of switching phenomena in nature, the underlying complex system abruptly changes from one state to another in a highly discontinuous fashion. For example, financial market fluctuations are characterized by many abrupt switchings creating increasing trends (“bubble formation”) and decreasing trends (“financial collapse”). Such switching occurs on time scales ranging from macroscopic bubbles persisting for hundreds of days to microscopic bubbles persisting only for a few seconds. We analyze a database containing 13,991,275 German DAX Future transactions recorded with a time resolution of 10 msec. For comparison, a database providing 2,592,531 of all S&P500 daily closing prices is used. We ask whether these ubiquitous switching phenomena have quantifiable features independent of the time horizon studied. We find striking scale-free behavior of the volatility after each switching occurs. We interpret our findings as being consistent with time-dependent collective behavior of financial market participants. We test the possible universality of our result by performing a parallel analysis of fluctuations in transaction volume and time intervals between trades. We show that these financial market switching processes have properties similar to those of phase transitions. We suggest that the well-known catastrophic bubbles that occur on large time scales—such as the most recent financial crisis—are no outliers but single dramatic representatives caused by the switching between upward and downward trends on time scales varying over nine orders of magnitude from very large (≈10² days) down to very small (≈10 ms).     

On the possible running of the cosmological “constant”

Despite the many outstanding cosmological observations leading to a strong evidence for a non-vanishing cosmological constant (CC) term Λ in the gravitational field equations, the theoretical status of this quantity seems to be lagging well behind the observational successes. It thus seems timely to revisit some fundamental aspects of the CC term in Quantum Field Theory (QFT). We emphasize that, in curved space–time, nothing a priori prevents this term from potentially having a mild running behavior associated to quantum effects. Remarkably, this could be the very origin of the dynamical nature of the Dark Energy, in contrast to many other popular options considered in the literature. In discussing this possibility, we also address some recent criticisms concerning the possibility of such running. Our conclusion is that, while there is no comprehensive proof of the CC running, there is no proof of the non-running either. The problem can be solved only through a deeper understanding of the vacuum contributions of massive quantum fields on a curved space–time background. We suggest that such investigations are at the heart of one of the most important endeavors of fundamental theoretical cosmology in the years to come.     

Fock space construction of the massless dipole field

We construct the Fock space representation of the free massless scalar dipole field in terms of creation and annihilation operators for the eigenvectors of the momentum operator. The Poincaré group is implemented unitarily only on a subspace of the full (positive metric) Hilbert space. The subspace possesses a hermitean, local, irreducible scalar field constructed out of the (non-hermitean) dipole field. Thus this subspace is a perfect candidate for a physical subspace of observable particles. We show that this possibility is however excluded by the fact that these particles interact with an external c-number source in a manner that violates unitarity. We illustrate our construction by applying it to the linearized Higgs model with external c-number source and examine the (non-trivial) dynamics of the dipole degrees of freedom in this case. An explicit separation of the physical degrees of freedom from the unphysical ones is presented for this interacting model.     

Transport in quantum wires

With a brief introduction to one-dimensional channels and conductance quantization in mesoscopic systems, we discuss some recent experimental puzzles in these systems, which include reduction of quantized conductances and an interesting odd-even effect in the presence of an in-plane magnetic field. We then discuss a recent non-homogeneous Luttinger liquid model proposed by us, which addresses and gives an explanation for the reduced conductances and the odd-even effect. We end with a brief summary and discussion of future projects.     

Search for superconductivity in hydrogen-implanted platinum

A recent theoretical calculation by Papaconstantopoulos predicted superconductivity in platinum hydride, possibly with a rather high critical temperature (about 10 K as in PdH). We have produced PtH_x (x ? 4.00) by ion-implantation, and find no evidence for superconductivity above 1.7 K. Residual resistivity measurements indicate that (in contrast to previous results on NiH_x and PdH_x) the Pt monohydride is not formed. We have considered the possibility that the dihydride might be formed.     

Robust methods of inclusive outlier analysis for structural health monitoring

The key novel element of this work is the introduction of robust multivariate statistical methods into the structural health monitoring (SHM) field through use of the minimum covariance determinant estimator (MCD) and the minimum volume enclosing ellipsoid (MVEE). In this paper, robust outlier statistics are investigated, focussed mainly on a high level estimation of the “masking effect” of inclusive outliers, not only for determining the presence or absence of novelty-something that is of fundamental interest but also to examine the normal condition set under the suspicion that it may already include multiple abnormalities. By identifying and detecting variability at an early stage, the prospects of achieving good generalisation and establishing a correct normal condition classifier may be increased. It is critical to highlight that there is no a priori division between the damaged and the undamaged condition data when the algorithms are implemented, offering a significant advantage over other methodologies. In summary, this paper introduces a new scheme for SHM by exploiting robust multivariate outlier statistics in order to investigate if the selected features are free from multiple outliers before such features can be selected for either supervised or unsupervised analysis.     

Bonding anomalies in the rare earth sesquioxides

Raman and IR vibrational spectra of C-type rare earth sesquioxides exhibit anomalous wavenumber decreases for europium (f⁶) and ytterbium (f¹³) oxide which have not been previously reported. Because of the large shifts observed for the Raman spectra, the IR band positions have been re-examined. Smaller, but definite decreases have been observed in the IR spectra as well. Scatter in the literature data had obscured these anomalies. The anomalies correlate with thermodynamic properties and indicate weaker bonding for these two oxides. It is suggested that these anomalies are caused by electron-phonon interaction, involving low-lying f-states which are below the Fermi level for europium and ytterbium. Therefore, these data provide another example of a system in which 4f electronic structure significantly influences chemical bonding. We suggest that it is due to the presence of the unique, low-lying electronic energy levels in these two elements, and not simply the result of the number of f electrons (half-full or full). Literature results suggest the anomalies are not due to “mixed valence” for Eu₂O₃ and Yb₂O₃; however, further research is needed to explain the origin more completely. Raman band positions are reported for the first time for several of the oxides.