Forschung

STATISTICAL PHYSICS OF COMPLEX SYSTEMS

What is a complex system?

There is no generally accepted definition, simply because the term is used in many different fields ranging from physics and chemistry over life science and medicine to economics and sociology.
schematic drawing of an atomic nucleus
​But even when we restrict the question to the field of physics, a full-fledged, comprehensive definition is lacking. Rather, one identifies various characterizing features as possible signatures of complexity. It is helpful to explain complexity by means of examples. Consider an atomic nucleus, as schematically shown in the picture.

​​

​The heavier nuclei consist of many nucleons, the protons (blue) and the neutrons (red). They all interact with each other and thus move in a very complicated way. A confusingly rich and entangled variety of nuclear excitations exists depending on how the system is probed. For example, while the individual nucleons move incoherently like a bunch of flies at some energies, they seem to suddenly conspire and move coherently like a school of fish at other energies. This emergence of collectivity has a large impact on statistical properties of the nuclear spectra, leading straight away to Many-Body Quantum Chaos. It may be viewed as one of the signatures of complexity. 

While nuclei are systems in traditional physics, just seen from the complexity viewpoint, a large, if not the largest part of research in complex systems is interdisciplinary. Financial markets, for example, are a fascinating challenge for a physics-style of analysis as they exhibit several signatures of complexity. 
  incoherently moving fish
​ A large but finite number of traders interact, there is incoherent motion of the stock prices in quiet periods and emergent collective phenomena during crises. The system is highly correlated and highly non-stationary. It is also open, there are internal (endogenous) mechanisms in the financial markets as well as external (exogenous) effects on them, such as political developments. Governing equations such as the Newton or Schrödinger equations in traditional physcis do not exist, neither for the individual trader's decisions nor for the financial market as a whole. The price formation results from rather involved rules encoded in the order book.​

​​ 

But is the study of such interdisciplinary systems physics? One may have different opinions, but our answer is a clear yes! Because for us the science of physics is, first, accumulating quantitative experimental or empirical information and, second, theoretical model building solely based on this information. For a physicist, this sounds trivial, but this is not the prime methodology in other sciences. The application of the physics approach has been impossible for a long time or is still impossible in many other sciences, simply because of lack of data and/or because the degree of complexity is too high, for example in medicine. 

​
collectively moving school of fish ​​
​When we do interdisciplinary research on systems such as financial markets or traffic or the human brain, we want to work as physicist with the same standards and ambitions as in the traditional fields of physics. We are physicists, we do not want to become economists, biologists, neuroscientists, etc.  It is of course essential to learn, discuss and collaborate across the disciplines.​
            a brain with its neurons
​​ The theoretical understanding and modeling of complex systems is a major challenge, there are too many degrees of freedom for a simple-minded reductionist treatment, but not enough for a statistical mechanics description. Hence, various approaches and techniques have to be used and combined, complementing each other, and, furthermore, the development of new theoretical tools and of new methods for data analysis is called for and a fantastic challenge, particularly for young scientists.

Current Research Interests

Many-Body Quantum Chaos and Driven, Interacting Spin Chains

Spin systems, in particular chains and fields, are a topical subject that enjoys considerable attention. First, experimentalists are capable of ever better realizing these systems in solids or on surfaces, second, there is a close connection to qubit systems in quantum information, third, there are connections that extend into field theory and, fourth, the analytical structure of these systems is so rich that the structure of these systems facilitates the study of deep questions about many-body chaos, integrability and exact solvability.
                 interacting birds on wires
​

The time evolution of driven, interacting spin chains exhibit a duality, i.e. partition and generating functions can be formulated via a time evolution operator of the spin chain for a fixed number of particles or via a "particle number enlargement operator" at fixed time. While the former is always unitary, this is true for the latter only under certain conditions, defining dual unitarity, see

In this vibrant field, there is a huge variety of interesting systems extending and generalizing the above described ones, for example driven, interacting arrays of spins, see Osipov, Krieger, Guhr, Gutkin Local correlations in partially dual-unitary lattice models Phys. Rev. B 109, 214302 (2024),  preprint:  arXiv:2312.03445  for different types of interactions.

Short-Term Memory in Finance and the Limits of the Efficient Market Hypothesis

​It is quite plausible that the stock market trading proceeds without memory effects, because if there were such effects to a sizeable extent, they would at least
                                    bull and bear
​ in a statistical sense result in a predictability of prices, which in turn would lead to riskless profits, i.e. to some kind of arbitrage in the economic terminology. Apart from the fact that this would be every investor's dream, its existence would imply the risk of market distortions and dramatic instabilities. The associated systemic risk is therefore of very fundamental interest. In fact, economic theories of the financial markets, in particular Fama's Efficient Market Hypothesis (Nobel Memorial Prize in 2013) and Zero Intelligence Trading are based on the assumption of memoryless trading.  But this cannot not be completely correct, since many traders work with strategies that amount to stretching out larger investments over time so that temporal correlations and thus memory effects are inevitably induced.

From a systemic point of view, it is therefore important to quantitatively analyze the undoubtedly existing memory effects and to quantitatively assess the resulting risk.  Some years ago, memory effects were measured with response functions in intraday trading of one stock [1]. By defining cross-response functions, we extended that to the whole correlated financial market, see

We found memory effects in intraday trading on time scales from 30 min to 120 min until they then disappear. After these time scales, market efficiency is restored. We had carried out these investigations for the year 2008.  To answer the important question whether this situation changed over the years,

The rules of the game are still the same, but it is played in a different, more risk averse style. This reflects the strong nonstationarity of the financial markets. 

[1] Bouchaud, Gefen, Potters, Wyart Fluctuations and response in financial markets: the subtle nature of ‘random’ price changes, Quantitative Finance 4 (2004)

Ultra Extreme Fast Events in Stock Market Trading and their Impact

A few years ago, reports circulated in the media about the the harmful influence of algorithmic trading, i.e. automatic trading by computers based on sophisticated programs and state-of-the-art modern methods such as artificial intelligence. The Frankfurter Allgemeine Zeitung published several articles such as [1,2,3], which dealt with the sudden crashes of the markets. With reference to some empirical analyses, the claim of these and other articles was that high-frequency algorithmic trading as such can cause serious problems, namely flash crashes, also known as Ultrafast Extreme Events (UEEs). On closer inspection, however, we had doubts about the statistical significance of these analyses. 
everything goes down the drain
​​

In our much more detailed study 

we showed that UEEs are by no means inevitable consequences of high-frequency algorithmic trading but rather follow predominantly normal market mechanisms.This analysis was carried out for the period 2007/2008 in which the financial and credit crisis set in. Furthermore, the fast development of computers and artificial intelligence might have changed the situation since, making a study of subsequent time periods highly desirable. We carried out such a data analysis

[1] Das systemische Risiko der Dummheit (15.10.2013)

[2] Der geheimnisvolle Schuldige am Flash Crash (22.04.2015)

[3] Flash Crash: Wenn der Blitz in die Finanzmärkte einschlägt (03.01.2019)

Random Matrix Theory

​
sheer randomness
​Random matrices provide powerful models for a rich variety of complex systems. Two types of Random Matrix Theory (RMT) have to be distinguished: Wigner's RMT for dynamical systems [1] and Wishart's RMT for large covariance or correlation matrices [2]

[1] Mehta Random Matrices Elsevier LTD, Oxford 1990

[2] Anderson An Introduction to Multivariate Statistical Analysis, New Jersey 2003,  

Wigner developed his idea for wave equations, originally for the Schrödinger equation in Quantum Mechanics. If the system is sufficiently complex due to a large number of coupled degrees of freedom or complicated interactions or both, the statistical fluctuation properties of the spectra are well described by large random matrices, i.e. by matrices with random entries and a large dimension which is eventually taken to infinity, Stochastic Quantum Scattering, was an early and is to the day a very fruitful application, with connections even to fields such as wireless communication [1]. Over the years it was found that Wigner's RMT also applies classical wave systems such as microwave cavities and reveberating objects, even though the correponding wave operators can be quite different from the Hamilton operator. Here is a brief review [2]  and here is a longer review [3] Although it is more than 25 years old it still gives an impression as of how broad the field RMT is, and in the last 25 years it become much broader. One of the new developments is Statistical Topology. 

 

[1] Tulino, Verdú Random Matrix Theory and Wireless Communications now, 2004

[2] Guhr Random Matrix Theory in Physics, 2005

[3] Guhr, Mueller-Groeling, Weidenmüller Random Matrix Theories in Quantum Physics: Common Concepts. Physics Reports 299, (1998)

Wishart's RMT is an important tool in statistical inference. The random matrices model time series of large correlated systems. The statistical properties of the corresponding large covariance or correlation matrices serve as null hypothesis. More recently, new interpretations of Wishart random matrices led to unexpected new applications for Nonstationarity in Correlated Complex Systems.

Credit Risk

The instability of the financial system as experienced in the period following the year 2007 is connected with to credit defaults, i.e., with the failure of obligors to make promised payments. Given the large number of credit contracts, this problem is amenable to be treated with approaches developed in statistical physics. We introduced a random matrix approach to model the nonstationarity of financial correlations, see Multivariate Distributions for Nonstationary Correlated Complex Systems. The fluctuations of the correlations lift the tails of the multivariate distribution entering the loss given default distributions which in turn become heavier. We quantitavely derive the generic features of credit risk and show how the often advertised concept of diversification, i.e., reducing the risk by distributing it, is deeply flawed when it comes to credit risk. The risk of extreme losses remains due to the ever present correlations, implying a substantial and persistent intrinsic danger to the financial system.
debts - and the clock is ticking
​​

We also investigated concurrent effects for creditors with overlapping portfolios 

Statistical Topology

​
Möbius structures
​Topological questions currently receive considerable attention because of their relevance in many areas of physics, including condensed matter matter, quantum optics and quantum chromodynamics. In a German-Israeli project we investigate generic topological properties of complex systems.  As statistical models, we use random matrices as they are known to reliably capture universal features of complex systems. In our first studies, we were interested in the universal statistics of of simple topological quantum numbers, the winding numbers. We also worked out the winding number densities and their correlations in chiral complex systems of the three symmetry classes orthogonal, unitary and symplectic. See 

as well as our results for ratios of determinants in 

  • Hahn, Kieburg, Gat, Guhr Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Determinants with Parametric Dependence J. Math. Phys. 64, 021901 (2023),  preprint: arXiv:2207.08612  the unitary and symplectic cases and in 
  • Hahn, Kieburg, Gat, Guhr Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Parametric Determinants in the Orthogonal Case J. Math. Phys. 64, 111902 (2023) ,  preprint:  arXiv:2306.12051> the orthogonal case. In the unitary case, we also calculated the universal correlations and the moments:
  • Hahn, Kieburg, Gat, Guhr Winding Number Statistics for Chiral Random Matrices: Universal Correlations and Statistical Moments in the Unitary Case, preprint: arXiv:2410.22808

Supersymmetry for Arbitrary Random Matrix Probability Densities

​Supersymmetry is an extension of analysis, linear algebra, group theory and related to spaces which comprise the ordinary commuting (bosonic) as well as anticommuting (fermionic) variables. These bosons and fermions work collaborate very well, providing new and powerful
bosonic and fermionic superpower
​mathematical tools [1]. Yet, two important remarks are in order. (1) There is a related but different notion of Supersymmetry in physics referring to a technique for calculating eigenvalues of operators. This is not what we have in mind here. (2) The above Supersymmetry that explicitly uses bosons and fermions has two very different applications. (2a) In high energy physics and field theory fermions given bosonic partners and vice versa which greatly improves the renornamalization properties. Here, the bosons and fermions have a direct physics meaning. Unfortunately, this Supersymmetry could not be confirmed in the recent experiments at CERN: (2b) In Random Matrix Theory and in the condensed matter physics of disordered systems, the bosons and fermions are not used as physical particles, rather they serve as bookkeeping devices to conveniently rewrite generating functions of correlators [2] which greatly facilitates ensemble averages. 

[1] Felix Alexandrovich Berezin Introduction to Superanalysis. Springer Verlag 1987

[2] Konstantin Efetov Supersymmetry in Disorder and Chaos Cambridge University Press 1996

This brief review: 

  • Guhr Supersymmetry invited contribution in: The Oxford Handbook of Random Matrix Theory, G. Akemann, J. Baik and P. Di Francesco (eds.), Oxford University Press, Oxford, 2011 Chapter 7, pages 135 - 154 , preprint: arXiv:1005.0979 

explains how that works.

We developed a new variant of the Supersymmetry method to also calculate certain distributions, see Stochastic Quantum Scattering. It had always been believed that the application of Supersymmetry requires Gaussian probability densities of the random matrices. Some time ago we gave the first proof that this is not the case. Supersymmetric representations of the generating functions exists for arbitrary probability densities 

In this paper which treats Hermitian random matrices it is also shown and illustrated by examples that, in general, the probability density in ordinary space and its formal equivalent in superspace do not have the same functional form. This is also seen from a series of other examples in a special class of probability densities: 

The problem was also solved in a related but different way for Hermitian, real-symmetric and self-dual random matrices in the paper [1].

We then also treated the cases of real-symmetric and self-dual random matrices,

The equivalance of our approach and that of [1] was debated, but we proved their mathematical equivalence in 

[1] Littelmann, Sommers, Zirnbauer Superbosonization of Invariant Random Matrix Ensembles, Communications in Mathematical Physics 283 (2008) 343 

Multivariate Distributions for Nonstationary Correlated Complex Systems

​
colorful spices
​We came across this problem in our study of financial markets, but our approach is a general solution applicable to a large variety of complex systems. In finance, already the investigation of the univariate return distributions is almost a field of its own within econophysics. The enormous wealth of available data facilitates detailed empirical studies. It turns out that the univariate return distributions are neither stable nor stationary. They can be fitted, but not fully captured by any dynamic or stochastic model which a physicist would find satisfactory. From a systemic point of view these univariate distributions are of utmost importance because they are often heavy-tailed, reflecting drastic risk in the financial markets, much larger than predicted by the popular Gaussian models. The reasons for the heavy tails are only partially understood. The order book is the platform that provides the same information to all traders, and its nontrivial dynamics plays an important role. In addition, the intrinsic nonstationarity of trading, the erratic succession of calm and hectic periods outmaneuvering any kind of equilibrium in the financial markets plays a relevant role, too.

However, if the univariate return distributions are already so complicated, the multivariate ones, which describe the fully correlated financial market must be even more complicated. Some years ago, we put forward a first model to capture the nonstationarity by separating time scales and by giving Wishart's Random Matrix Theory see Random Matrix Theory, a different and new interpretation.  We show that the nonstationary fluctuations of the correlations lift the tails of the multivariate return distributions. 

More recently, we considerably extended this by also employing algebraic distributions, particularly for the random matrices 

Our model works very well for the multivariate return distributions for high-frequency financial data 

Applications to other nonstationary correlated complex systems are in the making.

Traffic Networks

​
freeway interchanges at night​Together with the working group Schreckenberg we apply modern statistical analysis methods to flow and speed data of motorway network. The densely populated State of Nordrhein-Westfalen in Germany has a very large motorway network, the data are measured at with detectors at a large number of points o the network.  As a first result, we identified antipersistence of traffic flow time-series which impacts the duration of traffic congestion. There are universal characteristics on a wide range of time scales 

Correlations are of great interest, we used methods which my group had developed for financial markets. We identify dominant and subdominant collective effects. We solve a major challenge, namely the mapping of the correlation structure between the time series measured at the arbitrarily labeled detectors to the real network, i.e. to the motorway map. 

Traffic networks have a memory which is obvious as everybody makes this unpleasant experience when getting stuck in a traffic jam. To study the memory induced statistical dependencies, response functions, see Short-Term Memory in Finance and the Limits of the Efficient Market Hypothesis, are the ideal tool. We confirm the backwards traveling wave excited by a congestion in new and statistically highly significant way 

In a more general study on the whole network, we analyze the spatiotemporal statistical features as responses to congestions and find universal scaling 

When travelling by car from one location to another, our route is constrained by the road network. The network distance between the two locations is generally longer than the geodetic distance as the crow flies. We report a systematic relation between the statistical properties of these two distances. Empirically, we find a robust scaling between network and geodetic distance distributions for a variety of large motorway networks. A simple consequence is that we typically have to drive 1.3 ± 0.1 times longer than the crow flies. This scaling is not present in standard random networks; rather, it requires non-random adjacency. We develop a set of rules to build a realistic motorway network, also consistent with the above scaling. We hypothesise that the scaling reflects a compromise between two societal needs: high efficiency and accessibility on the one hand, and limitation of costs and other burdens on the other 

Windturbines and Windfarms

​

​​
offshore windfarm
​We contribute to new concepts and statistical methods for the analysis of offshore wind farm data, with the aim of early failure detection. We received a wealth of process data from the operating companies for various observables such as wind speed, torques, angle, temperatures, etc. in the form of high-frequency time series. We investigated the correlation structures of the observables and in particular their temporal dynamics. This followed on from our investigations of other, also highly non-stationary systems such as financial markets and Traffic Networks, making a transfer of methods possible. We analyzed the correlations with respect to spatial structures and found a sensitivity to the wind farm architecture 

We also identified with a clustering analysis, but otherwise inputfree, three distinct operational system states of wind turbines, predominantly determined by the wind speed 

 

Conditioned on these system states, we constructed stochastic process for the collective dynamics and also investigated the transitions 

  • Bette, Philipp, Wächter, Freund, Peinke, Guhr Dynamics of wind turbine operational states preprint: arXiv:2310.06098 

The non-stationarity assessed in this way can be employed to significantly improve Langevin power curve analysis 

Stochastic Quantum Scattering

Quantum scattering experiments are of great importance as they reveal microscopic processes in the target by mapping them on macroscopic scales in detectors. The associated theory was pioneered in nuclear physics. Amplitude and phase of an incoming wave in a scattering channel b are modified in the interaction zone containing the target. The outgoing wave leaves through a
cartoon of quantum scattering
​scattering channel a, . All information is encoded in the complex elements Sab(E) of the scattering matrix S, usually as function of energy E. The S matrix is unitary due to flux conservation and for M channels it has dimension M x M . The scattering matrix itself can only in some cases be measured directly, particularly in experiments with microwave cavities or reverberating elastic objects. In genuine quantum physics, only the cross sections are accessible. The off-diagonal ones are given by the squared moduli of the S matrix elements  σab=|Sab|2 . There are different time scales. Direct, non-random reactions due to channel-channel coupling occur on short time scales as for example in nuclear scattering.  On longer time scales, equilibration takes place and is accompanied by stochasticity. Compound nucleus scattering is a typical example, but equivalent situations frequently occur in many other systems. This stochasticity or, in a broad sense, chaoticity has various important features. Furthermore, the average resonance width Γ often grows with energy while the average resonance spacing D shrinks. At sufficiently high energies, Γ/D is large and the resonances strongly overlap. Scattering matrices and cross sections are random functions, implying that the distributions of the S matrix elements are Gaussian. This is the universal Ericson regime. Outside this regime the unitarity of the S matrix results in strong deviations from the Gaussians. The transition from isolated resonances to strongly overlapping ones is referred to as "Ericson transition''.

Random Matrix Theory provides the ideal model for the stochastic aspects in Quantum Scattering [1]. For a long time, only the correlation functions of the scattering matrix elements were accessible in analytical calculations by applying Supersymmetry [2]. Later, the full distributions for the diagonal scattering matrix elements Saa were obtained [3].

[1] Mitchell, Richter, Weidenmüller Random Matrices and Chaos in Nuclear Physics: Nuclear Reactions, Rev. Mod. Phys. 82, 2845 (2010)

[2] Verbaarschot, Weidenmüller, Zirnbauer Grassmann Integration in Stochastic Quantum Physics: The Case of Compound-Nucleus Scattering, Phys. Rep. 129, 367 (1985)

[3] Savin, Sommers Scattering, Reflection and Impedance of Waves in Chaotic and Disordered Systems with Absorption, J. Phys. A 38, 10731 (2005)

It remained an important challenge to compute the distributions of the off-diagonal scattering matrix elements Sab , a ≠ b because of their relevance for many experiments. We managed to solve this longstanding problem and derived exact formulae for the orthogonal and unitary cases which are valid in all regimes 

We compared with microwave experiments 

We also solved the considerably more involved problem of calculating the distributions of the off-diagonal cross sections

and compared with nuclear data

Quantum Chaos and Random Matrix Models

​Nucleus3

​​

Random matrices provide powerful models for a rich variety of complex systems. Here is a brief overview.  

A good example is the atomic nucleus shown to the right. It consists of many nucleons, the protons (blue) and the neutrons (red). They all interact with each other and move in a very complicated way. Large parts of the quantum mechanical excitation spectrum have statistical features which are well described by assuming that the interaction matrix elements can be replaced by random numbers. Hence, the Hamiltonian can be viewed as a random matrix. One often refers to this as Quantum Chaos in nuclei. 

In several different branches of physics, we study chaotically coupled systems or, equivalently, systems with symmetry breaking. The coupling or the symmetry breaking has a strong influence on the spectral statistics. In a recent work, we look at the time evolution of a certain energy localization effect which has been observed in coupled elastodynamical systems. Scar1 1ugerade

​​

Recently, we carried out detailed studies of the doorway mechanism in the framework of Random Matrix Theory. Doorways are distinct states in complex quantum systems which couple to a background of other states which often can be modeled by random matrices. Important conclusions about various systems can be drawn once the statistical implications of the doorway mechanism have been understood. The superscars in the barrier billiard provide an ideal model scenario for an in-depth investigation. In this context, we also calculated the distribution of the coupling coefficients to the doorway. The doorway mechanism is closely related to an important question of quantum information, namely how a prepared state is influenced by the mixing to its environment. We find a striking recovery and saturation of the survival probability. 

collaborators: Johan Grönqvist (PhD-student, now PhD), Dr. Heiner Kohler (now Madrid), Professor Hans-Jürgen Sommers 
external collaborators: Professor Sven Åberg at University of Lund, Professor Achim Richter and his group at Technical University of Darmstadt, Professor Hans-Jürgen Stöckmann at University of Marburg.

 

Many-Body Systems and Quantum Chaos

 

​Collective Motion-2​

As discussed above, nuclei very often show statistical features which are well described by random matrices. However, that is not the full story yet, because at some excitation energies, the nucleons move coherently in phase space. This leads, for example, to vibrations and rotations of the nucleus as a whole. Those collective excitations are not quantum chaotic, they rather tend to behave regularly. Proper Random Matrix Models can be used to distinguish collective from quantum chaotic motion, to some extent even purely from the measured data. We analyzed experimental data of magnetic scissors mode excitations and of electric pygmy dipole excitations. 

This also means that the strong arguments recently put forward for single-particle systems in favor of the Bohigas-Giannoni-Schmit conjecture cannot be carried over in a straightforward way to many-body systems. (The Bohigas-Giannoni-Schmit conjecture states that the spectral fluctuations of a quantum system are described by random matrices if the corresponding classical system is chaotic in the usual sense of classical mechanics.) We devote considerable efforts to the understanding of the interplay of incoherent single-particle and collective motion as well as its impact on the level statistics in many-body systems. We began with investigating spreading and its semiclassical interpretation in a model system in which we always have control over the individual particle dynamics. We derive, under rather general circumstances, that the collective observables of the quantum system relate to the classical dynamics. 

collaborators: Johannes Freese (Diplom-student), Prof. Dr. Boris Gutkin (junior research group leader), Jens Hämmerling (PhD-student) 
external collaborators: Professor Achim Richter and his group at Technical University of Darmstadt

 

Quantum Chaos Approach to QCD

 

​Qcdpic-2
Courtesy of Derek Leinweber (CSSM)
​​Quantum Chromodynamics (QCD) is the theory of the strong interaction. The quarks (the colored spheres) interact by exchanging gluons which are described by gauge fields (the green clouds). Unfortunately, QCD is so involved that analytical calculations are only feasible after drastic approximations. Thus, one resorts to demanding numerical simulations, referred to as lattice gauge calculations. We contributed to the statistical analysis of those lattice data which show that QCD has much in common with disordered mesoscopic systems in condensed matter physics. But QCD is even more complicated, in particular due to chirality and chiral symmetry breaking. We setup an effective stochastic field theory. Presently, we investigate the motion of the quarks in individual gauge field configurations which matches the way how lattice gauge calculations are done. As a first step, we identified a hierarchy of semiclassical limits in QCD. 

collaborator: Dr. Stefan Keppeler (Emmy Noether fellow, now at University of Tübingen) 
external collaborator: Professor Tilo Wettig at University of Regensburg.

 

Supersymmetry and Supergroups

​Cssl Fig2

​

Supersymmetry became a prominent tool in the Theory of Random Matrices and Disordered Systems. We developed the Graded Eigenvalue Method as a new and exact technique to solve Random Matrix Models. In the course of doing so we were led to address the general problem of harmonic analysis on superspaces and, in this context, the theory of supergroups. Here is an introduction to the main ideas. We showed that certain matrix Bessel functions (in mathematics referred to as Gelfand's spherical functions) are kernels of diffusion equations which drive systems with arbitrary spectral correlations into the chaotic regime. We derived various new and explicit results beyond the unitary case for these diffusion kernels in ordinary and super spaces. As a side result, we also proved a supersymmetric generalization of Harish-Chandra's famous integral formula. 

There is a close relation of these issues to Calogero-Sutherland models. We constructed a large class of those models in superspace. In a most natural way, models for two kinds of interacting particlesresult which even have a physical interpretation. 

For a long time, one thought that supersymmetry is restricted to Gaussian probability densities. We showed arbitrary probability densities can be treated with supersymmetry. Later on, this was also studied in the framework of superbosonization. We showed the equivalence. 

collaborators: Dr. Mario Kieburg (PhD-student, now PhD), Dr. Heiner Kohler (now Madrid), Professor Hans-Jürgen Sommers

 

Applications of Supersymmetry

 

Supersymmetry is nowadays indispensable for Random Matrix Theory, because it yields an exact representation of many Random Matrix Models in terms of supermatrix models in such a way that the number of degrees of freedom is drastically reduced. We use this in many applications. Some examples may illustrate that. 

Consider a Hamiltonian that depends on some parameter. The correlations in this parameter are known to have universal features. Another observable is fidelity. It measures the overlap between states, evolved with a perturbed and an unperturbed Hamiltonian, depending on the perturbation parameter. Hence, in some way, parametric correlations and fidelity must be related, but it was unclear how. Supersymmetry helped us to discover surprising relations between parametric level correlations and fidelity decay.

​Dataplot2

​​​When analyzing time series, for example of the electrodes attached to different places on the scalp in electroencephalography (EEG), the correlations coefficients between two time series provide important information about the system. The correlation coefficients are ordered in the correlation matrix. The Wishart model is a benchmark for statistical properties of such correlation matrices. It correctly incorporates the empirical eigenvalues, but is otherwise based on Gaussian distributed random time series. Formally, the Wishart model coincides with chiral Random Matrix Theory, if the empirical eigenvalues are all equal. An important observable is the density of eigenvalues of the correlation matrix which parametrically depends on the empirical eigenvalues. For complex correlation matrices, this eigenvalue density was known exactly, but in the real case, a deep mathematical reason made it impossible for a long time to derive exact closed form results. Using supersymmetry, we fully solved this problem and gave exact formulas for the eigenvalue density of real Wishart correlation matrices.

Structural insights are often also useful in applications. We discovered an unexpected connection between the building blocks of the correlation functions. Furthermore, we found supersymmetric structures without actually mapping to superspace. More precisely, we identified superspace Jacobians in the integral representations of correlation functions in ordinary space. This deep structural insight yields a powerful method to cast correlation functions into a determinant or Pfaffian form, depending on the underlying symmetric spaces.

collaborators: Johan Grönqvist (PhD-student, now PhD), Dr. Mario Kieburg (PhD-student, now PhD), Dr. Heiner Kohler (now Madrid), Christian Recher (Diplom-student, now PhD-student with Dr. Heiner Kohler) 
external collaborators: Dr. Francois Leyvraz and Dr. Thomas Seligman and their groups at UNAM, Cuernavaca (Mexico), Dr. Igor Smolyarenko at Brunel University, Uxbridge (England)

 

Elasticity, Wave Chaos and Ray Limit



​

 

​Slide15cut

​Classical wave phenomena are ubiquitous in technical applications. Quantum Chaos is the statistical theory of quantum waves. What can we learn from Quantum Chaos for classical wave phenomena? - We addressed this question by studying elastomechanical systems in a series of investigations over the years. This is non-trivial due to the presence of modes (pressure and shear) and due to the complicated boundary conditions. Among other things, we presented the first purely experimental and statistically highly significant study of parametric correlations. More recently, we showed that the elastic displacement field behaves statistically just like a quantum wave function. However, there is a beating phenomenon between the different modes. We now apply "semiclassical" ideas, that is, we study the ray limit of elastodynamics in which the wavelength is small compared to all system geometries. A first step was a Weyl approximation to the level density of the quartz sphere shown to the right. This is a monocrystal of grapefruit size whose spectra have been measured with fantastic precision. 

​Schalen

​​More ambitiously, we now work on understanding of elastodynamical spectra by relating fundamental features of those systems to geometric quantities such as periodic orbits. In particular, we became interested in shells because of their relevance for all kinds of applications, ranging from auto bodies to rockets. We considered a family of shells of revolution, with the disk and the hemisphere as limiting cases. We managed to explain a striking clustering effect in the spectra using periodic orbits. 

collaborators: Dr. Niels Søndergaard (postdoc, now researcher in industry) 
external collaborators: Professor Clive Ellegaard, Mikkel Avlund at the Niels Bohr Institute, Copenhagen (Denmark), Dr. Mark Oxborrow at the National Physical Laboratory, Teddington (England).

 

Micro- and Nanomechanics

 

​Cantilevers
Courtesy of Lars Montelius ​Hexagon

​Little is known about the mechanical properties of systems on the micrometer and nanometer scale. An example is the cantilevers shown to the left. Nonlinear or chaos-related effects are needed to gain deeper insight into those systems. We wish to apply our knowledge of chaos theory and elastodynamics to the micrometer and nanometer scale. In particular, we are interested in micro-electro-mechanical systems (MEMS) and nano-electro-mechanical systems (NEMS) which are nowadays so important in engineering. We used Euler's theory for rods to understand mechanical properties of cantilevers as shown in the picture. 

​The continuum theory of elasticity is bound to fail at those scales where the atomistic structure of matter becomes relevant. It is important to figure out on which scales this happens. We studied strain in semiconductor core--shell nanowires in an atomistic and in a continuum model. We found a remarkable robustness of the continuum theory.

collaborators: Johan Grönqvist (PhD-student, now PhD), Dr. Niels Søndergaard (postdoc, now researcher in industry)
external collaborators: Professor Lars Montelius and Professor Hongqi Xu and their groups at the Lund Nano Lab in Lund (Sweden).

 

Econophysics

 

​Boerse

​What is theoretical physics? - Our definition is the following: Theoretical physics is construction and analysis of mathematical models for the quantitative description of reproducible experiments. These models ought to be as compatible as possible with each other to yield an ever more unifying picture for the measurable features of our world. - There is no reason to exclude biology, sociology or economics from the list of interesting research topics. On the contrary, we think that we physicists would make a big strategic mistake by confining ourselves to the more established topics and by leaving, for example, the quickly expanding field of mathematical modeling in economics to the mathematicians. The physicists' expertise, in particular the data- and experiment-oriented approach has a truly high potential in this field. 

The time series of stocks are correlated, because the performance of the corresponding companies is mutually connected. The measurement of those financial correlations is very important for risk management. For various reasons, the empirically obtained correlations are dressed with noise. We developed methods to estimate and remove this noise, to begin with in a random matrix approach and then by inventing the power mapping as an alternative and supplement. These noise reduction techniques have direct applications in portfolio optimization and can greatly reduce risk in stock portfolios. 

​Dowjones

​An interesting feature of financial correlation is their behavior on short time scales. For relative price changes (or returns) on very short time intervals, the influence of asynchrony in trading and the finite tick size become increasingly important. They are the main statistical causes for the so-called Epps-effect, i.e., for decreasing correlations on short return intervals. 

In addition to market risk we also study credit risk. Since an obligor can go bankrupt or default in another way, credits are risky for the banks that issue them. It turns out that the distributions of the losses are highly asymmetric and have long tails. A better quantitative understanding by means of improved models is crucial for banks and regulators, and it is a challenge for statistical physics. 

collaborators: Dr. Rudi Schäfer, Per-Johan Andersson, Andreas Öberg, Markus Sjölin, Andreas Sundin, Michal Wolanski, Patrik Frisk, Johnny Pégeot, Juan Manuel Vázquez Montejo, Fredrik Nilsson and Per Berseus (Examen-students) 
external contact: Alexander Koivusalo (Danske Capital, Denmark), Dr. Axel Müller-Groeling (Centrosolar, Hamburg), Eduard Seligman (Osprey Asset Management, Geneva)