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Leo P. Kadanoff

Publications and source records attributed to Leo P. Kadanoff.

At least 19 recordsLinked to original sources
arXiv2016

Eigenpairs of Toeplitz and disordered Toeplitz matrices with a Fisher-Hartwig symbol

Toeplitz matrices have entries that are constant along diagonals. They model directed transport, are at the heart of correlation function calculations of the two-dimensional Ising model, and have applications in quantum information science. We derive their eigenvalues and eigenvectors when the symbol is singular Fisher-Hartwig. We then add diagonal disorder and study the resulting eigenpairs. We find that there is a "bulk" behavior that is well captured by second order perturbation theory of non-Hermitian matrices. The non-perturbative behavior is classified into two classes: Runaways type I leave the complex-valued spectrum and become completely real because of eigenvalue attraction. Runaways type II leave the bulk and move very rapidly in response to perturbations. These have high condition numbers and can be predicted. Localization of the eigenvectors are then quantified using entropies and inverse participation ratios . Eigenvectors corresponding to Runaways type II are most localized (i.e., super-exponential), whereas Runaways type I are less localized than the unperturbed counterparts and have most of their probability mass in the interior with algebraic decays. The results are corroborated by applying free probability theory and various other supporting numerical studies.

math-ph↗
arXiv2014

Innovations in Statistical Physics

In 1963-71, a group of people, myself included, formulated and perfected a new approach to physics problems, which eventually came to be known under the names of scaling, universality, and renormalization. This work formed the basis of a wide variety of theories ranging from its starting point in critical phenomena, and moving out to particle physics and relativity and then into economics and biology. This work was of transcendental beauty and of considerable intellectual importance. This left me with a personal problem. What next? Constructing the answer to that question would dominate the next 45 years of my professional life. I would try to: * Help in finding and constructing new fields of science * Do research and give talks on science/society borderline * Provide helpful, constructive criticism of scientific and technical work * Help students and younger scientists * Demonstrate scientific leadership

physics.hist-ph↗
arXiv2014

Reflections on Gibbs: From Statistical Physics to the Amistad

This note is based upon a talk given at a celebration in Austin Texas of the achievements of J. Willard Gibbs. J. Willard Gibbs, the younger, was the first American physical sciences theorist. He was one of the inventors of statistical physics. He introduced and developed the concepts of phase space, phase transitions, and thermodynamic surfaces in a remarkably correct and elegant manner. These three concepts form the basis of different areas of physics. The connection among these areas has been a subject of deep reflection from Gibbs' time to our own. This talk therefore tries to celebrate Gibbs by talking about modern ideas about how different parts of physics fit together. At the end of the talk, I shall get to a more personal note. Our own J. Willard Gibbs had all his achievements concentrated in science. His father, also J. Willard Gibbs, also a Professor at Yale, had one great achievement that remains unmatched in our day. I shall describe it.

physics.hist-ph↗
arXiv2014

Entropy is in Flux

The science of thermodynamics was put together in the Nineteenth Century to describe large systems in equilibrium. One part of thermodynamics defines entropy for equilibrium systems and demands an ever-increasing entropy for non-equilibrium ones. However, starting with the work of Ludwig Boltzmann in 1872, and continuing to the present day, various models of non-equilibrium behavior have been put together with the specific aim of generalizing the concept of entropy to non-equilibrium situations. This kind of entropy has been termed {\em kinetic entropy} to distinguish it from the thermodynamic variety. Knowledge of kinetic entropy started from Boltzmann's insight about his equation for the time dependence of gaseous systems. In this paper, his result is stated as a definition of kinetic entropy in terms of a local equation for the entropy density. This definition is then applied to Landau's theory of the Fermi liquid thereby giving the kinetic entropy within that theory. Entropy has been defined and used for a wide variety of situations in which a condensed matter system has been allowed to relax for a sufficient period so that the very most rapid fluctuations have been ironed out. One of the broadest applications of non-equilibrium analysis considers quantum degenerate systems using Martin-Schwinger Green's functions\cite{MS} as generalized of Wigner functions, $g^<$ and $g^>$. This paper describes once again these how the quantum kinetic equations for these functions give locally defined conservation laws for mass momentum and energy. In local thermodynamic equilibrium, this kinetic theory enables a reasonable local definition of entropy density. However, when the system is outside of local equilibrium, this definition fails. It is speculated that quantum entanglement is the source of this failure.

cond-mat.stat-mech↗
arXiv2014

The Early Years of Condensed Matter Physics at Illinois -- in Celebration of the 80th Birth Year of Charles P. Slichter -- Charlie Slichter & the gang at Urbana

The 1950s-- and perhaps also the 1960s-- were very special times for the development of solid-state/condensed-matter physics. The University of Illinois at Urbana was at the center of these activities. In areas like NMR and superconductivity, methods were developed which would form the basis for the next half century of science and technology. Experimentalists, including Charlie and John Wheatley, worked hand in hand with theorists, including the incomparable John Bardeen. They worked cooperatively to develop ideas, often born in Urbana, but with godparents at Harvard and Moscow and Paris. A characteristic style of broad collaboration and spirited exchange developed and spread from Illinois. This development was not an accident but the result of the vision of leaders like Wheeler Loomis, Fred Seitz, and later Gerald Almy1. The strong leadership saved the other scientists from expending their time on departmental decision-making. The style of the scientific activity was set by Fred, who strongly encouraged joint activities--especially the interaction between between theory and experiment and between physics and engineering. Fred encouraged comments from everyone, and helped everyone grow fast

physics.hist-ph↗
arXiv2013

Slippery Wave Functions V2.01

Superfluids and superconductors are ordinary matter that show a very surprising behavior at low temperatures. As their temperature is reduced, materials of both kinds can abruptly fall into a state in which they will support a persistent, essentially immortal, flow of particles. Unlike anything in classical physics, these flows engender neither friction nor resistance. A major accomplishment of Twentieth Century physics was the development of an understanding of this very surprising behavior via the construction of partially microscopic and partially macroscopic quantum theories of superfluid helium and superconducting metals. Such theories come in two parts: a theory of the motion of particle-like excitations, called quasiparticles, and of the persistent flows itself via a huge coherent excitation, called a condensate. Two people, above all others, were responsible for the construction of the quasiparticle side of the theories of these very special low-temperature behaviors: Lev Landau and John Bardeen. Curiously enough they both partially ignored and partially downplayed the importance of the condensate. In both cases, this neglect of the actual superfluid or superconducting flow interfered with their ability to understand the implications of the theory they had created. They then had difficulty assessing the important advances that occurred immediately after their own great work. Some speculations are offered about the source of this unevenness in the judgments of these two leading scientists.

physics.hist-ph↗
arXiv2013

Real Space Renormalization in Statistical Mechanics

This paper discusses methods for the construction of approximate real space renormalization transformations in statistical mechanics. In particular, it compares two methods of transformation: the "potential-moving" approach most used in the period 1975-1980 and the "rewiring method" as it has been developed in the last five years. These methods both employ a parameter, called χ, that measures the complexity of the localized stochastic variable forming the basis of the analysis. Both methods are here exemplified by calculations in terms of fixed points for the smallest possible values of χ. These calculations describe three models for two-dimensional systems: The Ising model solved by Onsager, the tricritical point of that model, and the three-state Potts model. The older method, often described as lower bound renormalization theory, provides a heuristic method giving reasonably accurate results for critical indices at the lowest degree of complexity, i.e. χ=2. In contrast, the rewiring method, employing "singular value decomposition", does not perform as well for low χvalues but offers an error that apparently decreases slowly toward zero as χis increased. It appears likely that no such improvement occurs in the older approach. A detailed comparison of the two methods is performed, with a particular eye to describing the reasons why they are so different. For example, the old method quite naturally employed fixed points for its analysis; these are hard to use in the newer approach. A discussion is given of why the fixed point approach proves to be hard in this context. In the new approach the calculated the thermal critical indices are satisfactory for the smallest values of χbut hardly improve as χis increased, while the magnetic critical indices do not agree well with the known theoretical values.

cond-mat.stat-mech↗
arXiv2012

Global and Local: Synchronization and Emergence

When a dynamical system contains several different modes of oscillations it may behave in a variety of ways: If the modes oscillate at their own individual frequencies, it exhibits quasiperiodic behavior; when the modes lock to one another it becomes synchronized, or, as a third possibility, complex chaotic behavior may emerge. With two modes present, like an internal oscillation coupled to a periodic external signal, one obtains a highly structured phase diagram that exhibits these possibilities. In essence, the details are related to the difference between rational and irrational numbers. Natural system can display fragments of this phase diagram, thereby offering insights into their dynamical mechanisms.

nlin.CD↗
arXiv2011

Relating Theories via Renormalization

The renormalization method is specifically aimed at connecting theories describing physical processes at different length scales and thereby connecting different theories in the physical sciences. The renormalization method used today is the outgrowth of one hundred and fifty years of scientific study of thermal physics and phase transitions. Different phases of matter show qualitatively different behavior separated by abrupt phase transitions. These qualitative differences seem to be present in experimentally observed condensed-matter systems. However, the "extended singularity theorem" in statistical mechanics shows that sharp changes can only occur in infinitely large systems. Abrupt changes from one phase to another are signaled by fluctuations that show correlation over infinitely long distances, and are measured by correlation functions that show algebraic decay as well as various kinds of singularities and infinities in thermodynamic derivatives and in measured system parameters. Renormalization methods were first developed in field theory to get around difficulties caused by apparent divergences at both small and large scales. The renormalization (semi-)group theory of phase transitions was put together by Kenneth G. Wilson in 1971 based upon ideas of scaling and universality developed earlier in the context of phase transitions and of couplings dependent upon spatial scale coming from field theory. Correlations among regions with fluctuations in their order underlie renormalization ideas. Wilson's theory is the first approach to phase transitions to agree with the extended singularity theorem. Some of the history of the study of these correlations and singularities is recounted, along with the history of renormalization and related concepts of scaling and universality. Applications are summarized.

cond-mat.stat-mech↗
arXiv2010

Theories of Matter: Infinities and Renormalization

This paper looks at the theory underlying the science of materials from the perspectives of physics, the history of science, and the philosophy of science. We are particularly concerned with the development of understanding of the thermodynamic phases of matter. The question is how can matter, ordinary matter, support a diversity of forms. We see this diversity each time we observe ice in contact with liquid water or see water vapor (steam) rise from a pot of heated water. The nature of the phases is brought into the sharpest focus in phase transitions: abrupt changes from one phase to another and hence changes from one behavior to another. This article starts with the development of mean field theory as a basis for a partial understanding of phase transition phenomena. It then goes on to the limitations of mean field theory and the development of very different supplementary understanding through the renormalization group concept. Throughout, the behavior at the phase transition is illuminated by an "extended singularity theorem", which says that a sharp phase transition only occurs in the presence of some sort of infinity in the statistical system. The usual infinity is in the system size. Apparently this result caused some confusion at a 1937 meeting celebrating van der Waals, since mean field theory does not respect this theorem. In contrast, renormalization theories can make use of the theorem. This possibility, in fact, accounts for some of the strengths of renormalization methods in dealing with phase transitions. The paper outlines the different ways phase transition phenomena reflect the effects of this theorem.

physics.hist-ph↗
arXiv2009

More is the Same; Phase Transitions and Mean Field Theories

This paper looks at the early theory of phase transitions. It considers a group of related concepts derived from condensed matter and statistical physics. The key technical ideas here go under the names of "singularity", "order parameter", "mean field theory", and "variational method". In a less technical vein, the question here is how can matter, ordinary matter, support a diversity of forms. We see this diversity each time we observe ice in contact with liquid water or see water vapor, "steam", come up from a pot of heated water. Different phases can be qualitatively different in that walking on ice is well within human capacity, but walking on liquid water is proverbially forbidden to ordinary humans. These differences have been apparent to humankind for millennia, but only brought within the domain of scientific understanding since the 1880s. A phase transition is a change from one behavior to another. A first order phase transition involves a discontinuous jump in a some statistical variable of the system. The discontinuous property is called the order parameter. Each phase transitions has its own order parameter that range over a tremendous variety of physical properties. These properties include the density of a liquid gas transition, the magnetization in a ferromagnet, the size of a connected cluster in a percolation transition, and a condensate wave function in a superfluid or superconductor. A continuous transition occurs when that jump approaches zero. This note is about statistical mechanics and the development of mean field theory as a basis for a partial understanding of this phenomenon.

physics.hist-ph↗
arXiv2009

Expansions for Eigenfunction and Eigenvalues of large-n Toeplitz Matrices

This note starts from work done by Dai, Geary, and Kadanoff (Hui Dai, Zachary Geary, and Leo P. Kadanoff, H. Dai, Z. Geary and L. P. Kadanoff, Journal of Statistical Mechanics, P05012 (2009)) on exact eigenfunctions for Toeplitz operators. It builds methods for finding convergent expansions for eigenvectors and eigenvalues of large-n Toeplitz matrices, using the infinite-n case as a starting point. One expansion is derived from operator equations having a two-dimensional continuous spectrum of eigenvalues, which include the eigenvalues of the finite-$n$ matrices. Another expansion is derived from the transpose equations, which have no eigenvalues at all. The two expansions work together to give an apparently convergent expansion with an expansion parameter expressed as an inverse power of n. A variational principle is developed which gives an approximate expression for determining eigenvalues. A consistency condition is generated, which gives to lowest order exactly the same condition for the eigenvalue.

math-ph↗
arXiv2009

Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices

A Toeplitz matrix is one in which the matrix elements are constant along diagonals. The Fisher-Hartwig matrices are much-studied singular matrices in the Toeplitz family. The matrices are defined for all orders, $N$. They are parametrized by two constants, $α$ and $β$. Their spectrum of eigenvalues has a simple asymptotic form in the limit as $N$ goes to infinity. Here we study the structure of their eigenvalues and eigenvectors in this limiting case. We specialize to the case $0<α<|β|<1$, where the behavior is particularly simple.

math-ph↗
arXiv2004

Exact Solutions for Loewner Evolutions

In this note, we solve the Loewner equation in the upper half-plane with forcing function xi(t), for the cases in which xi(t) has a power-law dependence on time with powers 0, 1/2 and 1. In the first case the trace of singularities is a line perpendicular to the real axis. In the second case the trace of singularities can do three things. If xi(t)=2*(kappa*t)^1/2, the trace is a straight line set at an angle to the real axis. If xi(t)=2*(kappa*(1-t))^1/2, the behavior of the trace as t approaches 1 depends on the coefficient kappa. Our calculations give an explicit solution in which for kappa<4 the trace spirals into a point in the upper half-plane, while for kappa>4 it intersects the real axis. We also show that for kappa=9/2 the trace becomes a half-circle. The third case with forcing xi(t)=t gives a trace that moves outward to infinity, but stays within fixed distance from the real axis. We also solve explicitly a more general version of the evolution equation, in which xi(t) is a superposition of the values +1 and -1.

math-ph↗
arXiv2004

Trace for the Loewner Equation with Singular Forcing

The Loewner equation describes the time development of an analytic map into the upper half of the complex plane in the presence of a "forcing", a defined singularity moving around the real axis. The applications of this equation use the trace, the locus of singularities in the upper half plane. This note discusses the structure of the trace for the case in which the forcing function, xi(t), is proportional to (-t)^beta with beta in the interval (0, 1/2). In this case, the trace is a simple curve, gamma(t), which touches the real axis twice. It is computed by using matched asymptotic analysis to compute the trajectory of the Loewner evolution in the neighborhood of the singularity, and then assuming a smooth mapping of these trajectories away from the singularity. Near the t=0 singularity, the trace has a shape given by [ Re(gamma(t)-gamma(0)) ]^(1-beta) ~ [ beta*Im(gamma(t)) ]^beta ~ O(xi(t))^(1-beta). A numerical calculation of the trace provides support for the asymptotic theory.

cond-mat.other↗
arXiv2003

Discrete charges on a two dimensional conductor

We investigate the electrostatic equilibria of N discrete charges of size 1/N on a two dimensional conductor (domain). We study the distribution of the charges on symmetric domains including the ellipse, the hypotrochoid and various regular polygons, with an emphasis on understanding the distributions of the charges, as the shape of the underlying conductor becomes singular. We find that there are two regimes of behavior, a symmetric regime for smooth conductors, and a symmetry broken regime for ``singular'' domains. For smooth conductors, the locations of the charges can be determined up to a certain order by an integral equation due to Pommerenke (1969). We present a derivation of a related (but different) integral equation, which has the same solutions. We also solve the equation to obtain (asymptotic) solutions which show universal behavior in the distribution of the charges in conductors with somewhat smooth cusps. Conductors with sharp cusps and singularities show qualitatively different behavior, where the symmetry of the problem is broken, and the distribution of the discrete charges does not respect the symmetry of the underlying domain. We investigate the symmetry breaking both theoretically, and numerically, and find good agreement between our theory and the numerics. We also find that the universality in the distribution of the charges near the cusps persists in the symmetry broken regime, although this distribution is very different from the one given by the integral equation.

cond-mat.stat-mech↗