Relaxation at critical points: Deterministic and stochastic theory

A generalized critical point can be characterized by non-linear dynamics. We formulate the deterministic and stochastic theory of relaxation at such a point. Canonical problems are used to motivate the general solutions. In the deterministic theory, we show that at the critical point certain modes have polynomial (rather than exponential) growth or decay. The stochastic relaxation rates can be calculated in terms of various incomplete special functions. Three examples are considered. First, a substrate inhibited reaction (marginal type dynamical system) is treated. Second, the relaxation of a mean field ferromagnet is considered. We obtain a result that generalizes the work of Griffiths et al. Third, we study the relaxation of a critical harmonic oscillator.     

Complex systems analysis of series of blackouts: cascading failure, critical points, and self-organization

We give an overview of a complex systems approach to large blackouts of electric power transmission systems caused by cascading failure. Instead of looking at the details of particular blackouts, we study the statistics and dynamics of series of blackouts with approximate global models. Blackout data from several countries suggest that the frequency of large blackouts is governed by a power law. The power law makes the risk of large blackouts consequential and is consistent with the power system being a complex system designed and operated near a critical point. Power system overall loading or stress relative to operating limits is a key factor affecting the risk of cascading failure. Power system blackout models and abstract models of cascading failure show critical points with power law behavior as load is increased. To explain why the power system is operated near these critical points and inspired by concepts from self-organized criticality, we suggest that power system operating margins evolve slowly to near a critical point and confirm this idea using a power system model. The slow evolution of the power system is driven by a steady increase in electric loading, economic pressures to maximize the use of the grid, and the engineering responses to blackouts that upgrade the system. Mitigation of blackout risk should account for dynamical effects in complex self-organized critical systems. For example, some methods of suppressing small blackouts could ultimately increase the risk of large blackouts.     

Bimodality: a sign of critical behavior in nuclear reactions

The recently discovered coexistence of multifragmentation and residue production for the same total transverse energy of light charged particles, which has been dubbed bimodality like it has been introduced in the framework of equilibrium thermodynamics, can be well reproduced in numerical simulations of heavy ion reactions. A detailed analysis shows that fluctuations (introduced by elementary nucleon-nucleon collisions) determine which of the exit states is realized. Thus, we can identify bifurcation in heavy ion reactions as a critical phenomenon. Also the scaling of the coexistence region with beam energy is well reproduced in these results from the quantum molecular dynamics simulation program.     

Topological order and conformal quantum critical points

We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary two-dimensional free boson, the two-dimensional Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously varying critical exponents separating phases with long-range order from a deconfined topologically ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-Abelian symmetry, the strong-coupling limit of 2+1-dimensional Yang–Mills gauge theory with a Chern–Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.     

Hydrodynamics of reactive systems: Chemical reactions near the critical points

The rate of a chemical reaction is expressed in terms of the correlation functions which are found from the full system of hydrodynamic equations in reactive systems. In the limiting cases one obtains all three types of behavior of reaction rates near the critical points found in experiments (slowing-down, speeding-up, insensitivity to criticality). This behavior depends on the number of reactive and inert components in a system, proximity and path of approach to the critical point.     

Phase equilibria,metastable states,and critical points in a simple one-component system

Stability of metastable phase states against infinitesimal perturbations in a simple one-component system is considered. The method of molecular dynamics simulation was used to determine the boundaries of essential instability of supersaturated vapor, a superheated liquid, and a superheated crystal. The absence of a spinodal from a supercooled liquid and the dependence of the boundary of essential instability of a superheated crystal on the character of deformation were established. It is shown that each of the three lines of phase equilibria in a one-component system has an endpoint of termination of phase coexistence. As distinct from the liquid–gas critical point, which is the point of phase identity and is located in the region of stable states, the endpoints of melting and sublimation lines are located in the region ofmetastable states. At these points, a critical (spinodal) state is achieved only for one of the coexisting phases.     

Bulk and boundary critical behavior at Lifshitz points

Lifshitz points are multicritical points at which a disordered phase, a homogeneous ordered phase, and a modulated ordered phase meet. Their bulk universality classes are described by natural generalizations of the standard φ⁴ model. Analyzing these models systematically via modern field-theoretic renormalization group methods has been a long-standing challenge ever since their introduction in the middle of 1970s. We survey the recent progress made in this direction, discussing results obtained via dimensionality expansions, how they compare with Monte Carlo results, and open problems. These advances opened the way towards systematic studies of boundary critical behavior atm-axial Lifshitz points. The possible boundary critical behavior depends on whether the surface plane is perpendicular to one of them modulation axes or parallel to all of them. We show that the semi-infinite field theories representing the corresponding surface universality classes in these two cases of perpendicular and parallel surface orientation differ crucially in their Hamiltonian’s boundary terms and the implied boundary conditions, and explain recent results along with our current understanding of this matter.     

Electron self-trapping at quantum and classical critical points

Using Feynman path integral technique estimations of the ground state energy have been found for a conduction electron interacting with order parameter fluctuations near quantum critical points. In some cases only singular perturbation theory in the coupling constant emerges for the electron ground state energy. It is shown that an autolocalized state (quantum fluctuon) can be formed and its characteristics have been calculated depending on critical exponents for both weak and strong coupling regimes. The concept of fluctuon is considered also for the classical critical point (at finite temperatures) and the difference between quantum and classical cases has been investigated. It is shown that, whereas the quantum fluctuon energy is connected with a true boundary of the energy spectrum, for classical fluctuon it is just a saddle-point solution for the chemical potential in the exponential density of states fluctuation tail.     

Excitation and entanglement transfer near quantum critical points

Recently, there has been growing interest in employing condensed matter systems such as quantum spin or harmonic chains as quantum channels for short distance communication. Many properties of such chains are determined by the spectral gap between their ground and excited states. In particular this gap vanishes at critical points of quantum phase transitions. In this article we study the relation between the transfer speed and quality of such a system and the size of its spectral gap. We find that the transfer is almost perfect but slow for large spectral gaps and fast but rather inefficient for small gaps. The text was submitted by the authors in English.     

Black holes and critical points in moduli space

We study the stabilization of scalars near a supersymmetric black hole horizon using the equation of motion of a particle moving in a potential and background metric. When the relevant 4-dimensional theory is described by special geometry, the generic properties of the critical points of this potential can be studied. We find that the extremal value of the central charge provides the minimal value of the BPS mass and of the potential under the condition that the moduli space metric is positive at the critical point. This is a property of a regular special geometry. We also study the critical points in all N 2 supersymmetric theories. We relate these ideas to the Weinhold and Ruppeiner metrics introduced in the geometric approach to thermodynamics and used for the study of critical phenomena.     

Bulk singularities at critical end points: a field-theory analysis

A class of continuum models with a critical end point is considered whose Hamiltonian [φ,ψ] involves two densities: a primary order-parameter field, φ, and a secondary (noncritical) one, ψ. Field-theoretic methods (renormalization group results in conjunction with functional methods) are used to give a systematic derivation of singularities occurring at critical end points. Specifically, the thermal singularity ∼ | t|^{2 - α} of the first-order line on which the disordered or ordered phase coexists with the noncritical spectator phase, and the coexistence singularity ∼ | t|^{1 - α} or ∼ | t|^β of the secondary density <ψ> are derived. It is clarified how the renormalization group (RG) scenario found in position-space RG calculations, in which the critical end point and the critical line are mapped onto two separate fixed points _CEP ^* and _λ ^*, translates into field theory. The critical RG eigenexponents of _CEP ^* and _λ ^* are shown to match. _CEP ^* is demonstrated to have a discontinuity eigenperturbation (with eigenvalue y = d), tangent to the unstable trajectory that emanates from _CEP ^* and leads to _λ ^*. The nature and origin of this eigenperturbation as well as the role redundant operators play are elucidated. The results validate that the critical behavior at the end point is the same as on the critical line. Received 18 January 2001     

Clusters and fluctuations at mean-field critical points and spinodals

We show that the structure of the fluctuations close to spinodals and mean-field critical points is qualitatively different from the structure close to non-mean-field critical points. This difference has important implications for many areas including the formation of glasses in supercooled liquids. In particular, the divergence of the measured static structure function in near-mean-field systems close to the glass transition is suppressed relative to the mean-field prediction in systems for which a spatial symmetry is broken.     

Cluster description of water-in-oil microemulsions near the critical and percolation points

The three-component ionic microemulsion system consisting of AOT/water/decane shows an unusual phase behavior in the vicinity of room temperature. The phase diagram in the temperature-volume fraction (of the dispersed phase) plane exhibits a lower consolute critical point at about 40 degrees centigrades and 10% volume fraction. A percolation line, starting from the vicinity of the critical point, cuts across the plane, extending to high volume fraction side at progressively lower temperatures. In this paper we review the evidence that allows to interpret the phase behavior of our system in terms of interacting spherical droplets. We also investigate the dynamics of droplets, below and approaching the critical point by dynamic light scattering. The first cumulant and time evolution of the droplet density correlation function can be quantitatively calculated by assuming the existence of polydispersed fractal clusters formed by the microemulsion droplets due to attraction. The relaxation phenomena observed in an extensive set of measurements of electrical conductivity and permittivity close to percolation is also reviewed and interpreted through the same cluster-forming mechanism, which reproduces the most relevant features of the frequency-dependent complex dielectric constant of this system. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.