Dynamical Singularity Research Articles

Dynamical singularity of the rate function for quench dynamics in finite-size quantum systems

The dynamical quantum phase transition is characterized by the emergence of nonanalytic behaviors in the rate function, corresponding to the occurrence of exact zero points of the Loschmidt echo in the thermodynamical limit. In general, exact zeros of the Loschmidt echo are not accessible in a finite-size quantum system except for some fine-tuned quench parameters. In this work, we study the realization of the dynamical singularity of the rate function for finite-size systems under the twist boundary condition, which can be introduced by applying a magnetic flux. By tuning the magnetic flux, we illustrate that exact zeros of the Loschmidt echo can be always achieved when the postquench parameter is across the underlying equilibrium phase transition point, and thus the rate function of a finite-size system is divergent at a series of critical times. We demonstrate our theoretical scheme by calculating the Su-Schrieffer-Heeger model and the Creutz model in detail and exhibit its applicability to more general cases. Our result unveils that the emergence of dynamical singularity in the rate function can be viewed as a signature for detecting dynamical quantum phase transition in finite-size systems. We also unveil that the critical times in our theoretical scheme are independent on the systems size, and thus it provides a convenient way to determine the critical times by tuning the magnetic flux to achieve the dynamical singularity of the rate function.

Finite-time singularities in swampland-related dark-energy models

In this work we shall investigate the singularity structure of the phase space corresponding to an exponential quintessence dark-energy model recently related to swampland models. The dynamical system corresponding to the cosmological system is an autonomous polynomial dynamical system, and by using a mathematical theorem we shall investigate whether finite-time singularities can occur in the dynamical system variables. As we demonstrate, the solutions of the dynamical system are non-singular for all cosmic times and this result is general, meaning that the initial conditions corresponding to the regular solutions, belong to a general set of initial conditions and not to a limited set of initial conditions. As we explain, a dynamical system singularity is not directly related to a physical finite-time singularity. Then, by assuming that the Hubble rate with functional form , is a solution of the dynamical system, we investigate the implications of the absence of finite-time singularities in the dynamical system variables. As we demonstrate, Big Rip and a Type-IV singularities can always occur if and , respectively. However, Type-II and Type-III singularities cannot occur in the cosmological system, if the Hubble rate we quoted is considered a solution of the cosmological system.

Four-point functions and the permutation groupS4

Four-point functions are at the heart of many interesting physical processes. A prime example is the light-by-light scattering amplitude, which plays an important role in the calculation of hadronic contributions to the anomalous magnetic moment of the muon. In the calculation of such quantities one faces the challenge of finding a suitable and well-behaved basis of tensor structures in coordinate and/or momentum space. Provided all (or many) of the external legs represent similar particle content, a powerful tool to construct and organize such bases is the permutation group ${S}_{4}$. We introduce an efficient notation for dealing with the irreducible multiplets of ${S}_{4}$, and we highlight the merits of this treatment by exemplifying four-point functions with gauge-boson legs such as the four-gluon vertex and the light-by-light scattering amplitude. The multiplet analysis is also useful for isolating the important kinematic regions and the dynamical singularity content of such amplitudes. Our analysis serves as a basis for future efficient calculations of these and similar objects.

Long-wavelength fluctuations lead to a model of the glass crossover

The effect of long-wavelength fluctuations on the Mode-Coupling-Theory (MCT) dynamical singularity at Tc in the β-regime is studied by means of the standard field-theoretical procedure for a genuine second-order phase transition. The resulting perturbative loop expansion can be resummed leading to an extension of the MCT equation for the critical correlator with local random fluctuations of the separation parameter. The corresponding model explains both qualitatively and quantitatively why the MCT dynamical singularity is transformed into a crossover from relaxational to activated dynamics. Dynamical Heterogeneities emerge naturally as the ergodicity-restoring mechanism instead of ad hoc hopping processes.

Dynamical arrest: interplay of glass and gel transitions.

The structural arrest of a polymeric suspension might be driven by an increase of the cross-linker concentration, which drives the gel transition, as well as by an increase of the polymer density, which induces a glass transition. These dynamical continuous (gel) and discontinuous (glass) transitions might interfere, since the glass transition might occur within the gel phase, and the gel transition might be induced in a polymer suspension with glassy features. Here we study the interplay of these transitions by investigating via event-driven molecular dynamics simulation the relaxation dynamics of a polymeric suspension as a function of the cross-linker concentration and the monomer volume fraction. We show that the slow dynamics within the gel phase is characterized by a long sub-diffusive regime, which is due both to the crowding as well as to the presence of a percolating cluster. In this regime, the transition of structural arrest is found to occur either along the gel or along the glass line, depending on the length scale at which the dynamics is probed. Where the two lines meet there is no apparent sign of higher order dynamical singularity. Logarithmic behavior typical of A3 singularity appears inside the gel phase along the glass transition line. These findings seem to be related to the results of the mode coupling theory for the F13 schematic model.

Bifurcation Dynamics: Theory, Methods, and Applications

Bifurcation theory is a subject with classical mathematical origin. The modern development of this subject dates back to the pioneering work of Poincare, and over the past four decades this theory has witnessed rapid developments with new ideas and methods from dynamical systems theory, singularity theory and many others. This subject relates to the mathematical study of possible changes in the qualitative or topological structure of parameter-dependent systems (families). Bifurcations occur in both continuous systems (described by ODEs, DDEs or PDEs), and discrete systems (described by maps). It is useful to place the subject of bifurcations in two principal classes such as local and global. Local bifurcations occur when variations in parameters cause changes in the stability of equilibrium (or fixed point). The topological changes in the phase portrait of the system may be confined to arbitrarily small neighborhoods of the bifurcating fixed points by moving the bifurcation parameter close to the bifurcation point (hence ‘local’). It is worth to recall that local bifurcations can produce changes in global behavior and that in the mathematical analysis of local bifurcations problems it is necessary to understand global dynamics, i.e., “punctual singularities can be a concentration of global information”. Global bifurcations occur when ‘larger’ invariant sets such as periodic orbits collide with equilibria. This brings changes in the topology of the trajectories in the phase space which cannot be confined to

Structural dynamics of supercooled water from quasielastic neutron scattering and molecular simulations

One of the outstanding challenges presented by liquid water is to understand how molecules can move on a picosecond time scale despite being incorporated in a three-dimensional network of relatively strong H-bonds. This challenge is exacerbated in the supercooled state, where the dramatic slowing down of structural dynamics is reminiscent of the, equally poorly understood, generic behavior of liquids near the glass transition temperature. By probing single-molecule dynamics on a wide range of time and length scales, quasielastic neutron scattering (QENS) can potentially reveal the mechanistic details of water's structural dynamics, but because of interpretational ambiguities this potential has not been fully realized. To resolve these issues, we present here an extensive set of high-quality QENS data from water in the range 253-293 K and a corresponding set of molecular dynamics (MD) simulations to facilitate and validate the interpretation. Using a model-free approach, we analyze the QENS data in terms of two motional components. Based on the dynamical clustering observed in MD trajectories, we identify these components with two distinct types of structural dynamics: picosecond local (L) structural fluctuations within dynamical basins and slower interbasin jumps (J). The Q-dependence of the dominant QENS component, associated with J dynamics, can be quantitatively rationalized with a continuous-time random walk (CTRW) model with an apparent jump length that depends on low-order moments of the jump length and waiting time distributions. Using a simple coarse-graining algorithm to quantitatively identify dynamical basins, we map the newtonian MD trajectory on a CTRW trajectory, from which the jump length and waiting time distributions are computed. The jump length distribution is gaussian and the rms jump length increases from 1.5 to 1.9 Å as the temperature increases from 253 to 293 K. The rms basin radius increases from 0.71 to 0.75 Å over the same range. The waiting time distribution is exponential at all investigated temperatures, ruling out significant dynamical heterogeneity. However, a simulation at 238 K reveals a small but significant dynamical heterogeneity. The macroscopic diffusion coefficient deduced from the QENS data agrees quantitatively with NMR and tracer results. We compare our QENS analysis with existing approaches, arguing that the apparent dynamical heterogeneity implied by stretched exponential fitting functions results from the failure to distinguish intrabasin (L) from interbasin (J) structural dynamics. We propose that the apparent dynamical singularity at ∼220 K corresponds to freezing out of J dynamics, while the calorimetric glass transition corresponds to freezing out of L dynamics.

Dynamical singularity resolution in spherically symmetric black hole formation

We study numerically the effects of loop quantum gravity motivated corrections on massless scalar field collapse in Painleve-Gullstrand coordinates. Near criticality the system exhibits Choptuik scaling with a mass gap and a new scaling relationship dependant upon the quantum length scale. Classical singularities are resolved by a radiationlike phase in the quantum collapse: the black hole consists of a compact region of spacetime bounded by a single, smooth trapping horizon. The 'evaporation' is not complete but leaves behind a small expanding shell that disperses to infinity.

A low-dimensional model of separation bubbles

In this work, motivated by the problem of model-based predictive control of separated flows, we identify the key variables and the requirements on a model-based observer and construct a prototype low-dimensional model to be embedded in control applications. Namely, using a phenomenological physics-based approach and dynamical systems and singularity theories, we uncover the low-dimensional nature of the complex dynamics of actuated separated flows and capture the crucial bifurcation and hysteresis inherent in separation phenomena. This new look at the problem naturally leads to several important implications, such as, firstly, uncovering the physical mechanisms for hysteresis, secondly, predicting a finite amplitude instability of the bubble, and, thirdly, to new issues to be studied theoretically and tested experimentally.

Singularities in core-level spectra near metallic surfaces

Calculations of the screening shift and dynamical singularity index of the structureless core hole placed near and inside the metallic slab are presented, where the screening mechanism is described using the quantum-mechanical response function calculated in the local density approximation. Core-hole screening shifts show oscillations which are compared with the experimental results. Spatial dependence of the static singularity index, besides experimentally observed enhancement at the surface, shows some new features (bulk oscillations) which were not yet experimentally observed. The dynamical singularity index is used to calculate complete core hole spectrum, including singular line shapes and bulk and surface plasmon losses, and the results explain very well some observed features in the experimental photoemission spectra.

Black hole head-on collisions and gravitational waves with fixed mesh-refinement and dynamic singularity excision

We present long-term-stable and convergent evolutions of head-on black-hole collisions and extraction of gravitational waves generated during the merger and subsequent ring-down. The new ingredients in this work are the use of fixed mesh-refinement and dynamical singularity excision techniques. We are able to carry out head-on collisions with large initial separations and demonstrate that our excision infrastructure is capable of accommodating the motion of the individual black holes across the computational domain as well as their merger. We extract gravitational waves from these simulations using the Zerilli-Moncrief formalism and find the ring-down radiation to be, as expected, dominated by the $\ensuremath{\ell}=2$, $m=0$ quasinormal mode. The total radiated energy is about $0.1%$ of the total Arnowitt-Deser-Misner mass of the system.

Glass-Glass Transition and New Dynamical Singularity Points in an Analytically Solvablep-Spin Glasslike Model

We introduce and analytically study a generalized p-spin glasslike model that captures some of the main features of attractive glasses, recently found by mode coupling investigations, such as a glass-glass transition line and dynamical singularity points characterized by a logarithmic time dependence of the relaxation. The model also displays features not predicted by the mode coupling scenario that could further describe the attractive glasses behavior, such as aging effects with new dynamical singularity points ruled by logarithmic laws or the presence of a glass spinodal line.

Damped finite-time singularity driven by noise.

We consider the combined influence of linear damping and noise on a dynamical finite-time singularity model for a single degree of freedom. We find that the noise effectively resolves the finite-time singularity and replaces it by a first-passage-time distribution or absorbing state distribution with a peak at the singularity and a long time tail. The damping introduces a characteristic cross-over time. In the early time regime the probability distribution and first-passage-time distribution show a power law behavior with scaling exponent depending on the ratio of the nonlinear coupling strength to the noise strength. In the late time regime the behavior is controlled by the damping. The study might be of relevance in the context of hydrodynamics on a nanometer scale, in material physics, and in biophysics.

A matching pursuit technique for computing the simplest normal forms of vector fields

This paper presents a matching pursuit technique for computing the simplest normal forms of vector fields. First a simple, explicit recursive formula is derived for general differential equations, which reduces computation to the minimum. Then a matching pursuit technique is introduced and applied to the Takens–Bogdanov dynamical singularity. It is shown that unlike other methods for computing normal forms, the technique using matching pursuit does not need any algebraic constraints which are required for the existence of the simplest normal form. The efficient method and matching pursuit technique, which have been implemented using Maple, can be “automatically” executed on various computer systems. A number of examples are presented to demonstrate the advantages of the technique.

2 + 1 black hole with a dynamical conical singularity

We find black hole solutions to Euclidean 2 + 1 gravity coupled to a relativistic particle which have a dynamical conical singularity at the horizon. These solutions mimic the tree level contribution to the partition function of gravity coupled to a quantum field theory. They are found to violate the standard area law for black hole entropy, their entropy being proportional to the total opening angle. Since each solution depends on the number of windings of the particle path around the horizon, the significance of their summation in the path integral is considered.

Fragile-glass behavior of a short-range p-spin model.

We propose a short-range generalization of the p-spin interaction spin-glass model. The model is well suited to test the idea that an entropy collapse is at the bottom line of the dynamical singularity encountered in structural glasses. The model is studied in three dimensions through Monte Carlo simulations, which put in evidence fragile glass behavior with stretched exponential relaxation and super-Arrhenius behavior of the relaxation time. Our data are in favor of a Vogel-Fulcher behavior of the relaxation time, related to an entropy collapse at the Kauzmann temperature. We, however, encounter difficulties analogous to those found in experimental systems when extrapolating thermodynamical data at low temperatures. We study the spin-glass susceptibility, investigating the behavior of the correlation length in the system. We find that the increase of the relaxation time is accompanied by a very slow growth of the correlation length. We discuss the scaling properties of off-equilibrium dynamics in the glassy regime, finding qualitative agreement with the mean-field theory. \textcopyright{} 1996 The American Physical Society.

Dynamics of a freely evolving, two-dimensional granular medium.

We consider the dynamics of an ensemble of identical, inelastic, hard disks in a doubly periodic domain. Because there is no external forcing the total energy of the system is monotonically decreasing so that this idealized granular medium is ``cooling down.'' There are three nondimensional control parameters: the coefficient of restitution r, the solid fraction \ensuremath{\nu}, and the total number of disks in the domain N. Our goal is a comprehensive description of the phenomenology of granular cooling in the (r,\ensuremath{\nu},N) parameter space. Previous studies have shown that granular cooling results in the formation of structures: both the mass and the momentum spontaneously become nonuniform. Four different regimes (kinetic, shearing, clustered, and collapsed) have been identified. Starting with the almost elastic case, in which r is just less than 1, the kinetic regime resembles a classical nondissipative gas in which there are no structures. When r is decreased (with fixed N and \ensuremath{\nu}) the system evolves into the shearing regime in which most of the energy and momentum resides in the gravest hydrodynamic shear mode. At still smaller values of r the clustered regime appears as an extended transient. Large clusters of disks form, collide, breakup, and reform. From the clustered state the gas eventually either evolves into the shearing regime or, alternatively, collapses. The collapsed regime is characterized by a dynamical singularity in which a group of particles collides infinitely often in a finite time. While each individual collision is binary, the space and time scales decrease geometrically with the cumulative number of collisions so that a multiparticle interaction occurs. The regime boundaries (i.e., the critical values of r) in the (N,\ensuremath{\nu}) plane have been delineated using event-driven numerical simulations. Analytic considerations show that the results of the simulations can be condensed by supposing that the critical values of r depend only on N and \ensuremath{\nu} through the optical depth, \ensuremath{\lambda}\ensuremath{\equiv}d\ensuremath{\surd}N\ensuremath{\pi}\ensuremath{\nu}/2 where d is the disk diameter. \textcopyright{} 1996 The American Physical Society.

A Cascade of Structure in a Drop Falling from a Faucet

A drop falling from a faucet is a common example of a mass fissioning into two or more pieces. The shape of the liquid in this situation has been investigated by both experiment and computer simulation. As the viscosity of the liquid is varied, the shape of the drop changes dramatically. Near the point of breakup, viscous drops develop long necks that then spawn a series of smaller necks with ever thinner diameters. Simulations indicate that this repeated formation of necks can proceed ad infinitum whenever a small but finite amount of noise is present in the experiment. In this situation, the dynamical singularity occurring when a drop fissions is characterized by a rough interface.

Dynamical singularity of the glass transition in molten lithium iodide

The liquid-glass transition in molten lithium iodide has been investigated by means of constant-pressure and constant-temperature molecular dynamics. Both static and dynamic properties were calculated, with a microscopic analysis of the ionic motions. In the vicinity of the glass transition, the mean-square displacement shows a non-linear time dependence and the incoherent scattering function is well fitted by a ‘stretched exponential’ function. The maximum value of the non-Gaussian parameter increases drastically on approaching the glass transition. The glass transition temperature is estimated by trapping diffusion theory.

On Genericity and Complements of Measure Zero Sets in Function Spaces

Generic properties of function spaces have been of particular interest in dynamical systems and singularity theory. The underlying assumption has been that the complement of a dense ${G_\delta }$ set is sparse enough to be considered unlikely. Nevertheless, in infinite dimensional spaces, even dense ${G_\delta }$’s may have measure zero. Since there is no one canonical measure on an infinite dimensional Fréchet space, notions of measure zero have not often been considered. Here we use a notion of Haar measure zero on abelian Polish groups due to Christensen [1]. We show that those sections of a finite dimensional vector bundle over a compact manifold whose jets are transverse to a submanifold of the jet bundle are complements of sets of Haar measure zero.