Classical Motion Research Articles

The stochastic Fisher-KPP Equation with seed bank and on/off branching coalescing Brownian motion

We introduce a new class of stochastic partial differential equations (SPDEs) with seed bank modeling the spread of a beneficial allele in a spatial population where individuals may switch between an active and a dormant state. Incorporating dormancy and the resulting seed bank leads to a two-type coupled system of equations with migration between both states. We first discuss existence and uniqueness of seed bank SPDEs and provide an equivalent delay representation that allows a clear interpretation of the age structure in the seed bank component. The delay representation will also be crucial in the proofs. Further, we show that the seed bank SPDEs give rise to an interesting class of “on/off”-moment duals. In particular, in the special case of the F-KPP Equation with seed bank, the moment dual is given by an “on/off-branching Brownian motion”. This system differs from a classical branching Brownian motion in the sense that independently for all individuals, motion and branching may be “switched off” for an exponential amount of time after which they get “switched on” again. On/off branching Brownian motion shows qualitatively different behaviour to classical branching Brownian motion and is an interesting object for study in itself. Here, as an application of our duality, we show that the spread of a beneficial allele, which in the classical F-KPP Equation, started from a Heaviside intial condition, evolves as a pulled traveling wave with speed sqrt{2}, is slowed down significantly in the corresponding seed bank F-KPP model. In fact, by computing bounds on the position of the rightmost particle in the dual on/off branching Brownian motion, we obtain an upper bound for the speed of propagation of the beneficial allele given by sqrt{sqrt{5}-1}approx 1.111 under unit switching rates. This shows that seed banks will indeed slow down fitness waves and preserve genetic variability, in line with intuitive reasoning from population genetics and ecology.

Classical nuclear motion: Does it fail to explain reactions and spectra in certain cases?

AbstractIs a classical description of nuclear motion sufficient when describing chemical reactions and spectra? This question is interesting because many researchers use a classical description of nuclear motion in molecular dynamics simulations. The present paper investigates some phenomena that were previously attributed to nuclear quantum effects. The question is if these phenomena can be modeled with traditional Car–Parrinello molecular dynamics, that is, with a method which treats nuclear motion classically and which is widely applied to the simulation of chemical reactions and spectra. We find that for the investigated system no additional paradigm is needed for describing chemical reactions. The special reactivity observed for carbenes can be attributed to the special environment represented by a noble gas matrix and to an additional transition state that was not considered before. Also the infrared spectrum of porphycene is perfectly modeled by traditional Car–Parrinello molecular dynamics. More studies are necessary to decide to what extent classical nuclear motion can replace the quantum mechanical description.

Effect of longitudinal and torsional ultrasound on corneal endothelial cells: experimental study in rabbit eyes.

To compare corneal endothelial damage from longitudinal and torsional ultrasound during phacoemulsification. John A. Moran Eye Center, University of Utah, Salt Lake City, Utah, USA. Experimental study. 9 New Zealand white rabbits underwent bilateral surgery. After incision, the Intrepid Balanced Tip of the Centurion Ozil handpiece was inserted into the anterior chamber, and the following settings were used: 50 mL/min flow, 70 mm Hg intraocular pressure, 600 mm Hg vacuum, and 60% longitudinal (1 eye) or torsional (contralateral eye) ultrasound for 30 seconds. Cumulative dissipated energy (CDE) was noted. After euthanasia and enucleation, the corneas were removed, stained with trypan blue/alizarin red, and photographed (400× photographs from 5 specific areas and 1 overview photograph from each corneal button). The ImageJ program was used to evaluate cell damage and loss in the photographs obtained from each cornea. Cavitation bubbles around the phacoemulsification tip were generally observed in the longitudinal group. CDE was 17.4 ± 0.58 and 6.93 ± 0.15 in the longitudinal and torsional groups, respectively (P = .003). The percentage of intact cells was statistically higher in the torsional group (94.22% ± 5.99% vs 56.85% ± 22.29%), and the percentage of lost cells was statistically higher in the longitudinal group (42.10% ± 22.02% vs 4.97% ± 6.23%) (P = .003). This was observed in the analysis of the 400× photographs and the overview photographs. This study suggests that torsional ultrasound was associated with significantly less corneal endothelial cell damage than classical longitudinal tip motion, providing further insight on mechanisms of corneal endothelial damage during phacoemulsification.

Critical indicators of dynamic stall vortex

During a dynamic stall process, various flow structures leave their pressure footprints on a wing surface, through which it is possible to understand the development of different flow stages and predict the critical flow events. Based on a classical airfoil ramp-up pitching motion, proper orthogonal decomposition analysis of pressure on the suction surface is carried out in this study. Accordingly, the surface pressure evolution during dynamic stall is summarized into three basic physical schemas, based on which several real-time critical indicators for predicting the flow events related to the dynamic stall vortex (DSV) are constructed. These critical indicators include the spatial distribution coefficient of pressure (SDCP), the high-order central moment of pressure (HCMP), the location of peak pressure (LPP) and the modulated location of peak pressure (MLPP). These indicators can predict the formation of laminar separation bubbles, DSV initiation, DSV centre position and the detachment of DSV. Therefore, the real-time whole-life monitoring of DSV has been realized. Moreover, the effectiveness of these indicators has also been confirmed under different parameters, and testing using wind tunnel experimental data proves their noise robustness. Studies show that the SDCP and HCMP may also be effective even if only two transducers are used. Finally, a modification method of SDCP based on Z-score standardized pressure is proposed. It is found that the modified SDCP can effectively reduce sensitivity to kinematic parameters and the Reynolds number. The critical indicators in this study can be used as a reference for the standardized mathematical and physical description of DSV evolution, thus laying a foundation for constructing a universal theoretical framework of dynamic stall.

Dirac approach in review of the classical and quantum motion on the surface of the hyperboloid of one sheet

Dirac approach in review of the classical and quantum motion on the surface of the hyperboloid of one sheet

Ghosts without Runaway Instabilities.

We present a simple class of mechanical models where a canonical degree of freedom interacts with another one with a negative kinetic term, i.e., with a ghost. We prove analytically that the classical motion of the system is completely stable for all initial conditions, notwithstanding that the conserved Hamiltonian is unbounded from below and above. This is fully supported by numerical computations. Systems with negative kinetic terms often appear in modern cosmology, quantum gravity, and high energy physics and are usually deemed as unstable. Our result demonstrates that for mechanical systems this common lore can be too naive and that living with ghosts can be stable.

Geometry of Krylov complexity

We develop a geometric approach to operator growth and Krylov complexity in many-body quantum systems governed by symmetries. We start by showing a direct link between a unitary evolution with the Liouvillian and the displacement operator of appropriate generalized coherent states. This connection maps operator growth to a purely classical motion in phase space. The phase spaces are endowed with a natural information metric. We show that, in this geometry, operator growth is represented by geodesics, and Krylov complexity is proportional to a volume. This geometric perspective also provides two novel avenues toward computation of Lanczos coefficients, and it sheds new light on the origin of their maximal growth. We describe the general idea and analyze it in explicit examples, among which we reproduce known results from the Sachdev-Ye-Kitaev model, derive operator growth based on SU(2) and Heisenberg-Weyl symmetries, and generalize the discussion to conformal field theories. Finally, we use techniques from quantum optics to study operator evolution with quantum information tools such as entanglement and Renyi entropies, negativity, fidelity, relative entropy, and capacity of entanglement.

On the extension of a physical body in classical motion. An analogy between a pseudo-velocity concept and Wiener’s process in (ideal) polymer solutions

A pseudo-velocity concept, based on the extension of a linear body, is defined by a special relativity experiment. It suggests an analogy with the covariance properties of Wiener’s process, ultimately implying that the scaling behavior of (Gaussian) polymer solutions can be derived from a Brownian Relativity theory, as it was formerly put forward. An ad-hoc statistical interpretation of the resulting spacetime transforms may be given by the central limit theorem.

Hydrodynamic Interpretation of Generic Squeezed Coherent States: A Kinetic Theory

The hydrodynamic interpretation of quantum mechanics treats a system of particles in an effective manner. In this work, we investigate squeezed coherent states within the hydrodynamic interpretation. The Hamiltonian operator in question is time dependent, n-dimensional and in quadratic order. We start with deriving a phase space Wigner probability distribution and an associated equilibrium entropy for the squeezed coherent states. Then, we decompose the joint phase space distribution into two portions: a marginal position distribution and a momentum distribution that is conditioned on the post-selection of positions. Our conditionally averaged momenta is shown to be equal to the Bohm's momenta whose connection to the weak measurements is already known. We also keep track of the corresponding classical system evolution by identifying shear, magnification and rotation components of the symplectic phase space dynamics. This allows us to pinpoint which portion of the underlying classical motion appears in which quantum statistical concept. We show that our probability distributions satisfy the Fokker-Planck equations exactly. They can be used to decompose the equilibrium entropy into the missing information in positions and in momenta as in the Sackur-Tetrode entropy of the classical kinetic theory. Eventually, we define a quantum pressure, a quantum temperature and a quantum internal energy which are related to each other in the same fashion as in the classical kinetic theory. We show that the quantum potential incorporates the kinetic part of the internal energy and the fluctuations around it. This allows us to suggest a quantum conditional virial relation. In the end, we show that the kinetic internal energy is linked to the fractional Fourier transformer part of the underlying classical dynamics similar to the case where the energy of a quantum oscillator is linked to its Maslov index.

On Mass Spectra of Primordial Black Holes

Evidence for the primordial black holes (PBH) presence in the early Universe renews permanently. New limits on their mass spectrum challenge existing models of PBH formation. One of the known models is based on the closed walls collapse after the inflationary epoch. Its intrinsic feature is the multiple production of small mass PBH which might contradict observations in the nearest future. We show that the mechanism of walls collapse can be applied to produce substantially different PBH mass spectra if one takes into account the classical motion of scalar fields together with their quantum fluctuations at the inflationary stage. Analytical formulas have been developed that contain both quantum and classical contributions.

Quantum-Classical Correspondence Principle for Heat Distribution in Quantum Brownian Motion.

Quantum Brownian motion, described by the Caldeira–Leggett model, brings insights to the understanding of phenomena and essence of quantum thermodynamics, especially the quantum work and heat associated with their classical counterparts. By employing the phase-space formulation approach, we study the heat distribution of a relaxation process in the quantum Brownian motion model. The analytical result of the characteristic function of heat is obtained at any relaxation time with an arbitrary friction coefficient. By taking the classical limit, such a result approaches the heat distribution of the classical Brownian motion described by the Langevin equation, indicating the quantum–classical correspondence principle for heat distribution. We also demonstrate that the fluctuating heat at any relaxation time satisfies the exchange fluctuation theorem of heat and its long-time limit reflects the complete thermalization of the system. Our research study justifies the definition of the quantum fluctuating heat via two-point measurements.

A dynamical version of the SYK model and theq-Brownian motion

We extend recent results on the asymptotic eigenvalue distribution of the SYK model to the multivariate case and relate the limit of a dynamical version of the SYK model with the q-Brownian motion, a non-commutative deformation of classical Brownian motion. Furthermore, we extend the results for fluctuations to the multivariate setting and treat also higher correlation functions. The structure of our results for the sparse SYK random matrices resembles the formulas for higher order freeness for ordinary GUE random matrices.

A Three-Dimensional Continuum Simulation Method for Grain Boundary Motion Incorporating Dislocation Structure

We develop a continuum model for the dynamics of grain boundaries in three dimensions that incorporates the motion and reaction of the constituent dislocations. The continuum model is based on a simple representation of densities of curved dislocations on the grain boundary. Illposedness due to nonconvexity of the total energy is fixed by a numerical treatment based on a projection method that maintains the connectivity of the constituent dislocations. An efficient simulation method is developed, in which the critical but computationally expensive long-range interaction of dislocations is replaced by another projection formulation that maintains the constraint of equilibrium of the dislocation structure described by the Frank’s formula. This continuum model is able to describe the grain boundary motion and grain rotation due to both coupling and sliding effects, to which the classical motion by mean curvature model does not apply. Comparisons with atomistic simulation results show that our continuum model is able to give excellent predictions of evolutions of low angle grain boundaries and their dislocation structures.

A novel kind of equations linking the quantum dynamics and the classical wave motions based on the catastrophe theory

Considering that the catastrophe theory could describe quantitatively any phase transition process, we adopt the folding and cusp catastrophe types as the potential functions in the Schrödinger equation to attempt to link the quantum dynamics and the classical wave motions. Thus, through the dimensionless analysis a novel kind of partial differential equations is derived out. When the scaling parameter of the novel equation is equal to the Planck's constant, this equation becomes a detailed time-independent Schrödinger equation, from which Bohr correspondence principle can be found. On the other hand, when the scaling parameter tends to zero, this equation could degenerate to the classical Helmholtz equation. Therefore, this novel kind of equations could describe quantitatively the variation process of the wave functions from the macroscopic level to the quantum size.

SPX Calibration of Option Approximations under Rough Heston Model

The volatility of stock return does not follow the classical Brownian motion, but instead it follows a form that is closely related to fractional Brownian motion. Taking advantage of this information, the rough version of classical Heston model also known as rough Heston model has been derived as the macroscopic level of microscopic Hawkes process where it acts as a high-frequency price process. Unlike the pricing of options under the classical Heston model, it is significantly harder to price options under rough Heston model due to the large computational cost needed. Previously, some studies have proposed a few approximation methods to speed up the option computation. In this study, we calibrate five different approximation methods for pricing options under rough Heston model to SPX options, namely a third-order Padé approximant, three variants of fourth-order Padé approximant, and an approximation formula made from decomposing the option price. The main purpose of this study is to fill in the gap on lack of numerical study on real market options. The numerical experiment includes calibration of the mentioned methods to SPX options before and after the Lehman Brothers collapse.

Exact Closed-Form Solutions of the Motion in Non-Inertial Reference Frames, Using the Properties of Lie Groups SO3 and SE3

The paper offers a general symbolic method to study the motion in a non-inertial reference frame. In order to achieve this, we use the algebraic and geometric properties of the Lie group of special orthogonal tensors, SO3, and the Lie group of the rigid body displacements, SE3. We obtain a simplified form of the initial value problem that models the non-inertial motion using a tensor instrument introduced in this paper. Thus, the study of the motion in a non-inertial reference frame is transferred into the study of a classical motion in an inertial reference frame. The applications of this method refer to solving the relative motion problem and deriving the straightforward solution to classical theoretical mechanics problems. The motion in a uniform gravitational force field in a rotating reference frame, the motion of a charged particle in non-stationary electric and magnetic fields, the exact solution of the relative rigid body motion in the non-inertial reference frame are studied. Using this symbolic method in studying the motion in a non-inertial reference frame reduces the number of computations. In addition, it offers, in some essential particular cases, exact closed-form coordinate-free analytical solutions.

Untying the Gordian KNOT: Unbiased Single Particle Tracking Using Point Clouds and Adaptive Motion Analysis.

Achieving mechanistic understanding of transport in complex environments such as inside cells or at polymer interfaces is challenging. We need better ways to image transport in 3-D and better single particle tracking algorithms to determine transport that are not systemically biased toward any classical motion model. Here we present an unbiased single particle tracking algorithm: Knowing Nothing Outside Tracking (KNOT). KNOT uses point clouds provided by iterative deconvolution to educate individual particle localizations and link particle positions between frames to achieve 2-D and 3-D tracking. Information from prior point clouds fuels an independent adaptive motion model for each particle to avoid global models that could introduce biases. KNOT competes with or surpasses other 2-D methods from the 2012 particle tracking challenge while accurately tracking adsorption dynamics of proteins on polymer surfaces and early endosome transport in live cells in 3-D. We apply KNOT to study 3-D endosome transport to reveal new physical insight into locally directed and diffusive transport in live cells. Our analysis demonstrates better accuracy in classifying local motion and its direction compared to previous methods, revealing intricate intracellular transport heterogeneities.

High-order geometric integrators for representation-free Ehrenfest dynamics.

Ehrenfest dynamics is a useful approximation for ab initio mixed quantum-classical molecular dynamics that can treat electronically nonadiabatic effects. Although a severe approximation to the exact solution of the molecular time-dependent Schrödinger equation, Ehrenfest dynamics is symplectic, is time-reversible, and conserves exactly the total molecular energy as well as thenormof the electronic wavefunction. Here, we surpass apparent complications due to the coupling of classical nuclear and quantum electronic motions and present efficient geometric integrators for "representation-free" Ehrenfest dynamics, which do not rely on a diabatic or adiabatic representation of electronic states and are of arbitrary even orders of accuracy in the time step. These numerical integrators, obtained by symmetrically composing the second-order splitting method and exactly solving the kinetic and potential propagation steps, are norm-conserving, symplectic, and time-reversible regardless of the time step used. Using a nonadiabatic simulation in the region of a conical intersection as an example, we demonstrate that these integrators preserve the geometric properties exactly and, if highly accurate solutions are desired, can be even more efficient than the most popular non-geometric integrators.

Hydrogen-bonded single-component organic ferroelectrics revisited by van der Waals density-functional theory calculations

We apply van der Waals density-functional theory (vdW-DFT) calculations to predict crystal structure parameters and spontaneous polarization values for seven hydrogen-bonded single-component organic ferroelectrics. The results show good agreement with experimental results, implying that an important step for the computational materials design of organic ferroelectrics has been achieved. This approach also enables the simulation of electromechanical responses. Calculations using the vdW-DFT method are performed for croconic acid (CRCA), 2-phenylmalondialdehyde (PhMDA), and 5,6-dichloro-2-methylbenzimidazole (DC-MBI) under uniaxial stresses or electric fields. Direct piezoelectric ${d}_{33}$ constants are evaluated from the polarization change as a function of stress, whereas converse piezoelectric ${d}_{33}$ constants are evaluated from the change in lattice parameter as a function of electric field. The obtained values show acceptable agreement with the experimental values if possible objective factors are considered. The stress-induced or electric-field-induced variation of polarization is analyzed considering two types of contributions. One is from proton transfer as a classical point charge motion and the other residual part corresponds to the redistribution of $\ensuremath{\pi}$ electrons.

Acoustic Emission Characterization of Transgranular Cracks in WC–Co Cemented Carbides During a One-way Scratch

The tribological behavior of tungsten carbide–cobalt materials is influenced by the cobalt content and the WC grains size. The main wear mechanisms in these materials are cobalt depletion, intergranular cracks, and WC grain cleavages. More specifically, coarse WC grains favor the apparition of transgranular cracks during sliding friction tests. A promising way to access in real-time blinded tribological contacts is the technique of acoustic emission (AE). This study clearly identifies transgranular cracks in AE signals. The AE energy and frequency of this mechanism were experimentally associated with the size of the transgranular cracks. A mechanical model based on the classical beam theory and harmonic motion equations confirms these relations. The AE centroid period (i.e., inverse of the centroid frequency) increases linearly with the size of the transgranular cracks. The AE energy increases linearly with the cube of the transgranular cracks length.