Classical Motion Research Articles

Subdiffusive option price model with Inverse Gaussian subordinator

The paper focuses on the option price subdiffusive model under the unusual behavior of the market, when the price may not be changed for some time, which is a quite common situation in modern illiquid financial markets or during global crises. In the model, the risk-free bond motion and classical geometrical Brownian motion (GBM) are time-changed by an inverted inverse Gaussian($\mathit{IG}$) subordinator. We explore the correlation structure of the subdiffusive GBM stock returns process, discuss option pricing techniques based on the martingale option pricing method and the fractal Dupire equation, and demonstrate how it applies in the case of the $\mathit{IG}$ subordinator.

Analysis of Velocity Autocorrelation Functions from Molecular Dynamics Simulations of a Small Peptide by the Generalized Langevin Equation with a Power-Law Kernel.

Internal motions play an essential role in the biological functions of proteins and have been the subject of numerous theoretical and spectroscopic studies. Such complex environments are associated with anomalous diffusion where, in contrast to the classical Brownian motion, the relevant correlation functions have power law decays with time. In this work, we investigate the presence of long memory stochastic processes through the analysis of atomic velocity autocorrelation functions. Analytical expressions of the velocity autocorrelation function spectrum obtained through a Mori-Zwanzig projection approach were shown to be compatible with molecular dynamics simulations of a small helical peptide (8-polyalanine).

Granger causality based on vector time series and quaternion algebra with possible applications to molecular dynamics data analysis.

Causal analysis plays a significant role in physics, chemistry, and biology. Dynamics of complex (bio)molecular and nanosystems, from the microscopic to the macroscopic scale, are characterized by time-dependent vectors such as positions, forces, momenta, angular momenta, or torques. Identification and analysis of causal relationships between these time-dependent signals is an important problem in the multidimensional time-series analysis and is of great practical importance in describing the properties of such dynamical systems, and to understanding their functionality. For linear stochastic systems characterized by multidimensional scalar signals, Granger proposed a simple procedure to detect causal relationships, called Granger causality. In this study we extended this formalism to vector signals representing physical vector quantities. For this purpose, we used quaternion algebra, where vector signals are treated as time-dependent quaternions. The developed analytical model is based on the autoregressive formalism. This formalism (Q-MVAR) and its numerical implementation were validated using two simple dynamic models: a rigid body model represented by a benzenelike molecular fragment, interacting with a short-range harmonic potential with a wall, as well as a system of three model atomic balls moving inside a soft spherical surface and interacting with long range electrostatic forces. Although the motivation to these studies was the analysis of classical motions in complex (bio)molecular systems, described with a mechanical model and based on molecular dynamics (MD) simulations, in particular coarse-grained ones, it should be noted that the developed extended formalism can be applied to any system composed of many rigid elements that interact with arbitrary potentials and are characterized by complex internal motions. A description of the detailed procedure for calculating causality measures is provided in the Appendicesof the Supplemental Material. This formalism and the prototype of its numerical implementation can be further developed and applied in many different fields of physical, natural, and engineering sciences.

Quantum and anharmonic effects in non-adiabatic transition state theory.

Quantitative descriptions of non-adiabatic transition rates at intermediate temperatures are challenging due to the simultaneous importance of quantum and anharmonic effects. In this paper, the interplay between quantum effects-for motion across or along the seam of crossing-and anharmonicity in the seam potential is considered within the weak coupling limit. The well-known expression for quantized 1-D motion across the seam (i.e., tunneling) in the linear terms approximation is derived in the thermal domain using the Lagrangian formalism, which is then applied to the case when tunneling is distributed along the seam of crossing (treating motion along the seam classically). For high-frequency quantum modes, a vibrationally adiabatic (VA) approach is developed that introduces to the non-adiabatic rate constant a factor associated with high-frequency wavefunction overlap; this approach treats the high-frequency motion along the seam quantum mechanically. To test these methodologies, the reaction N2O ↔ N2 + O(3P) was chosen. CCSD(T)-F12b/cc-pVTZ-F12 explorations of the 3A'-1A' seam of N2O revealed that seam anharmonicity has a strong effect on the rate constant (a factor of ∼20 at 2000K). Several quantum effects were found to be significant at intermediate/lower temperatures, including the quantum N-N vibration that was coupled with seam anharmonicity using the VA approach. Finally, a 1-D approximation to non-adiabatic instanton theory is presented to estimate the validity limit of the linear terms model at low temperatures (∼250 K for N2O). We recommend that the assumptions built into many statistical theories for non-adiabatic reactions-harmonic behavior, classical motion, linear terms, and weak coupling-should be verified on a case-by-case basis.

Global and local stability for ghosts coupled to positive energy degrees of freedom

Negative kinetic energies correspond to ghost degrees of freedom, which are potentially of relevance for cosmology, quantum gravity, and high energy physics. We present a novel wide class of stable mechanical systems where a positive energy degree of freedom interacts with a ghost. These theories have Hamiltonians unbounded from above and from below, are integrable, and contain free functions. We show analytically that their classical motion is bounded for all initial data. Moreover, we derive conditions allowing for Lyapunov stable equilibrium points. A subclass of these stable systems has simple polynomial potentials with stable equilibrium points entirely due to interactions with the ghost. All these findings are fully supported by numerical computations which we also use to gather evidence for stability in various nonintegrable systems.

NeI2 Photofragmentation Dynamics Through Quasiclassical and Semiclassical Studies.

The triatomic system NeI$_2$ has been studied under the consideration the diatom is found in an excited electronic state (B). The vibrational levels $(v=13,\ldots,23)$ are taken into account in this research. For such a study, we use two well-known theoretical procedure: the QCT (Quasi-Classical Trajectories) method, where the classical equations motion for nuclei are solved on a single potential energy surface (PES), unlike the TSH (Trajectory Surface Hopping) method, where they are solved in a bunch of crossed vibrational PES (diabatic representation). We have incorporated the Trajectory Surface Hopping - Fewest Switches (TSH-FS) in order to minimize the number of hoppings, allowing us calculate the probability of hopping between the different PES's, and the kinetic mechanism to track the path followed for the system until dissociation. By using these methods we are able to obtain several observables like the lifetimes, vibrational and rotational energies (I$_2$), dissociation channels, among others. Our results are compared with previous experimental and theoretical work.

Evolution of the wave function's shape in a time-dependent harmonic potential

Abstract An effective operational approach to quantum mechanics is to focus on the evolution of wave packets, for which the wave function can be seen in the semi-classical regime as representing a classical motion dressed with extra degrees of freedom describing the shape of the wave packet and its fluctuations. These quantum dressing are independent degrees of freedom, mathematically encoded in the higher moments of the wave function. We review how to extract the effective dynamics for Gaussian wave packets evolving according to the Schrödinger equation with time-dependent potential in a 1 + 1-dimensional spacetime, and derive the equations of motion for the quadratic uncertainty. We then show how to integrate the evolution of all the higher moments for a general wave function in a time-dependent harmonic potential.

Maximum principle for mean‐field controlled systems driven by a fractional Brownian motion

AbstractWe study a stochastic control problem of mean‐field controlled stochastic differential systems driven by a fractional Brownian motion with Hurst parameter . As a necessary condition of the optimal control we obtain a stochastic maximum principle. The associated adjoint mean‐field backward stochastic differential equation driven by a fractional Brownian motion and a classical Brownian motion. Applying the stochastic maximum principle to a mean‐field stochastic linear quadratic problem, we obtain the optimal control and prove that the necessary condition for the optimality of an admissible control is also sufficient under certain assumptions.

Composable energy policies for reactive motion generation and reinforcement learning

In this work, we introduce composable energy policies (CEP), a novel framework for multi-objective motion generation. We frame the problem of composing multiple policy components from a probabilistic view. We consider a set of stochastic policies represented in arbitrary task spaces, where each policy represents a distribution of the actions to solve a particular task. Then, we aim to find the action in the configuration space that optimally satisfies all the policy components. The presented framework allows the fusion of motion generators from different sources: optimal control, data-driven policies, motion planning, and handcrafted policies. Classically, the problem of multi-objective motion generation is solved by the composition of a set of deterministic policies, rather than stochastic policies. However, there are common situations where different policy components have conflicting behaviors, leading to oscillations or the robot getting stuck in an undesirable state. While our approach is not directly able to solve the conflicting policies problem, we claim that modeling each policy as a stochastic policy allows more expressive representations for each component in contrast with the classical reactive motion generation approaches. In some tasks, such as reaching a target in a cluttered environment, we show experimentally that CEP additional expressivity allows us to model policies that reduce these conflicting behaviors. A field that benefits from these reactive motion generators is the one of robot reinforcement learning. Integrating these policy architectures with reinforcement learning allows us to include a set of inductive biases in the learning problem. These inductive biases guide the reinforcement learning agent towards informative regions or improve collision safety while exploring. In our work, we show how to integrate our proposed reactive motion generator as a structured policy for reinforcement learning. Combining the reinforcement learning agent exploration with the prior-based CEP, we can improve the learning performance and explore safer.

A Note on Integrability Loss in Fuzzball Geometries

AbstractThe dynamics of certain string configurations in a class of fivebrane supertube backgrounds is studied. In the decoupling limit of the fivebranes, these solutions are known to admit an exact description in worldsheet string theory and string propagation is integrable. For the asymptotically flat solutions, by using analytic tools of classical Hamiltonian systems, the non‐integrability of classical string motion is proved. This suggests that string dynamics in circular supertube geometries exhibit a regime of chaotic behavior.

Electron Localization in Rydberg States

We discuss the possibility of localizing an electron in a highly excited Rydberg state. The second-order correlation of emitted photons is the tool for the determination of electron position. This second-order correlation of emitted radiation and, therefore, the correlation of operators describing the acceleration of the electron allows for a partial localization of the electron in its orbit. The correlation function is found by approximating the transition matrix elements by their values in the classical limit. It is shown that the second-order correlation, depending on two times, is a function of the time difference and is a periodic function of this argument with the period equal to the period of the corresponding classical motion. The function has sharp maxima corresponding to large electron acceleration in the vicinity of the "perihelion". This allows the localization of the electron in its consecutive approach to the perihelion point.

Implications of GMPE’s structure for multi-site seismic hazard

Previous research has discussed the implications of the structure of a classical ground motion prediction equation (GMPE) on single-site probabilistic seismic hazard analysis and disaggregation. Classical refers to a GMPE where local site conditions (or any other factor) are accounted for via constant (with respect to magnitude and source-to-site distance) terms added to the mean and that do not affect the distribution of the residuals. Herein, the implications of such a structure of the GMPE are briefly discussed with respect to multi-site hazard assessment that, typically, requires a large number of simulations of random fields of ground motion intensity measures. It is shown that this type of GMPEs enables to run the simulations only once, independently of the soil conditions (or any other factor modeled in a similar way) eventually assigned to each site, which can represent a significant computational advantage in the case of spatially distributed assets.

Asymptotic expansions for field moments of bound states

Asymptotic expansions are presented for the moments of bound states in one-dimensional anharmonic potentials. The results are derived by using the stable aggregate of flexible element (SAFE) method and include only the first non-zero wave-related correction to the familiar semi-classical approximation. Application to a couple of widely studied potentials that do not permit closed-form solutions is used to demonstrate surprising accuracy even in cases that are far from any asymptotic limit. We explore the absence of alternate terms in the asymptotic expansions as a way to explain the accuracy of the end results. Those results are expressed as definite integrals with integrands involving the parameter used in the SAFE method to control the extent of the associated elemental field contributions. Importantly, the integrals themselves are shown to be precisely independent of that parameter. Further, although the derivation proceeds by way of an asymptotic expansion for the wavefield that involves the associated classical motion, those entities do not appear in the end results which are expressed in terms of just the potential function and its first four derivatives.

Nuclear Motion Is Classical: Spectra of Hydrogen Chloride and Ammonia

The concept of classical nuclear motion is extremely successful in describing motion at the atomic scale. In describing chemical reactions, it is even far more convincing than the picture obtained by using the Schrödinger equation for time development. However, this theory must be subject to critical tests. In particular, it must be checked if vibrational and rotational spectra are obtained correctly. Particularly critical are the spectra of small molecules containing the light hydrogen atom, since they have a distinctive rotational structure. The present study presents computations of the spectra of ammonia and hydrogen chloride using ab initio molecular dynamics, that is, by describing nuclear motion classically.

Detecting moving targets in active sonar echograph of harbor environment using robust high-order flux tensor.

An important goal of an active sonar system is to detect and track underwater intruders such as frogmen, unmanned underwater vehicles, etc. Unfortunately, the intruders appear visually as a small fluctuating "blob" against the high-level fluctuating background caused by multipath propagation and reverberation in the harbor environment, making it difficult to be distinguished. Classical motion features well developed in computer vision cannot cope with an underwater environment. Thus, this paper presents a robust high-order flux tensor (RHO-FT) to characterize the small underwater moving targets against high-level fluctuating background. According to the dynamic behavior of active clutter from real-world harbor environment, we first classify it into two main types: (1) dynamic clutter but with relatively consistent spatial-temporal variation in a certain neighborhood; (2) sparkle clutter presenting completely random flashing. Then starting from the classical flux tensor, we develop a statistical high-order computation to handle the former followed by a spatial-temporal connected component to suppress the latter to achieve higher robustness. Experiments on a set of real-world harbor datasets demonstrate the effectiveness of our RHO-FT.

Interperformer coordination in piano-singing duo performances: phrase structure and empathy impact.

Musicians' body motion plays a fundamental role in ensemble playing, by supporting sound production, communication, and expressivity. This research investigates how Western classical musicians' head motion during ensemble performances relates to a piece's phrase structure and musicians' empathic perspective taking (EPT) profile. Twenty-four advanced piano and singing students took part in the study, and their EPT score was pre-assessed using the Interpersonal Reactivity Index. High and low EPT duos were formed, and musicians were paired with a co-performer from the same and the other EPT group. Musicians rehearsed Fauré's Automne and Schumann's Die Kartenlegerin, and performed the pieces one time before and three times after rehearsal. Motion capture data of the musicians' front head, audio, and MIDI recordings of the performances were collected and analysed. Similarity in musicians' head motion and tendency to lead/lag their co-performer were computed by extracting, respectively, power and phase difference of the cross-wavelet transforms of the velocity curves of each paired marker. Results demonstrate that the power of interperformer coordination corresponds to the piece's phrase levels and that singer's EPT can impact the leader-follower relationships between musicians, depending on piece and take number. In the Fauré piece, the higher the singer's EPT score, the higher the tendency for the singer to lead and pianist to follow in take 3, and the lower the tendency for the singer to lead and pianist to follow in take 2. These results contribute to a further understanding of the mechanisms underpinning social interactions, by revealing the complexity of the association between empathy and body motion in ensembles in promoting and diffusing leadership between musicians.

A review of mobile robot motion planning methods: from classical motion planning workflows to reinforcement learning-based architectures

A review of mobile robot motion planning methods: from classical motion planning workflows to reinforcement learning-based architectures

Generation of quasi-chirp-free isolated attosecond pulses from atoms under the action of orthogonal two-color combined pulse of fundamental frequency and higher intensity second harmonic fields

The intrinsic chirp of high-order harmonic generation is an important factor limiting the production of ultrashort attosecond pulses. Based on numerically solving the time-dependent Schrödinger equation, the generation process of high-order harmonic from the He atom under the action of orthogonal two-color combined pulse of fundamental frequency and higher intensity second harmonic fields is studied. In this paper, we propose to achieve quasi-chirp-free isolated attosecond pulses by superimposing a higher second-harmonic field on the orthogonal direction of the fundamental frequency field. It is found that the high-energy part of its harmonic emission exhibits small chirp characteristics, which can be used to synthesize isolated attosecond pulses. Through the analysis of the wave packets evolution and the classical motion trajectories of the electron, it is demonstrated that the quasi-chirp-free harmonic can be attributed to the simultaneous return of electrons ionized at different times to the parent particle. The influence of the relative phase of the two pulses on the harmonics is further analyzed, and it is observed that this phenomenon is sensitive to the relative phase, but it can still generate isolated attosecond pulses within a certain phase.

Lasting effects of static magnetic field on classical Brownian motion.

The Bohr-Van Leeuwen theorem states that an external static magnetic field does not influence the state of a classical equilibrium system: There is no equilibrium classical magnetism, since the magnetic field does not do work. We revisit this famous no-go result and consider a classical charged Brownian particle interacting with an equilibrium bath. We confirm that the Bohr-Van Leeuwen theorem holds for the long-time (equilibrium) state of the particle. But the external static, homogeneous magnetic field does influence the long-time state of the thermal bath, which is described via the Caldeira-Leggett model. In particular, the magnetic field induces an average angular momentum for the (uncharged) bath, which separates into two sets rotating in opposite directions. The effect relates to the bath going slightly out of equilibrium under the influence of the Brownian particle and persists for arbitrarily long times. In this context we studied the behavior of the two other additive integrals of motion, energy, and linear momentum. The situation with linear momentum is different, because it is dissipated away by (and from) the bath modes. The average energy of the bath mode retains the magnetic field as a small correction. Thus, only the bath angular momentum really feels the magnetic field for long times.

Chaotic string motion in a near pp-wave limit

We revisit classical string motion in a near pp-wave limit of AdS5 × S5. It is known that the Toda lattice models are integrable. But if the exponential potential is truncated at finite order, then the system may become non-integrable. In particular, when the exponential potential in a three-particle periodic Toda chain is truncated at the third order of the dynamical variables, the resulting system becomes a well-known non-integrable system, Henon-Heiles model. The same thing may happen in a near pp-wave limit of AdS5 × S5, on which the classical string motion becomes chaotic.