Classical Motion Research Articles

Impact of induced electric field on the motion of a charged particle in a uniform time-dependent magnetic field that is linearly increasing with time

The classical and quantum motion of a charged particle in a uniform constant magnetic field is widely studied in the scientific literature since it is a problem that allows an exact mathematical analytical solution. On the other hand the cases of a non-uniform, time-dependent or both non-uniform and time-dependent magnetic field lead to mathematical problems that do not have general exact analytical solutions except for a few very specific cases. We present in this work, the results for the two-dimensional motion of a charged particle in a uniform time-dependent magnetic field that is linearly increasing with time for a certain finite interval and then stabilizes to a constant value afterwards. The work draws attention to certain interesting features observed for the two-dimensional motion of the charged particle as a result of the interplay between the effects of an induced electric field and the time varying magnetic field.

Is Reinforcement Learning Good at American Option Valuation?

This paper investigates algorithms for identifying the optimal policy for pricing American Options. The American Option pricing is reformulated as a Sequential Decision-Making problem with two binary actions (Exercise or Continue), transforming it into an optimal stopping time problem. Both the least square Monte Carlo simulation method (LSM) and Reinforcement Learning (RL)-based methods were utilized to find the optimal policy and, hence, the fair value of the American Put Option. Both Classical Geometric Brownian Motion (GBM) and calibrated Stochastic Volatility models served as the underlying uncertain assets. The novelty of this work lies in two aspects: (1) Applying LSM- and RL-based methods to determine option prices, with a specific focus on analyzing the dynamics of “Decisions” made by each method and comparing final decisions chosen by the LSM and RL methods. (2) Assess how the RL method updates “Decisions” at each batch, revealing the evolution of the decisions during the learning process to achieve optimal policy.

Transient motion classification and segment analysis of diffusive trajectories of G proteins and coupled-receptors in a living cell

The molecular movement in single particle tracking (SPT) experiments shows a crucial role of diffusion in many biological processes such as signaling, cellular organization, transport mechanisms, and more. The SPT analysis detects not only classical Brownian motion but diffusion with other features. These include directed diffusion and confined motion. The behavior remains a challenging problem for several reasons. Due to the action of many physical processes, random trajectories of cellular molecules are segmented in different diffusive modes. Often their study requires sophisticated algorithms for the analysis of statistical properties. In this paper we consider the segment analysis for trajectories of G proteins and coupled-receptors in living cells. Their movement is often transient and switches among free diffusion, confined diffusion, directed diffusion, and immobility. Moreover, the confined segments can have both Gaussian and non-Gaussian statistics. The types of alternation of diffusive modes along the trajectories of G proteins and coupled-receptors are analyzed.

Independent-oscillator model and the quantum Langevin equation for an oscillator: a review

This review provides a brief and quick introduction to the quantum Langevin equation for an oscillator, while focusing on the steady-state thermodynamic aspects. A derivation of the quantum Langevin equation is carefully outlined based on the microscopic model of the heat bath as a collection of a large number of independent quantum oscillators, the so-called independent-oscillator model. This is followed by a discussion on the relevant ‘weak-coupling’ limit. In the steady state, we analyze the quantum counterpart of energy equipartition theorem which has generated a considerable amount of interest in recent literature. The free energy, entropy, specific heat, and third law of thermodynamics are discussed for one-dimensional quantum Brownian motion in a harmonic well. Following this, we explore some aspects of dissipative diamagnetism in the context of quantum Brownian oscillators, emphasizing upon the role of confining potentials and also upon the environment-induced classical-quantum crossover. We discuss situations where the system-bath coupling is via the momentum variables by focusing on a gauge-invariant model of momentum-momentum coupling in the presence of a vector potential; for this problem, we derive the quantum Langevin equation and discuss quantum thermodynamic functions. Finally, the topic of fluctuation theorems is discussed (albeit, briefly) in the context of classical and quantum cyclotron motion of a particle coupled to a heat bath.

Expanding wave packets and the quantum to classical transition

The standard description of the transition from quantum to classical mechanics presented in most text books is the proof that the quantum expectation values of position and momentum obey equations of Newtonian form. This is the Ehrenfest theorem. It is combined with the requirement that wave packets remain localised to describe a single particle moving according to classical mechanics. Hence, the natural spreading of wave packets is viewed as a quantum effect. In contradiction to this view, here it is argued that the spreading, where different momentum components separate, is the signature of the quantum to classical transition. The asymptotic spatial wave function becomes proportional to the initial momentum space wave function, which mirrors exactly the well-known far-field diffraction pattern in optics. Trajectories, defined as the locus of the normals to the expanding wave front, are used to illustrate the transition from quantum to classical motion. Again this is the analogue of the wave to beam transition in optics. It is suggested that this analysis of the quantum to classical transition should be incorporated routinely into introductory quantum mechanics courses.

Perturbative approach to time-dependent quantum solitons

Recently we have introduced a lightweight, perturbative approach to quantum solitons. Thus far, our approach has been largely limited to configurations consisting of a single soliton plus a finite number of mesons, whose classical limit is an isolated stationary or rigidly moving soliton. In this paper, with an eye to soliton collisions and oscillons, we generalize this approach to quantum states whose classical limits are genuinely time-dependent. More precisely, we use a unitary operator, inspired by the coherent state approach to solitons, to factor out the nonperturbative part of the state, which includes the classical motion. The solution for the quantum state and its evolution is then reduced to an entirely perturbative problem.

The role of Allee effects for Gaussian and Lévy dispersals in an environmental niche.

In the study of biological populations, the Allee effect detects a critical density below which the population is severely endangered and at risk of extinction. This effect supersedes the classical logistic model, in which low densities are favorable due to lack of competition, and includes situations related to deficit of genetic pools, inbreeding depression, mate limitations, unavailability of collaborative strategies due to lack of conspecifics, etc. The goal of this paper is to provide a detailed mathematical analysis of the Allee effect. After recalling the ordinary differential equation related to the Allee effect, we will consider the situation of a diffusive population. The dispersal of this population is quite general and can include the classical Brownian motion, as well as a Lévy flight pattern, and also a "mixed" situation in which some individuals perform classical random walks and others adopt Lévy flights (which is also a case observed in nature). We study the existence and nonexistence of stationary solutions, which are an indication of the survival chance of a population at the equilibrium. We also analyze the associated evolution problem, in view of monotonicity in time of the total population, energy consideration, and long-time asymptotics. Furthermore, we also consider the case of an "inverse" Allee effect, in which low density populations may access additional benefits.

On the Passage of a Relativistic Spinless Particle through a Potential Barrier

Contradictory descriptions of the passage of a relativistic particle through a high potential barrier are investigated. The conventional energy-momentum relation yields a real momentum for the particle within the barrier, suggesting classical particle motion. However, this contradicts the law of conservation of energy, which necessitates that the particle's momentum be imaginary, thereby rendering classical motion impossible. This paradox is resolved through the adoption of a novel energy-momentum relation.

Emergence of inertia in the low-Reynolds regime of self-diffusiophoretic motion.

For isotropic swimming particles driven by self-diffusiophoresis at zero Reynolds number (where particle velocity responds instantaneously to applied force), the diffusive timescale of emitted solute can produce an emergent quasi-inertial behavior. These particles can orbit in a central potential and reorient under second-order dynamics, not the first-order dynamics of classical zero-Reynolds motion. They are described by a simple effective model that embeds their history-dependent behavior as an effective inertia, this being the most primitive expression of memory. The system can be parameterized with dynamic quantities such as particle size and swimming speed, without detailed knowledge of the diffusiophoretic mechanism.

Dynamical chaos and level splitting under the channeling of the high energy positrons in [100] direction of the silicon crystal

The motion of charged particles in a crystal in the axial channeling regime can be both regular and chaotic. The chaos in quantum case manifests itself in the statistical properties of the energy levels set. These properties have been studied previously for the electrons channeling along [110] direction of the silicon crystal, in the case when the classical motion was completely chaotic, as well as for the ones channeling along [100] direction, when the classical motion can be both regular and chaotic for the same energy depending on the initial conditions. Here we study the positrons channeling in [100] direction. This case is of special interest due to the substantial tunneling probability between dynamically isolated regular motion domains in the phase space. The interaction of the energy levels via tunneling distinctly changes the level spacing statistics. All transverse motion energy levels as well as corresponding stationary wave functions are computed numerically for the 30 GeV positrons channeling in [100] direction of the silicon crystal. The values of the matrix elements for the tunnel transitions areextractedfrom these data. These results confirm the chaos assistance for the tunneling and the level splitting. These values will be used in the further researches of the quantum chaos manifestations in the channeling phenomenon.

Quantum control of classical motion: piston dynamics in a Rabi-coupled Bose–Einstein condensate

We develop a model and explore the dynamics of a hybrid classical-quantum system consisting of a classical piston and a self-interacting pseudospin 1/2 Bose–Einstein condensate with a time-dependent Rabi coupling. We investigate the mechanical work produced by the piston moving as a result of the quantum pressure of the condensate. The time-dependent Rabi field redistributes the condensate density between the spin components and, as a result, causes a time-dependent pressure acting on the piston. Correspondingly, the motion of the piston produces quantum evolution of the condensate mass- and spin density profiles. We show how by optimised design of the time-dependent direction of the Rabi field based on a quasi-stationary quantum pressure approximation, one can control both the position and velocity of the piston.

Constrained Motion Planning and Multi-Agent Path Finding on directed graphs

We discuss C-MP and C-MAPF, generalizations of the classical Motion Planning (MP) and Multi-Agent Path Finding (MAPF) problems on a directed graph G. Namely, we enforce an upper bound on the number of agents that occupy each member of a family of vertex subsets. For instance, this constraint allows maintaining a safety distance between agents. We prove that finding a feasible solution of C-MP and C-MAPF is NP-hard. Also, we propose a method to convert these problems to standard MP and MAPF by strengthening the constraints. The method consists in finding a subset of vertices W and a reduced graph GW, such that a feasible solution of MP and MAPF on GW provides, in polynomial time, a feasible solution of C-MP and C-MAPF on G. However, since the conversion into standard MP and MAPF is obtained by strengthening constraints, feasible solutions of C-MP and C-MAPF on G may exist even if MP and MAPF on GW do not admit any feasible solution. We also study the problem of finding W of maximum cardinality. First, we show that such problem is strongly NP-hard. Then, we propose a heuristic approach for its solution.

Quantum information scrambling and chemical reactions

The ultimate regularity of quantum mechanics creates a tension with the assumption of classical chaos used in many of our pictures of chemical reaction dynamics. Out-of-time-order correlators (OTOCs) provide a quantum analog to the Lyapunov exponents that characterize classical chaotic motion. Maldacena, Shenker, and Stanford have suggested a fundamental quantum bound for the rate of information scrambling, which resembles a limit suggested by Herzfeld for chemical reaction rates. Here, we use OTOCs to study model reactions based on a double-well reaction coordinate coupled to anharmonic oscillators or to a continuum oscillator bath. Upon cooling, as one enters the tunneling regime where the reaction rate does not strongly depend on temperature, the quantum Lyapunov exponent can approach the scrambling bound and the effective reaction rate obtained from a population correlation function can approach the Herzfeld limit on reaction rates: Tunneling increases scrambling by expanding the state space available to the system. The coupling of a dissipative continuum bath to the reaction coordinate reduces the scrambling rate obtained from the early-time OTOC, thus making the scrambling bound harder to reach, in the same way that friction is known to lower the temperature at which thermally activated barrier crossing goes over to the low-temperature activationless tunneling regime. Thus, chemical reactions entering the tunneling regime can be information scramblers as powerful as the black holes to which the quantum Lyapunov exponent bound has usually been applied.

New Classical Relativistic Theory of a Charged Particle in an Electric Field

A new relativistic theory of the classical motion of a charged particle in an electric field has been developed. The resulting equations characterize the kinematic and dynamic features of particle motion, demonstrating peculiar behavior in areas with high attractive potentials. This changes the existing paradigm for the interaction of charge with an electric field, entailing profound consequences. The new theory converges with the conventional theory of electricity under conditions of low potentials and nonrelativistic particle velocities. The possibility of experimental verification of the new theory is discussed.

Simulating two-level quantum systems via classical orbit motion: Adiabatic cyclic evolution and Berry phase

Simulating two-level quantum systems via classical orbit motion: Adiabatic cyclic evolution and Berry phase

Total angular momentum conservation in Ehrenfest dynamics with a truncated basis of adiabatic states.

We show that standard Ehrenfest dynamics does not conserve linear and angular momentum when using a basis of truncated adiabatic states. However, we also show that previously proposed effective Ehrenfest equations of motion [M. Amano and K. Takatsuka, "Quantum fluctuation of electronic wave-packet dynamics coupled with classical nuclear motions," J. Chem. Phys. 122, 084113 (2005) and V. Krishna, "Path integral formulation for quantum nonadiabatic dynamics and the mixed quantum classical limit," J. Chem. Phys. 126, 134107 (2007)] involving the non-Abelian Berry force do maintain momentum conservation. As a numerical example, we investigate the Kramers doublet of the methoxy radical using generalized Hartree-Fock with spin-orbit coupling and confirm that angular momentum is conserved with the proper equations of motion. Our work makes clear some of the limitations of the Born-Oppenheimer approximation when using abinitio electronic structure theory to treat systems with unpaired electronic spin degrees of freedom, and we demonstrate that Ehrenfest dynamics can offer much improved, qualitatively correct results.

Towards the generation of mechanical Kerr-cats: awakening the perturbative quantum Moyal corrections to classical motion

The quantum to classical transition is determined by the interplay of a trio of parameters: dissipation, nonlinearity, and macroscopicity. Why is nonlinearity needed to see quantum effects? And, is not an ordinary pendulum quite nonlinear already? In this manuscript, we discuss the parameter regime where the dynamics of a massive oscillator should be quantum mechanical in the presence of dissipation. We review the outstanding challenge of the dynamical generation of highly quantum mechanical cat states of a massive ‘pendulum’, known as Kerr-cats. We argue that state-of-the-art cold atom experiments may be in a position to reach such a nonlinear regime, which today singles out superconducting quantum circuits. A way to stabilize Schrödinger cat superpositions of a mechanical atomic oscillator via parametric squeezing and further protected by an unusual form of quantum interference is discussed. The encoding of a neutral atom Kerr-cat qubit is proposed.

The dynamic organicity theory of consciousness: how consciousness arises from the functionality of multiscale complexity in the material brain

The fine structure of consciousness is temporally experienced. This makes possible a dynamic organicity theory of consciousness through disunified order in the pre-, sub- and noncognitive levels of causal processes and dynamics. This multilevel approach is based on functional systems where space is implicitly grounded as changeable boundary conditions (due to organicity), and energy capture and storage under energy flow entail structuring intrinsic information (as both hidden thermodynamic energy and hidden thermodynamic information). The use of anaesthesia indicates consciousness disappears when there is an obliteration of intrinsic information within membrane protein amino acids, suggesting that quantum-thermal fluctuations of the electromagnetic field can be functional and non-vacuous. The origin of thermo-qubit syn-tax is derived from the principle of self-reference as the syntax of consciousness. In addition to the classical Brownian motion of macroscopically observable mean values, the negentropically derived quantum potential is an additional degree of freedom where the structuring of intrinsic information takes place by negentropic action from the microscopically random quantum-thermal fluctuations. Such functional fluctuations provide a way for intentionality to come in from spontaneous ordering, leading to path selection as an instruction to act. The resultant intentionality becomes part of information-based action when functional interactions are selected and boundary conditions change, causing interference pattern matching through temporal re-organization of informational redundancy structures (not used in functional interactions). The self-referential dynamic structures (of evolving informational holons) transform syntac-tic structures into experienceable forms. In comparing the interference patterns of functional interactions during restructuring informational redundancy structures, the functionality of multiscale complexity as functional systems patterns gives experienceable forms the potential to understand “meaning” by reducing uncertainty in the process of structuring intrinsic information. The self-reference principle establishes dynamical pathways from the microscale to the macroscale (this includes nonlocal pathways), in which diachronic causation and how the disunity of causal order in the redundancy creates a weak unity of consciousness through its temporal structure, which has an inferred purpose that gives rise to a sense of self.

Boundary driven turbulence on string worldsheet

We study the origin of turbulence on the string worldsheet with boundaries laid in anti de Sitter (AdS) spacetime. While the classical motion of a single closed string in AdS is integrable, it has recently been recognized that weak turbulence arises in the case of an open string suspended from the AdS boundary. In the open string case, it is necessary to impose boundary conditions on the worldsheet boundaries. We classify which boundary conditions preserve integrability. Based on this classification, we anticipate that turbulence may occur on the string worldsheet if integrability is not guaranteed by the boundary conditions. Numerical investigations of the classical open-string dynamics support that turbulence occurs when the boundary conditions are not integrable.

IR-finite thermal acceleration radiation

A charge accelerating in a straight line following the Schwarzschild–Planck moving mirror motion emits thermal radiation for a finite period. Such a mirror motion demonstrates quantum purity and serves as a direct analogy of a black hole with unitary evolution and complete evaporation. Extending the analog to classical electron motion, we derive the emission spectrum, power radiated, and finite total energy and particle count, with particular attention to the thermal radiation limit. This potentially opens the possibility of a laboratory analog of black hole evaporation.