Time-dependent External Force Research Articles

Global and Trajectory Attractors for a Nonlocal Cahn–Hilliard–Navier–Stokes System

The Cahn–Hilliard–Navier–Stokes system is based on a well-known diffuse interface model and describes the evolution of an incompressible isothermal mixture of binary fluids. A nonlocal variant consists of the Navier–Stokes equations suitably coupled with a nonlocal Cahn–Hilliard equation. The authors, jointly with P. Colli, have already proven the existence of a global weak solution to a nonlocal Cahn–Hilliard–Navier–Stokes system subject to no-slip and no-flux boundary conditions. Uniqueness is still an open issue even in dimension two. However, in this case, the energy identity holds. This property is exploited here to define, following J.M. Ball’s approach, a generalized semiflow which has a global attractor. Through a similar argument, we can also show the existence of a (connected) global attractor for the convective nonlocal Cahn–Hilliard equation with a given velocity field, even in dimension three. Finally, we demonstrate that any weak solution fulfilling the energy inequality also satisfies a dissipative estimate. This allows us to establish the existence of the trajectory attractor also in dimension three with a time dependent external force.

Energetics of Ortho-7 (oxime drug) translocation through the active-site gorge of tabun conjugated acetylcholinesterase.

Oxime drugs translocate through the 20 Å active-site gorge of acetylcholinesterase in order to liberate the enzyme from organophosphorus compounds’ (such as tabun) conjugation. Here we report bidirectional steered molecular dynamics simulations of oxime drug (Ortho-7) translocation through the gorge of tabun intoxicated enzyme, in which time dependent external forces accelerate the translocation event. The simulations reveal the participation of drug-enzyme hydrogen bonding, hydrophobic interactions and water bridges between them. Employing nonequilibrium theorems that recovers the free energy from irreversible work done, we reconstruct potential of mean force along the translocation pathway such that the desired quantity represents an unperturbed system. The potential locates the binding sites and barriers for the drug to translocate inside the gorge. Configurational entropic contribution of the protein-drug binding entity and the role of solvent translational mobility in the binding energetics is further assessed.

Full-counting statistics of heat transport in harmonic junctions: Transient, steady states, and fluctuation theorems

We study the statistics of heat transferred in a given time interval t_{M}, through a finite harmonic chain, called the center, which is connected to two heat baths, the left (L) and the right (R), that are maintained at two temperatures. The center atoms are driven by external time-dependent forces. We calculate the cumulant generating function (CGF) for the heat transferred out of the left lead, Q_{L}, based on the two-time quantum measurement concept and using the nonequilibrium Green's function method. The CGF can be concisely expressed in terms of Green's functions of the center and an argument-shifted self-energy of the lead. The expression of the CGF is valid in both transient and steady-state regimes. We consider three initial conditions for the density operator and show numerically, for a one-atom junction, how their transient behaviors differ from each other but, finally, approach the same steady state, independent of the initial distributions. We also derive the CGF for the joint probability distribution P(Q_{L},Q_{R}), and discuss the correlations between Q_{L} and Q_{R}. We calculate the CGF for total entropy production in the reservoirs. In the steady state we explicitly show that the CGFs obey steady-state fluctuation theorems. We obtain classical results by taking ℏ→0. We also apply our method to the counting of the electron number and electron energy, for which the associated self-energy is obtained from the usual lead self-energy by multiplying a phase and shifting the contour time, respectively.

Quantum behaviors of the particles affected by the linear-damping and time-dependent external force

In this paper, we handle the Hamiltonian of the particles that are affected by the linear-damping and time-dependent external force given by two different canonical transformations by using the theory of generalized quantum linear transformation. We give the rigorous solution of evolution operator, and the expectation values of coordinate and momentum of the particles quantum fluctuations. Results show that 1) the two regular translations are equivalent; 2) linear damping has a squeezing effect on the momentum of particle, and the deviation of the momentum attenuates with time t according to the rule of negative exponent, and the bigger the damping coefficient, the faster the attenuation is; 3) the expectation values of coordinate and momentum of the particle are equal to their classical values correspondingly.

Random Long-Range Interaction Induced Synchronization in Coupled Networks of Inertial Ratchets

We investigate how the random long-range interactions affect the synchronization features in networks of inertial ratchets, where each ratchet is driven by a periodic time-dependent external force, under the influence of an asymmetric periodic potential. It is found that for a given coupling strength C, the synchronization of the coupled ratchets is induced as the fraction of random long-range interactions p increases and the ratchet networks reach full synchronization for a larger p. It is also found that the system reaches synchronization more effectively for a stronger coupling strength.

Quantization and instability of the damped harmonic oscillator subject to a time-dependent force

We consider the one-dimensional motion of a particle immersed in a potential field U ( x ) under the influence of a frictional (dissipative) force linear in velocity ( − γ x ̇ ) and a time-dependent external force ( K ( t ) ). The dissipative system subject to these forces is discussed by introducing the extended Bateman’s system, which is described by the Lagrangian: ℒ = m x ̇ y ̇ − U ( x + 1 2 y ) + U ( x − 1 2 y ) + γ 2 ( x y ̇ − y x ̇ ) − x K ( t ) + y K ( t ) , which leads to the familiar classical equations of motion for the dissipative (open) system. The equation for a variable y is the time-reversed of the x motion. We discuss the extended Bateman dual Lagrangian and Hamiltonian by setting U ( x ± y / 2 ) = 1 2 k ( x ± y / 2 ) 2 specifically for a dual extended damped–amplified harmonic oscillator subject to the time-dependent external force. We show the method of quantizing such dissipative systems, namely the canonical quantization of the extended Bateman’s Hamiltonian ℋ . The Heisenberg equations of motion utilizing the quantized Hamiltonian ℋ ̂ surely lead to the equations of motion for the dissipative dynamical quantum systems, which are the quantum analog of the corresponding classical systems. To discuss the stability of the quantum dissipative system due to the influence of an external force K ( t ) and the dissipative force, we derived a formula for transition amplitudes of the dissipative system with the help of the perturbation analysis. The formula is specifically applied for a damped–amplified harmonic oscillator subject to the impulsive force. This formula is used to study the influence of dissipation such as the instability due to the dissipative force and/or the applied impulsive force.

Finite-momentum Bose-Einstein condensates in shaken two-dimensional square optical lattices

We consider ultracold bosons in a two-dimensional square optical lattice described by the Bose-Hubbard model. In addition, an external time-dependent sinusoidal force is applied to the system, which shakes the lattice along one of the diagonals. The effect of the shaking is to renormalize the nearest-neighbor-hopping coefficients, which can be arbitrarily reduced, can vanish, or can even change sign, depending on the shaking parameter. Therefore, it is necessary to account for higher-order-hopping terms, which are renormalized differently by the shaking, and to introduce anisotropy into the problem. We show that the competition between these different hopping terms leads to finite-momentum condensates with a momentum that may be tuned via the strength of the shaking. We calculate the boundaries between the Mott insulator and the different superfluid phases and present the time-of-flight images expected to be observed experimentally. Our results open up possibilities for the realization of bosonic analogs of the Fulde, Ferrel, Larkin, and Ovchinnikov phase describing inhomogeneous superconductivity.

Pullback exponential attractors for nonautonomous equations Part II: Applications to reaction–diffusion systems

The existence of a pullback exponential attractor being a family of compact and positively invariant sets with a uniform bound on their fractal dimension which at a uniform exponential rate pullback attract bounded subsets of the phase space under the evolution process is proved for the nonautonomous logistic equation and a system of reaction–diffusion equations with time-dependent external forces including the case of the FitzHugh–Nagumo system.

Single-file diffusion of externally driven particles

We study one-dimensional diffusion of N hard-core interacting Brownian particles driven by the space- and time-dependent external force. We give the exact solution of the N-particle Smoluchowski diffusion equation. In particular, we investigate the nonequilibrium energetics of two interacting particles under the time-periodic driving. The hard-core interaction induces entropic repulsion which differentiates the energetics of the two particles. We present exact time-asymptotic results which describe the mean energy, the accepted work and heat, and the entropy production of interacting particles, and we contrast these quantities against the corresponding ones for the noninteracting particles.

Multi-soliton solutions of the forced variable-coefficient extended Korteweg–de Vries equation arisen in fluid dynamics of internal solitary waves

Under investigation in this paper, with symbolic computation, is a forced variable-coefficient extended Korteweg–de Vries equation, which can describe the weakly-nonlinear long internal solitary waves (ISWs) in the fluid with the continuous stratification on density. By virtue of the Hirota bilinear method, multi-soliton solutions for such an equation with the external force term have been derived. Furthermore, effects are discussed with the aid of the characteristic line: (I) Inhomogeneities of media and nonuniformities of boundaries, depicted by the variable coefficients, play a role in the soliton behavior; (II) Solitons change their initial propagation direction on the compact shock or anti-shock wave background in the presence of the external time-dependent force, and the results present an extended view compared with that for the linear theory; (III) Combined effects of the inhomogeneities and external force are regarded as the nonlinear composition of the independent influence induced by the two factors. Those results could be expected to be helpful for the investigation on the dynamics of the ISWs in an ocean or atmosphere stratified fluid.

The one-dimensional damped forced harmonic oscillator revisited

In this paper we give a general solution to the problem of the damped harmonic oscillator under the influence of an arbitrary time-dependent external force. We employ simple methods accessible for beginners and useful for undergraduate students and professors in an introductory course of mechanics.

Quantum walks and quantum simulations with Bloch-oscillating spinor atoms

We propose a scheme for the realization of a quantum walker and a quantum simulator for the Dirac equation with ultracold spinor atoms in driven optical lattices. A precise control of the dynamics of the atomic matter wave can be realized using time-dependent external forces. If the force depends on the spin state of the atoms, the dynamics will entangle the inner and outer degrees of freedom which offers unique opportunities for quantum information and quantum simulation. Here, we introduce a method to realize a quantum walker based on the state-dependent transport of spinor atoms and a coherent driving of the internal state. In the limit of weak driving the dynamics is equivalent to that of a Dirac particle in 1+1 dimensions. Thus it becomes possible to simulate relativistic effects such as Zitterbewegung and Klein tunneling.

Trajectory attractors for binary fluid mixtures in 3D

Two different models for the evolution of incompressible binary fluid mixtures in a three-dimensional bounded domain are considered. They consist of the 3D incompressible Navier-Stokes equations, subject to time-dependent external forces and coupled with either a convective Allen-Cahn or Cahn-Hilliard equation. Such systems can be viewed as generalizations of the Navier-Stokes equations to two-phase fluids. Using the trajectory approach, the authors prove the existence of the trajectory attractor for both systems.

A quantum model of option pricing: When Black–Scholes meets Schrödinger and its semi-classical limit

The Black–Scholes equation can be interpreted from the point of view of quantum mechanics, as the imaginary time Schrödinger equation of a free particle. When deviations of this state of equilibrium are considered, as a product of some market imperfection, such as: Transaction cost, asymmetric information issues, short-term volatility, extreme discontinuities, or serial correlations; the classical non-arbitrage assumption of the Black–Scholes model is violated, implying a non-risk-free portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the Black–Scholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new Black–Scholes–Schrödinger model based on the endogenous arbitrage option pricing formulation introduced by Contreras et al. (2010) [2]. Hence, we derive a more general quantum model of option pricing, that incorporates arbitrage as an external time dependent force, which has an associated potential related to the random dynamic of the underlying asset price. This new resultant model can be interpreted as a Schrödinger equation in imaginary time for a particle of mass 1 / σ 2 with a wave function in an external field force generated by the arbitrage potential. As pointed out above, this new model can be seen as a more general formulation, where the perfect market equilibrium state postulated by the Black–Scholes model represent a particular case. Finally, since the Schrödinger equation is in place, we can apply semiclassical methods, of common use in theoretical physics, to find an approximate analytical solution of the Black–Scholes equation in the presence of market imperfections, as it is the case of an arbitrage bubble. Here, as a numerical illustration of the potential of this Schrödinger equation analogy, the semiclassical approximation is performed for different arbitrage bubble forms (step, linear and parabolic) and compare with the exact solution of our general quantum model of option pricing.

The solutions of the generalized classical and quantum harmonic oscillators with time dependent mass, frequency, two-photon parameter and external force: The squeezing effects

The exact analytical solutions of a generalized classical harmonic oscillator with time dependent mass, frequency, two-photon parameter and external forces are obtained. By using the invariance property of the scaled Wronskian, these solutions are used to obtain the solutions of the quantum mechanical counterpart of the oscillator under Heisenberg picture. In order to discuss the applications of these solutions of the quantum mechanical oscillator, we calculate the exact analytical expressions for the second order variances of both the canonically conjugate quadratures in terms of the time dependent mass, frequency and two-photon parameter. However, these variances calculated in terms of the initial coherent state do not depend on the time dependent driven terms. We argued that the time dependent frequency is on the way of the exact analytical solutions and hence it is kept constant throughout the investigation. We, however, discuss few situations of physical interest where the mass is varying in time. The special circumstance where all the parameters are time independent is used to discuss the squeezing effect in both the quadratures. It is seen that the parameter g involving the two-photon interaction term produces squeezing effects. With the increase of interaction time, the squeezing in both the quadrature components exhibit collapse and revival phenomena for g < ω (frequency of the oscillator). The squeezing of X-quadrature is completely ruled out for g > ω. However, the squeezing for P-quadrature is possible for small interaction time. The squeezing patterns of the X and P-quadratures are also discussed for pulsating mass, and for mass increasing with time. The squeezing is also discussed when the mass is increasing exponentially in time. It is envisage that the solutions could be used in the investigation of quantum statistical properties of the radiation field coupled to the said oscillator.

Three faces of the second law. I. Master equation formulation

We propose a formulation of stochastic thermodynamics for systems subjected to nonequilibrium constraints (i.e. broken detailed balance at steady state) and furthermore driven by external time-dependent forces. A splitting of the second law occurs in this description leading to three second-law-like relations. The general results are illustrated on specific solvable models. The present paper uses a master equation based approach.

P-Splines Using Derivative Information

Time series associated with single-molecule experiments and/or simulations contain a wealth of multiscale information about complex biomolecular systems. We demonstrate how a collection of Penalized-splines (P-splines) can be useful in quantitatively summarizing such data. In this work, functions estimated using P-splines are associated with stochastic differential equations (SDEs). It is shown how quantities estimated in a single SDE summarize fast-scale phenomena, whereas variation between curves associated with different SDEs partially reflects noise induced by motion evolving on a slower time scale. P-splines assist in "semiparametrically" estimating nonlinear SDEs in situations where a time-dependent external force is applied to a single-molecule system. The P-splines introduced simultaneously use function and derivative scatterplot information to refine curve estimates. We refer to the approach as the PuDI (P-splines using Derivative Information) method. It is shown how generalized least squares ideas fit seamlessly into the PuDI method. Applications demonstrating how utilizing uncertainty information/approximations along with generalized least squares techniques improve PuDI fits are presented. Although the primary application here is in estimating nonlinear SDEs, the PuDI method is applicable to situations where both unbiased function and derivative estimates are available.

Theoretical study regarding the modes shapes of wood beam having different boundary conditions using the Eigenvalues problems.

In engineering practice, however, beams problems often involve consideration of dynamic disturbances, produced by time-dependent external forces or displacements. Structural dynamics deals with time-dependent motions of structures and analyzes the internal forces associated with them. Thus, its objective is to determine the effect of vibrations on the performance of the structure. It can be started from the case of a homogeneous beam with different boundary conditions at the ends. By applying the principle of D’Alembert for an infinitesimal beam element, the equation of free transverse vibrations of the beam. The equation of the normal vibration modes, along with the homogeneous boundary conditions describe a type Sturm-Liouville problem. Solving type Sturm –Liouville problems allows the determination of inherent values , and the functions of the vibration shapes of beams. The normal vibration modes for several/more combinations of conditions at the ends of the beam are determined.

Flow of a film on a plate with periodical jumps at the time-dependent viscosity and external force

Flow of a film on a plate with periodical jumps at the time-dependent viscosity and external force

Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems

Fluctuation theorems (FTs), which describe some universal properties of nonequilibrium fluctuations, are examined from a quantum perspective and derived by introducing a two-point measurement on the system. FTs for closed and open systems driven out of equilibrium by an external time-dependent force, and for open systems maintained in a nonequilibrium steady-state by nonequilibrium boundary conditions, are derived from a unified approach. Applications to fermion and boson transport in quantum junctions are discussed. Quantum master equations and Green's functions techniques for computing the energy and particle statistics are presented.