Brownian Harmonic Oscillator Research Articles

Inertial spin model of flocking with position-dependent forces.

We propose an extension to the inertial spin model (ISM) of flocking and swarming. The model has been introduced to explain certain dynamic features of swarming (second sound, a lower than expected dynamic critical exponent) while preserving the mechanism for onset of order provided by the Vicsek model. The inertial spin model (ISM) has only been formulated with an imitation ("ferromagnetic") interaction between velocities. Here we show how to add position-dependent forces in the model, which allows to consider effects such as cohesion, excluded volume, confinement, and perturbation with external position-dependent field, and thus study this model without periodic boundary conditions. We study numerically a single particle with an harmonic confining field and compare it to a Brownian harmonic oscillator and to a harmonically confined active Browinian particle, finding qualitatively different behavior in the three cases.

Langevin original approach and Ornstein–Uhlenbeck-type processes

This paper applies Langevin idea to describe the Brownian motion of a particle characterized by an Ornstein–Uhlenbeck-type process. The original and clever method proposed by Langevin is based on Newton’s second law plus a fluctuating force whose solution for the mean square displacement consists in separating the fluctuating force from his equation to obtain a deterministic equation for the relevant physical variable. In this work the Langevin original idea is applied to calculate the mean square velocity for a field free particle case; then it is extended for a charged particle in a constant magnetic field. In a similar way, the strategy is also applied to calculate the mean square displacement for a Brownian harmonic oscillator in the overdamped regime, and also when a magnetic field is present. In particular, it is shown in the field free case that Langevin’s original strategy leads to the same results as those obtained using the statistical properties of a Gaussian white noise. All the theoretical results are compared with both the numerical simulation of the Langevin equation, and numerical solution of the corresponding deterministic differential equations.

Reply to "Comment on 'Non-Markovian harmonic oscillator across a magnetic field and time-dependent force fields' ".

Recently in a paper by Hidalgo-Gonzalez and Jiménez-Aquino [Phys. Rev. E 100, 062102 (2019)PREHBM2470-004510.1103/PhysRevE.100.062102], the generalized Fokker-Planck equation (GFPE) for a Brownian harmonic oscillator in a constant magnetic field and under the action of time-dependent force fields, has been explicitly calculated using the characteristic function method. Although the problem is linear it is not easy to solve, however, the method of the characteristic function is effective and allows to obtain an exact and precise solution of the problem. Our theoretical result has been compared with the one reported by Das etal. in a recently submitted paper [arXiv:2011.09771] using another solution method. The proposed method consists in constructing the GFPE and then calculating each time-dependent coefficient associated with this equation. However, in a more complicated case, one cannot know a priori the exact number of terms that this equation must contain. The precise number is further provided by the characteristic function method.

Comment on "Non-Markovian harmonic oscillator across a magnetic field and time-dependent force fields".

In a recent paper Das etal. [J. Chem. Phys. 147, 164102 (2017)JCPSA60021-960610.1063/1.4999408] proposed the Fokker-Planck equation (FPE) for the Brownian harmonic oscillator in the presence of a magnetic field and the non-Markovian thermal bath, respectively. This system has been studied very recently by Hidalgo-Gonzalez and Jiménez-Aquino [Phys. Rev. E 100, 062102 (2019)PREHBM2470-004510.1103/PhysRevE.100.062102] and the Fokker-Planck equation was derived using the characteristic function. It includes a few extra terms in the FPE and the authors conclude that their method is accurate compared to the calculation by Das etal. Then we reexamine our calculation and which is present in this comment. The revised calculation shows that both methods give the same result.

Fast Bayesian inference of the multivariate Ornstein-Uhlenbeck process.

The multivariate Ornstein-Uhlenbeck process is used in many branches of science and engineering to describe the regression of a system to its stationary mean. Here we present an O(N) Bayesian method to estimate the drift and diffusion matrices of the process from N discrete observations of a sample path. We use exact likelihoods, expressed in terms of four sufficient statistic matrices, to derive explicit maximum a posteriori parameter estimates and their standard errors. We apply the method to the Brownian harmonic oscillator, a bivariate Ornstein-Uhlenbeck process, to jointly estimate its mass, damping, and stiffness and to provide Bayesian estimates of the correlation functions and power spectral densities. We present a Bayesian model comparison procedure, embodying Ockham's razor, to guide a data-driven choice between the Kramers and Smoluchowski limits of the oscillator. These provide novel methods of analyzing the inertial motion of colloidal particles in optical traps.

Energetics of a driven Brownian harmonic oscillator

We provide insight into the energetics of a Brownian oscillator in contact with a heat bath and driven by an external unbiased time-periodic force that takes the system out of thermodynamic equilibrium. Solving the corresponding Langevin equation, we compute the average kinetic and potential energies in the long-time stationary state. We also derive the energy balance equation and study the energy flow in the system. In particular, we identify the energy delivered by the external force, the energy dissipated by a thermal bath and the energy provided by thermal equilibrium fluctuations. Next, we illustrate the Jarzynski work-fluctuation relation and consider the stationary state fluctuation theorem for the total work done on the system by external force. Finally, by determining time scales in the system, we analyze the strong damping regime and discuss the problem of overdamped dynamics when inertial effects can be neglected.

Brownian Emitters

A Brownian harmonic oscillator, which dissipates energy either by friction or via emission of electromagnetic radiation, is considered. This Brownian emitter is driven by the surrounding thermo-quantum fluctuations, which are theoretically described by the fluctuation–dissipation theorem. It is shown how the Abraham–Lorentz force leads to dependence of the half-width on the peak frequency of the oscillator amplitude spectral density. It is found that for the case of a charged particle moving in vacuum at zero temperature, its root-mean-square velocity fluctuation is a universal constant, equal to roughly 1/18 of the speed of light. The relevant Fokker–Planck and Smoluchowski equations are also derived.

Non-Markovian work fluctuation theorem in crossed electric and magnetic fields.

The validity of the transient work fluctuation theorem for a charged Brownian harmonic oscillator embedded in a non-Markovian heat bath and under the action of crossed electric and magnetic fields is investigated. The aforementioned theorem is verified to be valid within the context of the generalized Langevin equation with an arbitrary memory kernel and arbitrary dragging in the potential minimum. The fluctuation-dissipation relation of the second kind is assumed to be valid and shows that the non-Markovian stochastic dynamics associated with the particle, in the absence of the external time-dependent electric field, reaches an equilibrium state, as is precisely demanded by such a relation. The Jarzynski equality in this problem is also analyzed.

Hamilton–Jacobi and Fokker–Planck equations for the harmonic oscillator in the inertial regime

In this work we use Feynman’s path integral formalism to show the strict equivalence between the Hamilton–Jacobi (HJ) and Fokker–Planck (FP) equations, for a Brownian harmonic oscillator characterized by a Langevin equation within the inertial regime. In this case, the Lagrangian function is Gaussian and then the path integration which defines the conditional probability density can be replaced by the extremal path. This extremal action principle allows us to derive in a straightforward way not only the HJ differential equation, but also its solution, the extremal action. The probability density related to the extremal action is also the solution of the FP equation and shown to be exactly the same as those obtained by Chandrasekhar in his celebrated paper (Chandrasekhar, 1943) and Wang and Uhlenbeck (1945).

Hamilton–Jacobi and Fokker–Planck equations for the harmonic oscillator

Using Feynman’s path integral formalism applied to stochastic classical processes, we show the equivalence between the Hamilton–Jacobi (HJ) and Fokker–Planck (FP) equations, associated with an overdamped Brownian harmonic oscillator. In this case, the Langevin equation leads to a Gaussian Lagrangian function and then the path integration which defines the conditional probability density can be replaced by the extremal path. Due to this fact and following the classical dynamics formalism, we prove the strict equivalence between the HJ and FP equations. We do this first for an ordinary Brownian harmonic oscillator and then the proof is extended to an electrically charged Brownian particle under the action of force fields: magnetic field and additional time-dependent force fields. We observe that this extremal action principle allows us to derive in a straightforward way not only the HJ differential equation, but also its solution, the extremal action.

Power-fluctuation theorem for a Brownian oscillator in a thermal bath

In this paper we prove the validity of the total power fluctuation theorem for a Brownian particle in a harmonic trap when it is dragged out of equilibrium, taking into account the inertia term in the Langevin equation. We first consider the theorem for an ordinary Brownian harmonic oscillator embedded in a thermal bath. As a second case we assume the presence of an electromagnetic field acting on the charged Brownian oscillator. In both cases, the theorem is proved for two examples in the trap motion, linear and oscillatory drag on the potential minimum.

Power fluctuation theorem for a Brownian harmonic oscillator

In this paper we study the validity of the total power fluctuation theorem spent on a Brownian harmonic oscillator when the system is driven out of equilibrium through the drag of the potential minimum. The theorem is first proved for an ordinary harmonic oscillator in two cases: The first one considers the particle in a thermal bath under the action of Gaussian white noise, and in the second one the drift is provided by an additional external Gaussian colored noise satisfying the characteristics of an Ornstein-Uhlenbeck process. We go further, by considering a charged harmonic oscillator under the action of an electromagnetic field. The theorem is also proven as in the two cases given above. In both of those cases, we illustrate the theorem for a uniform motion of the trap potential minimum and show that in the presence of external colored noise, the theorem is only valid in the stationary state.

Work-fluctuation theorem for a charged harmonic oscillator

The solution of the Langevin equation including the inertia term is used to prove the transient work-fluctuation theorem for an electrically charged Brownian harmonic oscillator in an electromagnetic field. The theorem is proved for the physical situation in which the system is driven out of equilibrium by an arbitrary time-dependent dragging of the trap potential minimum. The proof is first given for a harmonic oscillator in the absence of an electromagnetic field, and then is extended to include the case of a charged harmonic oscillator under the action of crossed electric and magnetic fields. The case of a linear motion for the potential minimum is explicitly addressed.

Multipoint correlation functions for continuous-time random walk models of anomalous diffusion

Recursive relations are developed for computing the multipoint correlation functions of a particle undergoing a biased continuous-time random walk (CTRW) in an external potential. Two- and three-point correlation functions are calculated for waiting-time distributions with an anomalous power-law profile t(-alpha-1), 0 < alpha < 1, on intermediate time scales with a crossover to an exponential long time decay. Comparison of the CTRW with the Brownian harmonic oscillator model (Gaussian process) illustrates how higher-order correlation functions may be used to distinguish between dynamical models that have the same two-point correlation function.

Two-dimensional Raman and infrared vibrational spectroscopy for a harmonic oscillator system nonlinearly coupled with a colored noise bath.

Multidimensional vibrational response functions of a harmonic oscillator are reconsidered by assuming nonlinear system-bath couplings. In addition to a standard linear-linear (LL) system-bath interaction, we consider a square-linear (SL) interaction. The LL interaction causes the vibrational energy relaxation, while the SL interaction is mainly responsible for the vibrational phase relaxation. The dynamics of the relevant system are investigated by the numerical integration of the Gaussian-Markovian Fokker-Planck equation under the condition of strong couplings with a colored noise bath, where the conventional perturbative approach cannot be applied. The response functions for the fifth-order nonresonant Raman and the third-order infrared (or equivalently the second-order infrared and the seventh-order nonresonant Raman) spectra are calculated under the various combinations of the LL and the SL coupling strengths. Calculated two-dimensional response functions demonstrate that those spectroscopic techniques are very sensitive to the mechanism of the system-bath couplings and the correlation time of the bath fluctuation. We discuss the primary optical transition pathways involved to elucidate the corresponding spectroscopic features and to relate them to the microscopic sources of the vibrational nonlinearity induced by the system-bath interactions. Optical pathways for the fifth-order Raman spectroscopies from an "anisotropic" medium were newly found in this study, which were not predicted by the weak system-bath coupling theory or the standard Brownian harmonic oscillator model.

Representation of Quantum Brownian Motion in the Collective Coordinate Method

We consider two explicitly solvable models of quantum random processes described by the Langevin equation, namely, those for a “free” quantum Brownian particle and for a quantum Brownian harmonic oscillator. The Hamiltonian (string) realization of the models reveals a soliton-like structure of “classical” solutions. Accordingly, the zero-mode collective coordinate method turns out to be an adequate means for describing the quantum dynamics of the models.

Multi-dimensional vibrational spectroscopy measured from different phase-matching conditions

We developed a theoretical method that can systematically treat the phase-matching condition of nonlinear infrared or Raman measurements. This method might be a rational tool for the analysis of observed signals from various directions based on the response function especially under non-impulsive excitation. Model calculations of the third-order nonlinear IR spectroscopy for a Brownian harmonic oscillator system are performed. We discuss the variation of signals that depend on the observation direction and the temporal width of excitation pulse in comparison with the behavior within the impulsive limit approximation.

Quantum control of dissipative systems: Exact solutions

Optimal quantum control theory, which predicts the tailored light fields that best drive a system to a desired target, is applied to the quantum dissipative dynamics of systems linearly coupled to a Gaussian bath. To calculate the material response function required for optimizing the light field, the analytical solution is derived for the two-level Brownian harmonic oscillator model and the recently developed method for directly simulating the Gaussian force is implemented for anharmonic Brownian oscillators. This study confirms the feasibility of quantum control in favorable condensed phase environments and explores new quantum control features in the presence of dissipation, including memory effects and temperature dependence.

Stochastic Wess-Zumino-Witten model over a symplectic manifold

Over the path space of a symplectic manifold with end points in two Lagrangian submanifolds, we define a measure and a stochastic symplectic action in the simply connected case. We define a regularized Wess-Zumino-Witten Laplacian over the forms of finite degree over the path space. We perform a short time asymptotic near the critical points and find a limit Brownian harmonic oscillator: we can diagonalize it explicitly, and find the limit ground state of the Laplacian. We define a stochastic Witten complex, and its algebraic counterpart at the level of Chen forms.

Stationarity of time correlation functions for globally linear classical systems and its consequences

A necessary and sufficient condition for the stationarity of all time correlation functions associated with a given globally linear classical dynamical system is rigorously established from basic principles. Since stationarity of time correlation functions is a physical requirement that must be satisfied, the necessary and sufficient condition obtained for its realization represents a universal dynamical constraint on globally linear classical dynamical models intended to describe the execution of spontaneous fluctuations about a stationary state. This dynamical constraint is shown to (i) impose restrictions on the symmetry properties of the transition operator appearing in the global propagator for a system; (ii) represent a universal operator relation that embodies detailed balance and microscopic reversibility, giving rise to their traditional formulations; and (iii) imply the existence of certain generalized symmetry relations for time correlation functions and their Laplace and Fourier transforms. Apart from elucidating some fundamental symmetries of classical dynamical systems, the reported theory has the advantage of providing a simple model independent framework for treating classical time correlation functions via the extraction and utilization of dynamically embedded information. This is demonstrated in a concrete way by exploiting the mathematical apparatus of dual Lanczos transformation theory to determine the advanced and retarded components of the elements of the correlation matrices for first and second moment coordinate and momentum fluctuations for the Brownian harmonic oscillator. The Laplace transforms of the retarded components of the time correlation functions and the Fourier transforms of the full-time correlation functions are also obtained.