Nonconservative Force Field Research Articles

Bayesian Estimation of Experimental Parameters in Stochastic Inertial Systems: Theory, Simulations, and Experiments with Objects Levitated in Vacuum

High-quality nanomechanical oscillators can sensitively probe force, mass, or displacement in experiments bridging the gap between the classical and quantum domain. Dynamics of these stochastic systems is inherently determined by the interplay between acting external forces, viscous dissipation, and random driving by the thermal environment. The relevance of inertia then dictates that both position and momentum must, in principle, be known to fully describe the system, which makes its quantitative experimental characterization rather challenging. We introduce a general method of Bayesian inference of the force field and environmental parameters in stochastic inertial systems that operates solely on the time series of recorded noisy positions of the system. The method is first validated on simulated trajectories of model stochastic harmonic and anharmonic oscillators with damping. Subsequently, the method is applied to experimental trajectories of particles levitating in tailored optical fields and used to characterize the dynamics of particle motion in a nonlinear Duffing potential, a static or time-dependent double-well potential, and a nonconservative force field. The presented inference procedure does not make any simplifying assumptions about the nature or symmetry of the acting force field and provides robust results with trajectories 2 orders of magnitude shorter than those typically required by alternative inference schemes. In addition to being a powerful tool for quantitative data analysis, it can also guide experimentalists in choosing appropriate sampling frequency (at least 20 measured points per single characteristic period) and length of the measured trajectories (at least 10 periods) to estimate the force field and environmental characteristics with a desired accuracy and precision.

Intelligent dissipative particle dynamics: Bridging mesoscopic models from microscopic simulations via deep neural networks

We develop a bottom-up coarse graining framework to construct mesoscopic force fields directly from microscopic dynamics by a machine learning technique. This scheme, called the intelligent dissipative particle dynamics (IDPD), holds various consistencies, including configuration, phase, static and dynamic consistencies that are incorporated into a deep neural network. The network is trained with full atomic data in a special way for preserving the natural symmetries of the system, and generates both the conservative and non-conservative force fields in the mesoscopic system. Subsequently, the force fields are used in the dissipative particle dynamics (DPD) that is considered as the effective mesoscopic dynamics resulting from the Mori–Zwanzig projection of the underlying atomic dynamics. To show the validity and applicability of IDPD, we coarse grain star polymer and methane fluids respectively, from full molecular dynamics (MD) simulations. Quantitative comparisons between the MD and IDPD indicate that the IDPD is able to reproduce both the static and dynamic properties of underlying MD system, with the aids of a pressure correction and a diffusion rescaling, even when coarse graining multiple molecules into a particle. It follows that the IDPD is competent for preserving the dynamic properties and coarse graining non-bonded molecules, which are common challenges for most of coarse graining methods.

Geometric decomposition of entropy production into excess, housekeeping, and coupling parts.

For a generic overdamped Langevin dynamics driven out of equilibrium by both time-dependent and nonconservative forces, the entropy production rate can be decomposed into two positive terms, termed excess and housekeeping entropy. However, this decomposition is not unique: There are two distinct decompositions, one due to Hatano and Sasa, the other one due to Maes and Netočný. Here we establish the connection between these two decompositions and provide a simple, geometric interpretation. We show that this leads to a decomposition of the entropy production rate into three positive terms, which we call the excess, housekeeping, and coupling part, respectively. The coupling part characterizes the interplay between the time-dependent and nonconservative forces. We also derive thermodynamic uncertainty relations for the excess and housekeeping entropy in both the Hatano-Sasa and Maes-Netočný decomposition and show that all quantities obey integral fluctuation theorems. We illustrate the decomposition into three terms using a solvable example of a dragged particle in a nonconservative force field.

On (non-)conservative body forces, vorticity generation and energy conversion in ideal flows

Usually, the load on lifting bodies in incompressible, inviscid flow is determined by integration of the pressure on the body surface, once the flow is solved. Prandtl (Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch- Physikalische Klasse, vol. 1918, pp. 451–477) proposed an opposite method in which the body force field is the source term in the equation of motion to solve the flow. This force field method has not been used intensively but has regained importance in modern wind energy research. However, an analysis of which type of body force field generates vorticity and converts energy, and which body force field does not, is lacking. Prandtl's method is adopted here, but with the addition that the force field is allowed to be conservative or non-conservative. The relation between conservative/non-conservative body forces, vorticity generation, potential and kinetic energy and Helmholtz's vorticity theorems are derived. Similarly, the load on lifting bodies in two and three dimensions is classified as (non-)conservative, with some examples. To show that the force field method is consistent with the method where the load is output of an analysis, the expression for the Kutta–Joukowsky load and the relation between bound and trailing vorticity of a wing have been rederived using the force field method. The analysis confirms that this relation between bound and trailing vorticity is not governed by Helmholtz's theorems, as is often assumed, but by the non-conservative force field generating vorticity.

Steady state of overdamped particles in the non-conservative force field of a simple non-linear model of optical trap

Optically trapped particles are often subject to a non-conservative scattering force arising from radiation pressure. In this paper, we present an exact solution for the steady state statistics of an overdamped Brownian particle subjected to a commonly used force field model for an optical trap. The model is the simplest of its kind that takes into account non-conservative forces. In particular, we present the exact results for certain marginals of the full three-dimensional steady state probability distribution, in addition to results for the toroidal probability currents that are present in the steady state, as well as for the circulation of these currents. Our analytical results are confirmed by numerical solution of the steady state Fokker–Planck equation.

Cases of Integrability Which Correspond to the Motion of a Pendulum in the Three-dimensional Space

We systematize some results on the study of the equations of spatial motion of dynamically symmetric fixed rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of a spatial motion of a free rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint, or the center of mass of the body moves rectilinearly and uniformly; this means that there exists a nonconservative couple of forces in the system

Enhanced force-field calibration via machine learning

The influence of microscopic force fields on the motion of Brownian particles plays a fundamental role in a broad range of fields, including soft matter, biophysics, and active matter. Often, the experimental calibration of these force fields relies on the analysis of the trajectories of the Brownian particles. However, such an analysis is not always straightforward, especially if the underlying force fields are non-conservative or time-varying, driving the system out of thermodynamic equilibrium. Here, we introduce a toolbox to calibrate microscopic force fields by analyzing the trajectories of a Brownian particle using machine learning, namely, recurrent neural networks. We demonstrate that this machine-learning approach outperforms standard methods when characterizing the force fields generated by harmonic potentials if the available data are limited. More importantly, it provides a tool to calibrate force fields in situations for which there are no standard methods, such as non-conservative and time-varying force fields. In order to make this method readily available for other users, we provide a Python software package named DeepCalib, which can be easily personalized and optimized for specific force fields and applications. This package is ideal to calibrate complex and non-standard force fields from short trajectories, for which advanced specific methods would need to be developed on a case-by-case basis.

Acoustokinetics: Crafting force landscapes from sound waves

Factoring the pressure field of a harmonic sound wave into its amplitude and phase profiles provides the foundation for an analytical framework for studying acoustic forces that not only provides novel insights into the forces exerted by specified sound waves, but also addresses the inverse problem of designing sound waves to implement desired force landscapes. We illustrate the benefits of this acoustokinetic framework through case studies of purely nonconservative force fields, standing waves, pseudo-standing waves, and tractor beams.

Subfemtonewton force fields measured with ergodic Brownian ensembles

We demonstrate that radiation pressure force fields can be measured and reconstructed with a resolution of $0.3$ fN (at a $99.7\%$ confidence level) using an ergodic ensemble of overdamped colloidal particles. The outstanding force resolution level is provided by the large size of the statistical ensemble built by recording all displacements from all diffusing particles, regardless of trajectory and time. This is only possible because the noise driving the particles is thermal, white and stationary, so that the colloidal system is ergodic, as we carefully verify. Using an ergodic colloidal dispersion for performing ultra-sensitive measurements of external forces is not limited to non-conservative optical force fields. Our experiments therefore give way to interesting opportunities in the context of weak force measurements in fluids.

Using the fluctuation-dissipation theorem for nonconservative forces

An equilibrium system which is perturbed by an external potential relaxes to a new equilibrium state, a process obeying the fluctuation-dissipation theorem. In contrast, perturbing by nonconservative forces yields a nonequilibrium steady state, and the fluctuation-dissipation theorem can in general not be applied. Here we exploit a freedom inherent to linear response theory: Force fields which perform work that does not couple statistically to the considered observable can be added without changing the response. Using this freedom, we demonstrate that the fluctuation-dissipation theorem can be applied for certain nonconservative forces. We discuss the case of a nonconservative force field linear in particle coordinates, where the mentioned freedom can be formulated in terms of symmetries. In particular, for the case of shear, this yields a response formula, which we find advantageous over the known Green-Kubo relation in terms of statistical accuracy.

Near-field optical trapping in a non-conservative force field

The force-field generated by a near-field optical trap is analyzed. A C-shaped engraving on a gold film is considered as the trap. By separating out the conservative component and the solenoidal component of the force-field using Helmholtz-Hodge decomposition, it was found that the force is non-conservative. Conventional method of calculating the optical potential from the force-field is shown to be inaccurate when the trapping force is not purely conservative. An alternative method is presented to accurately estimate the potential. The positional statistics of a trapped nanoparticle in this non-conservative field is calculated. A model is proposed that relates the position distribution to the conservative component of the force. The model is found to be consistent with numerical and experimental results. In order to show the generality of the approach, the same analysis is repeated for a plasmonic trap consisting of a gold nanopillar. Similar consistency is observed for this structure as well.

New Examples of Integrable Systems with Dissipation on the Tangent Bundles of Multidimensional Spheres

In many problems of multidimensional dynamics, systems appear whose state spaces are spheres of finite dimension. Clearly, phase spaces of such systems are tangent bundles of these spheres. In this paper, we examine nonconservative force fields in the dynamics of a multidimensional rigid body in which the system possesses a complete set of first integrals that can be expressed as finite combinations of elementary transcendental functions. We consider the case where the moment of nonconservative forces depends on the tensor of angular velocity.

Image enhancement with PDEs and nonconservative advection flow fields.

We propose a new method for the progressive enhancement of images degraded by noise and blur using shock filters. The originality of our approach lies in the iterative exploitation of nonconservative edge-based force fields normally designed for segmentation to determine both locations and amplitudes of the sharpening action. The proposed nonconservative shock filter (NCSF) produces images with strong discontinuities that can be used as an additional pre-processing step to facilitate higher level tasks such as edge detection or segmentation. NCSF is especially useful when coupled with an anisotropic diffusion equation in noisy configurations. Using the proposed method, edges that are not well defined prior to filtering can be iteratively recovered. We show processing results on various types of images, focusing on medical transrectal ultrasound and brain PET imaging.

ON A PENDULUM MOTION IN MULTI-DIMENSIONAL SPACE. PART 3. DEPENDENCE OF FORCE FIELDS ON THE TENSOR OF ANGULAR VELOCITY

In the proposed cycle of work, we study the equations of motion of dynamically symmetric fixedn-dimensional rigid bodiespendulums located in a nonconservative force fields. The form of these equationsis taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium.In parallel, we study the problem of motion of a free n-dimensional rigid body also located in a similarforce fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under actionof this force, either the magnitude of the velocity of some characteristic point of the body remains constant,which means that the system possesses a nonintegrable servo constraint. In this work, we studythat case when the force fields linearly depend on the tensor of angular velocity.

Eigenmode orthogonality breaking and anomalous dynamics in multimode nano-optomechanical systems under non-reciprocal coupling

Thermal motion of nanomechanical probes directly impacts their sensitivities to external forces. Its proper understanding is therefore critical for ultimate force sensing. Here, we investigate a vectorial force field sensor: a singly-clamped nanowire oscillating along two quasi-frequency-degenerate transverse directions. Its insertion in a rotational optical force field couples its eigenmodes non-symmetrically, causing dramatic modifications of its mechanical properties. In particular, the eigenmodes lose their intrinsic orthogonality. We show that this circumstance is at the origin of an anomalous excess of noise and of a violation of the fluctuation dissipation relation. Our model, which quantitatively accounts for all observations, provides a novel modified version of the fluctuation dissipation theorem that remains valid in non-conservative rotational force fields, and that reveals the prominent role of non-axial mechanical susceptibilities. These findings help understand the intriguing properties of thermal fluctuations in non-reciprocally-coupled multimode systems.

Noise induced aperiodic rotations of particles trapped by a non-conservative force.

We describe a mechanism whereby random noise can play a constructive role in the manifestation of a pattern, aperiodic rotations, that would otherwise be damped by internal dynamics. The mechanism is described physically in a theoretical model of overdamped particle motion in two dimensions with symmetric damping and a non-conservative force field driven by noise. Cyclic motion only occurs as a result of stochastic noise in this system. However, the persistence of the cyclic motion is quantified by parameters associated with the non-conservative forcing. Unlike stochastic resonance or coherence resonance, where noise can play a constructive role in amplifying a signal that is otherwise below the threshold for detection, in the mechanism considered here, the signal that is detected does not exist without the noise. Moreover, the system described here is a linear system.

Integrable Variable Dissipation Systems on the Tangent Bundle of a Multi-Dimensional Sphere and Some Applications

This paper is a survey of integrable cases in dynamics of a multi-dimensional rigid body under the action of a nonconservative force field. We review both new results and results obtained earlier. Problems examined are described by dynamical systems with so-called variable dissipation with zero mean. The problem of the search for complete sets of transcendental first integrals of systems with dissipation is quite topical; a large number of works are devoted to it. We introduce a new class of dynamical systems that have a periodic coordinate. Due to the existence of nontrivial symmetry groups of such systems, we can prove that these systems possess variable dissipation with zero mean, which means that on the average for a period with respect to the periodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either energy pumping or dissipation can occur. Based on the facts obtained, we analyze dynamical systems that appear in the dynamics of a multi-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions, which can be expressed through a finite combination of elementary functions. As applications, we study dynamical equations of motion arising in the study of the plane and spatial dynamics of a rigid body interacting with a medium and also a possible generalization of the obtained methods to the study of general systems arising in the qualitative theory of ordinary differential equations, in the theory of dynamical systems, and also in oscillation theory.

Memory effects in active particles with exponentially correlated propulsion.

The Ornstein-Uhlenbeck particle (OUP) model imagines a microscopic swimmer propelled by an active force which is correlated with itself on a finite time scale. Here we investigate the influence of external potentials on an ideal suspension of OUPs, in both one and two spatial dimensions, with particular attention paid to the pressure exerted on "confining walls." We employ a mathematical connection between the local density of OUPs and the statistics of their propulsion force to demonstrate the existence of an equation of state in one dimension. In higher dimensions we show that active particles generate a nonconservative force field in the surrounding medium. A simplified far-from-equilibrium model is proposed to account for OUP behavior in the vicinity of potentials. Building on this, we interpret simulations of OUPs in more complicated situations involving asymmetrical and spatially curved potentials, and characterize the resulting inhomogeneous stresses in terms of competing active length scales.

ON A PENDULUM MOTION IN MULTI-DIMENSIONAL SPACE. PART 2. INDEPENDENCE OF FORCE FIELDS ON THE TENSOR OF ANGULAR VELOCITY

In the proposed cycle of work, we study the equations of the motion of dynamically symmetric fixed n-dimensional rigid bodiespendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of the motion of a free n-dimensional rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. In this work, we study the case of independence of force fields on the tensor of angular velocity.

Langevin Dynamics with Variable Coefficients and Nonconservative Forces: From Stationary States to Numerical Methods

Langevin dynamics is a versatile stochastic model used in biology, chemistry, engineering, physics and computer science. Traditionally, in thermal equilibrium, one assumes (i) the forces are given as the gradient of a potential and (ii) a fluctuation-dissipation relation holds between stochastic and dissipative forces; these assumptions ensure that the system samples a prescribed invariant Gibbs-Boltzmann distribution for a specified target temperature. In this article, we relax these assumptions, incorporating variable friction and temperature parameters and allowing nonconservative force fields, for which the form of the stationary state is typically not known a priori. We examine theoretical issues such as stability of the steady state and ergodic properties, as well as practical aspects such as the design of numerical methods for stochastic particle models. Applications to nonequilibrium systems with thermal gradients and active particles are discussed.