Classical Dynamics Research Articles

Machine learning stochastic differential equations for the evolution of order parameters of classical many-body systems in and out of equilibrium

Abstract We develop a machine learning algorithm to infer the emergent stochastic equation governing the evolution of an order parameter of a many-body system. We train our neural network to independently learn the directed force acting on the order parameter as well as an effective diffusive noise. We illustrate our approach using the classical Ising model endowed with Glauber dynamics, and the contact process as test cases. For both models, which represent paradigmatic equilibrium and nonequilibrium scenarios, the directed force and noise can be efficiently inferred. The directed force term of the Ising model allows us to reconstruct an effective potential for the order parameter which develops the characteristic double-well shape below the critical temperature. Despite its genuine nonequilibrium nature, such an effective potential can also be obtained for the contact process and its shape signals a phase transition into an absorbing state. Also, in contrast to the equilibrium Ising model, the presence of an absorbing state renders the noise term dependent on the value of the order parameter itself.

Determination of the density of dark matter in the Solar system

A method based on the Newtonian and relativistic theories of body motion is proposed for calculating the density of dark matter, which, like visible (baryonic) matter, creates a gravitational field. Experimental data obtained by the Pioneer 10 and Pioneer 11 spacecraft and a variety of astronomical observations are used to detect and establish the mass of dark matter in the solar system, which turned out to be approximately equal to the mass of the Sun. Using the equations of motion of test bodies in the Newtonian and post-Newtonian approximations of the general theory of relativity, calculation formulas are obtained for calculating the density of dark matter in three cases: 1) baryonic and dark matter are uniformly distributed in space (their density is constant); 2) they are distributed according to spherically symmetrical laws; 3) baryonic matter is distributed spherically symmetrically, while dark matter is uniformly distributed. In the volume of a sphere with radius of 45 a. u. with the center in the center of gravity of the Sun, on the basis of known experimental data, the average density of the gas-dust and relict matter located in it is calculated, equal to 1,26 · 10–16 g · cm–3. In the same volume, the density of dark matter in all three cases varies according to the derived calculation formulas in the range from 3,38 · 10–16 to 3,34 · 10–16 g · cm–3, which gives the superiority of dark matter over baryonic one from 2.68 to 2.72 times. The given numerical estimates may change when the experimental data used change. The paper also contains a brief discussion of other methods for calculating the density of dark matter in space and a comparison with our results.

Physics-guided neural network-based framework for 3D modeling of slope stability

Physics-informed neural networks (PINN) have gradually attracted attention in the field of geotechnical engineering. This paper proposes a novel PINN-based framework for the three-dimensional (3D) stability analysis of soil slopes. Based on the fundamental theorem of plasticity and limit analysis, the partial differential equations (PDE) with regard to slope collapse are derived and integrated into the physics-guided loss function. A kinematically admissible failure mechanism that rigorously satisfies the Mohr-Coulomb associated flow rule is obtained by minimizing the loss function, thereby circumventing complex mathematical calculations. The discrete points generated by PINN are selected and refined through a series of procedures to represent the failure block of slopes using a two-dimensional matrix. The entire training process of the PINN-based framework is conducted without the need for any labeled data. The resulting discretized failure mechanism is meshfree and capable of accommodating spatially discrete data. A validation exercise is performed to verify the proposed framework by comparing it with the classical 3D rotational failure mechanism. To further consider the impact of external excitation on slope stability, a hybrid PINN framework is developed to assess the stability of slopes subjected to complex external environments. In addition to the PINN to generate a failure mechanism, a parallel PINN is employed to acquire the corresponding spatially discrete data of specified external excitations. The hybrid PINN framework for seismic stability assessment of slopes is demonstrated by way of example, indicating favorable feasibility and applicability of the developed approach. The proposed PINN-based framework provides innovative and promising avenues for 3D slope stability analysis.

Critical Review of Physical-Mechanical Principles in Geostructure-Soil Interface Mechanics

AbstractDue to the relatively different mechanical and physical properties of soils and structures, the interface plays a critical role in the transfer of stress and strain between them. The stability and safety of geotechnical structures are thus greatly influenced by the behavior at the soil–structure interface. It is therefore important to focus on the unique characteristics that set the interface apart from other geomaterials while examining the interface behaviour. Understanding the physical mechanism and modelling principles of these interfaces becomes a crucial step for the secure design and investigation of soil-structure interaction (SSI) issues. Moreover, to deal with this soil-environment interaction problem, the classical soil mechanics formulation must be progressively generalised in order to incorporate the effects of new phenomena and new variables on SSI behaviour. Considering the variety of energy geostructures that are emerging nowadays, it is crucial to comprehend the thermo-hydro-mechanical (THM) behaviour of the interface. The objective of this study is to fill this information gap as concisely as possible. A critical review is provided along with the state-of-the-art information on the thermo-hydro-mechanical behaviour of the soil-structure interface, including testing tools and measurement methods, basic principles and deformation mechanisms, constitutive models, as well as their applications in numerical simulations. This study explains how loading influences the mechanisms at the interface and critically examines the effects of boundary conditions, soil properties, environmental factors, and structure type on the THM behaviour of interface zones between soils and structural elements. The validity and reliability of the interface shear stress-displacement models are also covered in this paper. Lastly, the trends and recent advancements are also recommended for the interface research.

Enabling large-scale and high-precision fluid simulations on near-term quantum computers

Quantum computational fluid dynamics (QCFD) offers a promising alternative to classical computational fluid dynamics (CFD) by leveraging quantum algorithms for higher efficiency. This paper introduces a comprehensive QCFD method, including an iterative method “Iterative-QLS” that suppresses error in quantum linear solver, and a subspace method to scale the solution to a larger size. We implement our method on a superconducting quantum computer, demonstrating successful simulations of steady Poiseuille flow and unsteady acoustic wave propagation. The Poiseuille flow simulation achieved a relative error of less than 0.2%, and the unsteady acoustic wave simulation solved a 5043-dimensional matrix. We emphasize the utilization of the quantum–classical hybrid approach in applications of near-term quantum computers. By adapting to quantum hardware constraints and offering scalable solutions for large-scale CFD problems, our method paves the way for practical applications of near-term quantum computers in computational science.

Complex nonlinear dynamics of a multidirectional energy harvester with hybrid transduction

Abstract Mechanical energy harvesting has increasing scientific and technological interests due to novel energetic challenges. A critical issue in classical cantilever-based mechanical energy harvesting systems is the lack of multidirectional energy conversion capabilities and, due to that, deviations from the excitation source can drastically reduce their performance. This limitation has led to the development of energy harvesters with attached pendula, serving as a direction coupling mechanism. Nevertheless, the pendulum structure itself can act as an energy absorber, drastically reducing the harvester performance in certain scenarios. In order to overcome this issue, a hybrid transduction multidirectional pendulum-based energy harvester is proposed integrating a piezoelectric element, to capture energy from the principal direction, and an electromagnetic transducer, to harness rotational energy from the pendulum. This paper presents an in-depth analysis of the hybrid multidirectional pendulum-based energy harvester using a nonlinear dynamics perspective to evaluate the energy harvesting performance. A reduced-order model is proposed to represent the essential characteristics of such systems. A parametric analysis using a nonlinear dynamics perspective is carried out to map the system dynamics and performance. The emergence of complex and rich dynamics is observed, including chaos and hyperchaos. Results reveal the most and least effective combinations of structural parameters in terms of energy conversion. Additionally, the dynamical responses and patterns associated with high performance are identified. These responses are often characterized by a blend of irregular complex behaviors, coupled with a mix of oscillatory and rotational patterns of motion, resulting in wider bandwidth systems.

Accounting of imperfect interfaces in multilayer modeling

Multilayer systems are widely used in industry for their performances as compared with homogeneous materials. The layers are usually glued together but the bonding may be imperfect (air bubble, detachment). The interfaces are generally modeled as sliding or bonded. These two behaviors can be significantly different and the actual behavior is generally somewhere in between. The vibroacoustic behavior of the structure is thus modified by the imperfect interface. Several models have been implemented to describe the dynamic behavior of multilayer systems with imperfect interfaces. In this paper, we propose an analytical model of imperfect interfaces based on the Transfer Matrix Method (TMM). Two computations are performed in parallel (one with a perfectly bonded interface and the other with a sliding interface). Instead of mixing impedances or admittances matrices, a mixing law on the state variables is applied, resulting in an equivalent global matrix which can be coupled in any multi-layer. The model is compared in terms of the sound absorption or sound transmission loss with the existing ones and is applied on different classical multilayer systems with multi-layered solid plates and a multilayer composed of a porous layer with a heavy layer which is widely used in the automotive industry.

Application of the frozen-modes approximation to classical harmonic oscillator systems

The problems of open classical systems usually correspond to a motion of a test particle that interacts with a large number of bath oscillators. Often, the test particle itself can be considered a harmonic oscillator. For such composite systems, exact numerical solutions are available, but they can become increasingly costly for a large number of bath oscillators. Here we take inspiration from the recent work on open quantum systems and investigate the applicability of the frozen-modes approximation to such classical systems. This approach assumes that some part of the low-frequency bath modes are frozen, thus only their initial values need to be considered. We show that by applying the frozen-modes approximation one can significantly increase the accuracy of the perturbative multiple-scales solution, especially for slow baths. This approach provides a good accuracy even for strong system–bath couplings, a regime that is not accessible to straightforward applications of the perturbation theory. We also suggest a rule for the splitting of spectral density to the fast and slow bath modes. We find that our approach gives excellent results for the ohmic spectral density, but it could be applied for other similar spectral densities as well.

Rydberg excitons in cuprous oxide: A two-particle system with classical chaos.

When an electron in a semiconductor gets excited to the conduction band, the missing electron can be viewed as a positively charged particle, the hole. Due to the Coulomb interaction, electrons and holes can form a hydrogen-like bound state called the exciton. For cuprous oxide, a Rydberg series up to high principle quantum numbers has been observed by Kazimierczuk et al. [Nature 514, 343 (2014)] with the extension of excitons up to the μm-range. In this region, the correspondence principle should hold and quantum mechanics turn into classical dynamics. Due to the complex valence band structure of Cu2O, classical dynamics deviates from a purely hydrogen-like behavior. The uppermost valence band in cuprous oxide splits into various bands resulting in yellow and green exciton series. Since the system exhibits no spherical symmetry, the angular momentum is not conserved. Thus, the classical dynamics becomes non-integrable, resulting in the possibility of chaotic motion. Here, we investigate the classical dynamics of the yellow and green exciton series in cuprous oxide for two-dimensional orbits in the symmetry planes as well as fully three-dimensional orbits. Our analysis reveals substantial differences between the dynamics of the yellow and green exciton series. While it is mostly regular for the yellow series, large regions in phase space with classical chaos do exist for the green exciton series.

Self-gravitating anisotropic fluid. II: Newtonian theory

Self-gravitating anisotropic fluid. II: Newtonian theory

A derivation of the Schrödinger equation from Feynman’s path-integral formulation of quantum mechanics

The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.

A New model of gas-water two-phase seepage based on Newton's law of motion

The pore throat size, structure distribution, and lithology of porous media in gas reservoirs are varied, and the gas-water two-phase seepage law is complex, making it difficult to describe the seepage model. Newton's law of motion is a basic law of motion in classical mechanics, and its application in gas-water two-phase seepage modeling is an innovative practice of classical mechanics in seepage mechanics systems. Based on Newton's three laws of motion, a gas-water two-phase seepage model was established from the force analysis of fluid, and the relationship between velocity and pressure difference, pipe radius, water film thickness, viscosity, etc. was derived under the condition that the model reached a stable state, i.e., force balance. The relationship is referred to as the steady-state model. Subsequently. the equivalent permeability representation model was derived based on the steady-state model to calculate the permeability of rock samples with different channel distribution characteristics. The results were consistent with the measured values, and there is a good correspondence between channel distribution characteristics and permeability. The relative permeability calculation method was established based on the steady-state model and capillary pressure curve. The obtained relative permeability curve aligned with the two-phase seepage law and coincides with the relative permeability curve obtained by Poiseuille's law. Finally, a new productivity equation was derived based on the steady-state model, and the Inflow Performance Relationship (IPR) curve calculated by the gas well example was consistent with the traditional equation. The research results were based on the force analysis of fluid in porous media and fundamentally explored the law of fluid flow. The derived seepage model can be used to calculate reservoir permeability, the gas-water relative permeability curve, and gas well productivity analysis and to effectively guide gas reservoir development. The study was a successful application of the basic theory of mechanics in gas-water two-phase seepage in gas reservoirs.

On removing the classical-quantum boundary

We argue that it is adherence to the axiom of counterfactual definiteness and not to that of locality and realism that results in Bell inequality violations. Furthermore, this axiom is not supported in classical mechanics because of deeper implications that arise from the principle of stationary action. This means that the Bell inequality fails classically, effectively removing the classical-quantum boundary–a conclusion prophesized by Bell himself. An implication here is that a local hidden variable theory, in the configuration space of classical mechanics, may not be ruled out. Basing our idea on Jacobi’s “initial variable” framework, we propose that a classical path of stationary action that recedes into the infinite past and stretches into an infinite future implies a reality that lacks counterfactual definiteness. We then corroborate this with recent experimental results, through which it could be understood that our world could be such a reality.

Semiclassical perturbations of single-degree-of-freedom Hamiltonian systems II: Nonintegrability

Continuing from Paper I [Ohsawa and Yagasaki, J. Math. Phys. 65, 102706 (2024)], we study semiclassical perturbations of single-degree-of-freedom analytic Hamiltonian systems and provide a sufficient condition for its meromorphic nonintegrability such that the first integrals depend on the small parameter meromorphically. Our approach is based on a generalization due to Ayoul and Zung of the Morales-Ramis theory, which enables us to show the meromorphic nonintegrability of dynamical systems by using the differential Galois theory. We remark that standard systems of Hagedorn and Heller for the semiclassical Gaussian wave packet dynamics are analytically integrable as well as the corresponding classical systems. We illustrate our theory for a bounded potential.

Semiclassical perturbations of single-degree–of–freedom Hamiltonian systems I: Separatrix splitting

We study semiclassical perturbations of single-degree-of-freedom Hamiltonian systems possessing hyperbolic saddles with homoclinic orbits, and provide a sufficient condition for the separatrices to split, using a Melnikov-type approach. The semiclassical systems give approximations of the expectation values of the positions and momenta to the semiclassical Schrödinger equations with Gaussian wave packets as the initial conditions. The occurrence of separatrix splitting explains a mechanism for the existence of trajectories to cross the separatrices on the classical phase plane in the expectation value dynamics. Such separatrix splitting does not occur in standard systems of Hagedorn and Heller for the semiclassical Gaussian wave packet dynamics as well as in the classical systems. We illustrate our theory for the potential of a simple pendulum and give numerical computations for the stable and unstable manifolds in the semiclassical system as well as solutions crossing the separatrices.

Modeling conformational changes in alginic acid oligomers induced by external forces

In this study, the mechanism and nature of mechanical force-induced conformational transitions of alginate oligomers with different ratios of β-D-mannuronic acid (M unit) and α-L-guluronic acid (G unit) units were investigated. The influence of the type of glycosidic linkage in either homo- or heterooligomers on the nature of conformational transitions was also considered. For this purpose, two different theoretical methods were used: quantum mechanics (QM) at the DFT level with the EGO (Enforced Geometry Optimization) approach previously tested also for other saccharide systems, and molecular dynamics (MD) simulations within hybrid interaction potentials, which take into account both the ab initio (QM) level of theory and classical molecular mechanics (MM) force fields. This allowed to characterize in detail the structural and energetic properties of the conformational transition occurring upon the influence of external, mechanical forces (e.g. ring conformations at the path of ring-inversion process as well as the energies corresponding to initial, final and intermediate states). The results indicate qualitatively various responses against the applied force, depending on the G:M ratio, which have their source in differing topologies of glycosidic linkage in either G or M units. This is of potential relevance for determining the content of naturally heterogeneous alginate chains by the AFM experimental studies. The effects of explicit solvent and non-zero temperature are not of primarily importance in the context of determined stretching properties.

Solving the Hele–Shaw flow using the Harrow–Hassidim–Lloyd algorithm on superconducting devices: A study of efficiency and challenges

The development of quantum processors for practical fluid flow problems is a promising yet distant goal. Recent advances in quantum linear solvers have highlighted their potential for classical fluid dynamics. In this study, we evaluate the Harrow–Hassidim–Lloyd (HHL) quantum linear systems algorithm (QLSA) for solving the idealized Hele–Shaw flow. Our focus is on the accuracy and computational cost of the HHL solver, which we find to be sensitive to the condition number, scaling exponentially with problem size. This emphasizes the need for preconditioning to enhance the practical use of QLSAs in fluid flow applications. Moreover, we perform shots-based simulations on quantum simulators and test the HHL solver on superconducting quantum devices, where noise, large circuit depths, and gate errors limit performance. Error suppression and mitigation techniques improve accuracy, suggesting that such fluid flow problems can benchmark noise mitigation efforts. Our findings provide a foundation for future, more complex application of QLSAs in fluid flow simulations.

Hydrodynamic modulation instability triggered by a two-wave system.

The modulation instability (MI) is responsible for the disintegration of a regular nonlinear wave train and can lead to strong localizations in the form of rogue waves. This mechanism has been studied in a variety of nonlinear dispersive media, such as hydrodynamics, optics, plasma, mechanical systems, electric transmission lines, and Bose-Einstein condensates, while its impact on applied sciences is steadily growing. It is well-known that the classical MI dynamics can be triggered when a pair of small-amplitude sidebands are excited within a particular frequency range around the main peak frequency. That is, a three-wave system, consisting of the carrier wave together with a pair of unstable sidebands, is usually adopted to initiate the wave focusing process in a numerical or laboratory experiment. Breather solutions of the nonlinear Schrödinger equation (NLSE) revealed that MI can generate much more complex localized structures, beyond the three-wave system initialization approach or by means of a continuous spectrum. In this work, we report an experimental study for deep-water surface gravity waves asserting that a MI process can be triggered by a single unstable sideband only, and thus, initialized from a two-wave process when including the contribution of the peak frequency. The experimental data are validated against fully nonlinear hydrodynamic numerical wave tank simulations and show very good agreement. The long-term evolution of such unstable wave trains shows a distinct shift in the recurrent Fermi-Pasta-Ulam-Tsingou focusing cycles, which are captured by the NLSE and fully nonlinear hydrodynamic simulations with some distinctions.

Utilization of cryogenic phases of helium as “magic” working fluids in laboratory experiments: Pushing the boundaries of fluid dynamics

A surprisingly large part of the valuable scientific contributions of K. R. Sreenivasan—known to his friends and coworkers as Sreeni—to fluid dynamics is directly connected with precisely controlled benchmark experiments with cryogenic phases of helium. These experiments have been performed in a number of leading low temperature laboratories using conventional viscous 4He and 3He gases, normal liquid 4He (He I), as well as liquid 3He–4He mixtures and superfluid phases of 4He (He II) and 3He-B. We review their extraordinary physical properties and describe selected experiments, focusing on various forms of thermal convection, which push the boundaries of classical and quantum fluid dynamics.

Fluorescence labeling-based differential scanning fluorimetry, an effective method for protein thermal stability and protein-compound binding analysis

Differential scanning fluorimetry (DSF) is widely used to assess protein thermal stability and protein-ligand interaction. However, its utility is often limited by the presence of detergents, which can affect hydrophobic binding. To tackle this issue, we developed an effective fluorescence-labeled DSF (FL-DSF) technique that tracks protein denaturation by monitoring the labeling fluorescence decrease, thus overcoming challenges typically encountered with traditional DSF methods. In this research, FL-DSF was first validated using Peroxisome Proliferators-Activated Receptor γ (PPARγ), Retinoid X Receptor α (RXRα), and Lysozyme, confirming its accuracy in determining melting curves. Expectedly, FL-DSF also exhibited strong compatibility with detergents in our investigations. Besides this, a new calculation method was proposed to characterize the protein denaturation process and evaluate protein-ligand binding. This mathematical model goes beyond traditional approaches, which simply treated the melting temperature (TM) shift as a concentration-dependent variable. Instead, it comprehensively incorporates the influence of irreversible denaturation-induced native protein loss on the equilibrium of protein-ligand binding. This methodology was successfully applied into the evaluation of binding affinity for 2 classical binding systems of PPARγ-Rosiglitazone and RXRα-CD3254. It was also utilized for the following binding screening studies, leading to the discovery of promising ligands for PPARγ, RXRα, and Lysozyme.