Class Of Physical Problems Research Articles

Simulating unsteady flows on a superconducting quantum processor

Recent advancements of quantum technologies have triggered tremendous interest in exploring practical quantum advantage. The simulation of fluid dynamics, a highly challenging problem in classical physics but vital for practical applications, emerges as a potential direction. Here, we report an experiment on the digital simulation of unsteady flows with a superconducting quantum processor. The quantum algorithm is based on the Hamiltonian simulation using the hydrodynamic formulation of the Schrödinger equation. With the median fidelities of 99.97% and 99.67% for parallel single- and two-qubit gates respectively, we simulate the dynamics of a two-dimensional (2D) compressible diverging flow and a 2D decaying vortex with ten qubits. Note that the former case is an inviscid potential flow, and the latter one is an artificial vortical flow with an external body force. The experimental results well capture the temporal evolution of averaged density and momentum profiles, and qualitatively reproduce spatial flow fields with moderate noises. This work demonstrates the potential of quantum computing in simulating more complex flows, such as turbulence, for practical applications.

Statistical Dynamics and Subgrid Modelling of Turbulence: From Isotropic to Inhomogeneous

Turbulence is the most important, ubiquitous, and difficult problem of classical physics. Feynman viewed it as essentially unsolved, without a rigorous mathematical basis to describe the statistical dynamics of this most complex of fluid motion. However, the paradigm shift came in 1959, with the formulation of the Eulerian direct interaction approximation (DIA) closure by Kraichnan. It was based on renormalized perturbation theory, like quantum electrodynamics, and is a bare vertex theory that is manifestly realizable. Here, we review some of the subsequent exciting achievements in closure theory and subgrid modelling. We also document in some detail the progress that has been made in extending statistical dynamical turbulence theory to the real world of interactions with mean flows, waves and inhomogeneities such as topography. This includes numerically efficient inhomogeneous closures, like the realizable quasi-diagonal direct interaction approximation (QDIA), and even more efficient Markovian Inhomogeneous Closures (MICs). Recent developments include the formulation and testing of an eddy-damped Markovian anisotropic closure (EDMAC) that is realizable in interactions with transient waves but is as efficient as the eddy-damped quasi-normal Markovian (EDQNM). As a similarly efficient closure, the realizable eddy-damped Markovian inhomogeneous closure (EDMIC) has been developed. Moreover, we present subgrid models that cater for the complex interactions that occur in geophysical flows. Recent progress includes the determination of complete sets of subgrid terms for skilful large-eddy simulations of baroclinic inhomogeneous turbulent atmospheric and oceanic flows interacting with Rossby waves and topography. The success of these inhomogeneous closures has also led to further applications in data assimilation and ensemble prediction and generalization to quantum fields.

Identifying regions of importance in wall-bounded turbulence through explainable deep learning

Despite its great scientific and technological importance, wall-bounded turbulence is an unresolved problem in classical physics that requires new perspectives to be tackled. One of the key strategies has been to study interactions among the energy-containing coherent structures in the flow. Such interactions are explored in this study using an explainable deep-learning method. The instantaneous velocity field obtained from a turbulent channel flow simulation is used to predict the velocity field in time through a U-net architecture. Based on the predicted flow, we assess the importance of each structure for this prediction using the game-theoretic algorithm of SHapley Additive exPlanations (SHAP). This work provides results in agreement with previous observations in the literature and extends them by revealing that the most important structures in the flow are not necessarily the ones with the highest contribution to the Reynolds shear stress. We also apply the method to an experimental database, where we can identify structures based on their importance score. This framework has the potential to shed light on numerous fundamental phenomena of wall-bounded turbulence, including novel strategies for flow control.

APPLICATION OF THERMAL ANEMOMETRY FOR TURBULENT FLOW ASSESSMENT

This paper is devoted to a review of the physical principle and effective methodology of thermal anemometry. Its uniqueness lies in the ability to realize a comprehensive high-frequency assessment of the dynamic and thermal state of the turbulent flow. It is worth noting that the American Nobel Prize winner in physics Richard Feynman called turbulence "the most important unsolved problem of classical physics", since there is no direct description of this phenomenon according to classical principles. Therefore, its physical interpretation is still considered one of the six most important mathematical issues of our time. Thus, despite its long history, thermal anemometry remains one of the leading techniques for studying of turbulent flow. Which has a significant impact on hydromechanical and heat-mass exchange processes, particularly in food technology. One of the most common types of thermal anemometry is the Hot-Wire anemometer, where a thin platinum wire is used as a sensor. The principle of operation of the anemometer is to maintain a constant heating of the wire while it is cooled by the surrounding flow of liquid or gas. Thus, the power required to compensate for the thermal state of the sensor wire correlates with the flow rate. The first part of the paper is devoted to the description of the physical principles of thermal anemometry, its advantages and limitations. In particular, special attention is paid to the mathematical interpretation of the heat transfer process between the incremental element of the sensor wire and the surrounding flow. After that, the paper provides a detailed analysis of the design features and practical application of various types of wire sensors. Finally, the last section discusses the calibration methodology and various approaches to the linearization procedure of the obtained calibration curves. Among them, the linearization method based on the Collis-Williams law deserves special attention, since it provides highly accurate interpolation of calibration data and takes into account the temperature compensation of the sensor.

Metasurfaces for Amplitude-Tunable Superposition of Plasmonic Orbital Angular Momentum States.

The superposition of orbital angular momentum (OAM) in a surface plasmon polariton (SPP) field has attracted much attention in recent years for its potential applications in classical physics problems and quantum communications. The flexible adjustment of the amplitudes of two OAM states can provide more freedom for the manipulation of superposed states. Here, we propose a type of plasmonic metasurface consisting of segmented spiral-shaped nanoslits that not only can generate the superposition of two OAM states with arbitrary topological charges (TCs), but also can independently modulate their relative amplitudes in a flexible manner. The TCs of two OAM states can be simultaneously modulated by incident light, the rotation rate of the nanoslits, and the geometric parameters of the segmented spiral. The relative amplitudes of the two OAM states are freely controllable by meticulously tuning the width of the nanoslits. Under a circularly polarized beam illumination, two OAM states of opposite TCs can be superposed with various weightings. Furthermore, hybrid superposition with different TCs is also demonstrated. The presented design scheme offers an opportunity to develop practical plasmonic devices and on-chip applications.

Multipole infinite series expansion of the Coulomb repulsion term using analytical spherical Bessel-like functions

Special functions are particular mathematical functions which arise in the solutions of various classical problems in physics. The functions are quite useful in mathematical analysis. Mostly, they find applications in applied mathematics, theoretical physics, and engineering. Special functions are usually uniquely described by a generating function, a Rodrigues’ formula, and a recurrence relation. In this work, we derive analytical spherical Bessel-like functions corresponding to the infinite degree power series expansion of the functions obtained in the multipole series expansion of the Coulomb repulsion term. The convergence of the truncated power series expansion of the functions with the corresponding derived analytical functions, as well as the comparison of analytical functions of the first and second kinds, is analyzed. Rodrigues’ formula and a recurrence relation for obtaining these analytical functions are also presented. It is shown that the higher order even and odd spherical Bessel-like functions of the first kind are derivable from the lowest order functions of the first and second kinds, respectively.

Producing and imaging quantum turbulence via pair-breaking in superfluid He3−B

Turbulence remains one of the last unsolved problems of classical physics. Here, the authors present insights from the nucleation and evolution of the simpler quantum turbulence. They generate turbulence mechanically, and image the dynamics nondestructively through illumination of thermal quasiparticles. At low temperatures, the ballistic quasiparticles are Andreev-reflected by the vortex flow fields, producing a shadow which shows that the tangle is asymmetrical. This suggests that the vortex rings - precursors of quantum turbulence - are unevenly created by the generator.

N-Body Simulation Inspired by Metaheuristics Optimization

The N-body problem in classical physics, is the calculation of force of gravitational attraction of heavenly bodies towards each other. Solving this problem for many heavenly bodies has always posed a challenge to physicists and mathematicians. Large number of bodies, huge masses, long distances and exponentially increasing number of equations of motion of the bodies have been the major hurdles in solving this problem for large and complex galaxies. Advent of high performance computational machines have mitigated the problem to much extent, but still for large number of bodies it consumes huge amount of resources and days for computation. Conventional algorithms have been able to reduce the computational complexity from to by splitting the space into a tree or mesh network, researchers are still looking for improvements. In this research work we propose a novel solution to N-body problem inspired by metaheuristics algorithms. The proposed algorithm is simulated for various time periods of selected heavenly bodies and analyzed for speed and accuracy. The results are compared with that of conventional algorithms. The outcomes show about 50% time saving with almost no loss in accuracy. The proposed approach being a metaheuristics optimization technique, attempts to find optimal solution to the problem, searching the entire space in a unique and efficient manner in a very limited amount of time.

Cavitation upon low-speed solid\u2013liquid impact

When a solid object impacts on the surface of a liquid, extremely high pressure develops at the site of contact. Von Karman’s study of this classical physics problem showed that the pressure on the bottom surface of the impacting body approaches infinity for flat impacts. Yet, in contrast to the high pressures found from experience and in previous studies, we show that a flat-bottomed cylinder impacting a pool of liquid can decrease the local pressure sufficiently to cavitate the liquid. Cavitation occurs because the liquid is slightly compressible and impact creates large pressure waves that reflect from the free surface to form negative pressure regions. We find that an impact velocity as low as ~3 m/s suffices to cavitate the liquid and propose a new cavitation number to predict cavitation onset in low-speed solid-liquid impact-scenarios. These findings imply that localized cavitation could occur in impacts such as boat slamming, cliff jumping, and ocean landing of spacecraft.

Hertz elastic dynamics of two colliding elastic spheres

This paper revisits a classic problem in physics — Hertz elastic dynamics of two colliding elastic spheres. This study obtains impact period in terms of hypergeometric function and successfully combines Deresiewicz’s three segmental solutions into one single solution. Our numerical investigation confirms that Deresiewicz’s inversion is a good approximation. As an essential part of this study, a general Maple code is provided.

Classical and Quantum Mechanical Models of Many-Particle Systems

The collective behaviour of many-particle systems is a common denominator in the challenges of a highly diverse range of applications: from classical problems in Physics (gas dynamics e.g. Boltzmann’s equation, plasma dynamics e.g. various Vlasov equations, semiconductors, quantum mechanics) to current models in biology (kinetic models for collective interaction e.g. swarming, evolution of trait-structured species) to rising topics in social sciences (opinion formation, crowding phenomena) and economics (wealth distribution, mean-field games). Key mathematical questions concern the analysis (global-in-time wellposedness, regularity), rigorous scaling resp. macroscospic limits (model reduction from many-particle models to mean-field/mesoscopic descriptions to macroscopic evolutions), efficient and asymptotic preserving numerical methods and qualitative results (e.g. large-time equilibration).

Gödel’s Incompleteness Theorem and Problems Outside Quantum Mechanics

Gödel’s Incompleteness Theorem is considered a significant finding in logic and mathematics and has many indications in physics. This paper analyzes and concludes that Gödel’s Incompleteness theorem potentially parallels problems of reference frame in classical physics and other issues considering self-inclusion with regard to consciousness, which not necessarily relevant to quantum mechanics. It also demonstrates further investigations on Gödel’s Incompleteness theorem, which can add to our understanding of problems in classical physics and philosophy of mind that still require in-depth research.

Variable-Order Fractional Models for Wall-Bounded Turbulent Flows.

Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with relatively slow progress in the last few decades beyond the log law, which only describes the intermediate region in wall-bounded turbulence, i.e., 30–50 y+ to 0.1–0.2 R+ in a pipe of radius R. Here, we propose a fundamentally new approach based on fractional calculus to model the entire mean velocity profile from the wall to the centerline of the pipe. Specifically, we represent the Reynolds stresses with a non-local fractional derivative of variable-order that decays with the distance from the wall. Surprisingly, we find that this variable fractional order has a universal form for all Reynolds numbers and for three different flow types, i.e., channel flow, Couette flow, and pipe flow. We first use existing databases from direct numerical simulations (DNSs) to lean the variable-order function and subsequently we test it against other DNS data and experimental measurements, including the Princeton superpipe experiments. Taken together, our findings reveal the continuous change in rate of turbulent diffusion from the wall as well as the strong nonlocality of turbulent interactions that intensify away from the wall. Moreover, we propose alternative formulations, including a divergence variable fractional (two-sided) model for turbulent flows. The total shear stress is represented by a two-sided symmetric variable fractional derivative. The numerical results show that this formulation can lead to smooth fractional-order profiles in the whole domain. This new model improves the one-sided model, which is considered in the half domain (wall to centerline) only. We use a finite difference method for solving the inverse problem, but we also introduce the fractional physics-informed neural network (fPINN) for solving the inverse and forward problems much more efficiently. In addition to the aforementioned fully-developed flows, we model turbulent boundary layers and discuss how the streamwise variation affects the universal curve.

Fractional values of orbital angular momentum in problems of classical physics

Classical field theory problems for which the presence of nontrivial topological Pauli phase is essential (i.e. fractional values of the orbital angular momentum are possible in two-dimensional case) are discussed within the two-dimensional Helmholtz equation. As examples, we consider the “wedge problem” — determination of the field created by a point charge placed between two conducting half-planes — and the Fresnel diffraction from knife edge.

On essential self-adjointness for first order differential operators on domains in $\mathbb R^d$

We consider general symmetric systems of first order linear partial differential operators on domains \Omega \subset \mathbb R^d , and we seek sufficient conditions on the coefficients which ensure essential self-adjointness. The coefficients of the first order terms are only required to belong to C^1 (\Omega) and there is no ellipticity condition. Our criterion writes as the completeness of an associated Riemannian structure which encodes the propagation velocities of the system. As an application we obtain sufficient conditions for confinement of energy for certain wave propagation problems of classical physics.

Multiple states in turbulence

Turbulence is ubiquitous in nature, and is known as one of the unresolved problems in classical physics. The classical theory of turbulence assumed that turbulence is ergodic, which means that when turbulence is in a stationary state, even though the instantaneous properties are sensitive to initial conditions, the statistical averages of the instantaneous properties, such as the mean profiles and the skin friction in wall-bounded turbulence, are unique at any fixed set of parameters. This classical ergodic theory is the foundation of turbulence theory and modeling, and makes it possible to extrapolate data from lower Reynolds number turbulence to higher ones. However, in recent years, a few experimental and numerical evidences showed that multiple states exist in several flow problems. That is, the turbulent statistics and flow structures are not the same even at the same control parameters. In this paper, several published flow problems with multiple states are reviewed, including Rayleigh-Benard convection (RBC), von Karman flow (VKF), Taylor-Couette flow (TCF), spherical Couette flow (SCF), rotating homogeneous turbulence with Taylor-Green forcing (TGF), and the spanwise rotating plane Couette flows (RPCF). Two different categories of multiple states can be observed from these flow problems. One is in RBC, where several research groups found that the system will have different flow states and it will switch between different states as time evolves. The other can be generally seen from other five flows, where initial conditions or hysteresis effect can be observed. For example, hysteresis loops were reported in the experiments of TCF for global torque and local velocities at very high Reynolds numbers, while in RPCF, different flow statistics and flow structures were obtained with different initial flow fields based on the same code, the same computational domain and the same grid resolution at the same control parameters. Whether the second type of multiple states will finally turn into the first one in a long enough time is an open question which deserves further investigations. The underlying mechanism of multiple states is not clear at the present moment, which also demands continued effort and studies. In the six flow problems mentioned above, large-scale flow structures persist, which may highlight the importance of the coherent structures and their selectability in turbulence, as concluded by Huisman et al. Another possible guess from myself is the multiple competing flow mechanisms in the flow problems with multiple states. For example, in rotating homogeneous turbulence with Taylor-Green forcing (TGF), there are two competing mechanisms, one is the viscosity dissipation which causes the forward cascade, the other is the system rotation which results in the inverse cascade. The two competing mechanisms will make the system has more equilibrium points and multiple states. I believe that there will be more and more flow problems with multiple states reported in the future, together with a better understanding of the underling mechanism.

Rigorous Hamiltonian and Lagrangian analysis of classical and quantum theories with minimal length

GUP is a phenomenological model aimed for a description of a minimal length in quantum and classical systems. However, the analysis of problems in classical physics is usually approached preferring a different formalism than the one used for quantum systems, and vice versa. Potentially, the two approaches can result in inconsistencies. Here, we eliminate such inconsistencies proposing particular meanings and relations between the variables used to describe physical systems, resulting in a precise form of the Legendre transformation. Furthermore, we introduce two different sets of canonical variables and the relative map between them. These two sets allow for a complete and unambiguous description of classical and quantum systems.

Ubiquity of the Sturm-Liouville problem in multilayer systems

We postulate a general Lagrangian density which leads to the equations of motion of the well-known cases of the electromagnetic field, the theory of elasticity, and some other particular problems in classical physics. When the mentioned cases are studied in multilayer systems, i.e., when the constitutive parameters depend on one of the Cartesian coordinates, all of them lead us to a matrix Sturm-Liouville problem whose equation of motion, in its most general form, can be derived from the postulated Lagrangian density too. These results demonstrate the ubiquitous character of the Sturm-Liouville matrix problem and consequently the relevance of its study from the mathematical, physical and numerical point of view. The consequences of this curious result are briefly analyzed.

An Eulerian projection method for quasi-static elastoplasticity

A well-established numerical approach to solve the Navier–Stokes equations for incompressible fluids is Chorin's projection method [1], whereby the fluid velocity is explicitly updated, and then an elliptic problem for the pressure is solved, which is used to orthogonally project the velocity field to maintain the incompressibility constraint. In this paper, we develop a mathematical correspondence between Newtonian fluids in the incompressible limit and hypo-elastoplastic solids in the slow, quasi-static limit. Using this correspondence, we formulate a new fixed-grid, Eulerian numerical method for simulating quasi-static hypo-elastoplastic solids, whereby the stress is explicitly updated, and then an elliptic problem for the velocity is solved, which is used to orthogonally project the stress to maintain the quasi-staticity constraint. We develop a finite-difference implementation of the method and apply it to an elasto-viscoplastic model of a bulk metallic glass based on the shear transformation zone theory. We show that in a two-dimensional plane strain simple shear simulation, the method is in quantitative agreement with an explicit method. Like the fluid projection method, it is efficient and numerically robust, making it practical for a wide variety of applications. We also demonstrate that the method can be extended to simulate objects with evolving boundaries. We highlight a number of correspondences between incompressible fluid mechanics and quasi-static elastoplasticity, creating possibilities for translating other numerical methods between the two classes of physical problems.

Classical and quantum framing of the Now

Mermin replies: Understanding the Now was not the main purpose of my March 2014 commentary. My primary point was that a new way of resolving the puzzles and paradoxes of quantum mechanics, called QBism by Christopher Fuchs and Rüdiger Schack, was also able to settle that longstanding problem in classical physics. After my earlier commentary on QBism (Physics Today, July 2012, page 8), I was concerned that many of the letters about it (December 2012, page 8) had sounded the same theme: that the letter writer’s own way of looking at quantum mechanics was already perfectly satisfactory. I hoped that by applying QBist thinking to a strictly classical problem, I could disengage those readers from their favorite interpretation of quantum mechanics and help them think about QBism on its own merits. I recently emphasized in Nature11. N. D. Mermin, Nature 507, 421 (2014). https://doi.org/10.1038/507421a that QBism sheds light on classical physics, too, but there I discussed quantum applications of QBism as well as classical applications (“CBism”).The letters here all address only the problem of the Now but not the fact that I deal with it through a classical application of QBism. While I’m disappointed that they say nothing about QBism or CBism, I’m pleased that they all agree that the problem of the Now is indeed a problem. Not everybody does.Berge Tatian and B. K. Ridley both criticize me for taking the Now to be a point. But I don’t. I call the Now an event “whose duration and location are restricted enough that it can usefully be represented as a point in space and time.” I say that “the events we experience are complex, extended entities” and that “to represent our actual experiences as a collection of mathematical points … is a brilliant strategic simplification, but we ought not to confuse a cartoon … with the experience itself.”Ridley’s comment “Nor could one person’s Now be exactly the same as another’s” suggests that I say it could be. What I do say is “The commonality of my Now and your Now whenever we are together requires that our Nows must coincide at each of two consecutive meetings.” “Commonality” or “coincide” mean only that our two private Now experiences happen at a common place and time, not that they are identical. Indeed, the personal experiences of different people are incomparable, except through the imperfect medium of language.I agree with James Hartle on much in the paper he cites, but we have important differences. He takes spacetime to be objective and fundamental; I take it to be an abstract tool used by an agent to organize her experience. He uses the notion of a point in spacetime uncritically; I regard it as an approximate representation of an agent’s spatially and temporally extended experience. He takes an agent’s experience to be an objective property of the agent, like the contents of a register. I take an agent’s experience to be private and self-evident to that agent and to be the fundamental basis for her inference of an external world; the experience of each agent plays a special role for that—and only that—agent, analogous to the special role played by “the classical domain” in the quantum mechanics of Lev Landau and Evgeny Lifshitz.Rudolf Peierls wrote to John Bell in 1980, “In my view, a description of the laws of physics consists in giving us a set of correlations between successive observations. By observations I mean … what our senses can experience. That we have senses and can experience such sensations is an empirical fact, which has not been deduced (and in my opinion cannot be deduced) from current physics.”22. S. Lee, ed., Sir Rudolf Peierls: Selected Private and Scientific Correspondence, vol. 2, World Scientific, River Edge, NJ (2009), p. 807. If “us” is expanded to “each of us,” then nobody has ever put QBism and CBism more concisely than that.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. N. D. Mermin, Nature 507, 421 (2014). https://doi.org/10.1038/507421a, Google ScholarCrossref, ISI2. S. Lee, ed., Sir Rudolf Peierls: Selected Private and Scientific Correspondence, vol. 2, World Scientific, River Edge, NJ (2009), p. 807. Google ScholarCrossref© 2014 American Institute of Physics.