Hasil untuk "Analytic mechanics"

B. Budiansky, J. Hutchinson, A. Evans

L. Meirovitch

A. Sahbani, S. El-Khoury, P. Bidaud

P. Charbonneau, J. Kurchan, G. Parisi et al.

Glasses are amorphous solids whose constituent particles are caged by their neighbours and thus cannot flow. This sluggishness is often ascribed to the free energy landscape containing multiple minima (basins) separated by high barriers. Here we show, using theory and numerical simulation, that the landscape is much rougher than is classically assumed. Deep in the glass, it undergoes a ‘roughness transition’ to fractal basins, which brings about isostaticity and marginal stability on approaching jamming. Critical exponents for the basin width, the weak force distribution and the spatial spread of quasi-contacts near jamming can be analytically determined. Their value is found to be compatible with numerical observations. This advance incorporates the jamming transition of granular materials into the framework of glass theory. Because temperature and pressure control what features of the landscape are experienced, glass mechanics and transport are expected to reflect the features of the topology we discuss here. Potential energy landscape models are often used to describe transitions in the glassy state. Here, the authors report that the landscape is much rougher than usually assumed, and demonstrate that it undergoes a transition to fractal basins before the jamming point is reached.

Fiona P. McDonald, emilie isch, Suzi Asa et al.

In qualitative research, what happens when reflexivity is made explicit? In some instances, researchers may encounter phenomena that run counter to the cultural norms and expectations that shape their everyday lives. We present in this article a qualitative protocol titled A Method for Creative Reflexive Data in Autoethnography—an experimental method for creative reflexive data that enables researchers to follow a step-by-step protocol using drawing. This is achieved by presenting the analytical mechanics underlying each step of the experimental method, thereby making a researcher's reflexive process more visible in data production. We share two objectives in this article: First, to present a programmatic case for the protocol as a (more-than-reproducible) way of making any qualitative method more explicit; second, to report on the specifics of a new method that helps render the analytic process visible at each step.

S.B. Malyshev, S.V. Sudoplatov

A series of geometrical and topological properties induced by structures, including degeneration, modularity, local modularity, projectivity, local finiteness, etc., play an important role in clarifying structural relationships and in classifying basic and derivative semantical and syntactical objects related to a given class of structures and their valuable characteristics. It is natural to turn to the family of all structures on a given finite or infinite universe, which allows us to represent all possible structures of a given cardinality up to the definability and to describe relationships, possibilities of preserving and violating structural properties during enrichments and restrictions of structures within the framework of the chosen family. This paper studies the behavior of pregeometry types (degenerate, locally finite, modular) within the Boolean algebra B(M) of regular expansions and reducts of a relational structure M. We establish criteria for the inheritance of pregeometry properties under Boolean operations, proving that degeneracy and local finiteness are preserved under intersections. In contrast, we show through counterexamples that modularity generally fails to be preserved, as does local finiteness under unions. We formulate a sufficient condition of linear growth of the closure operator under which the union of locally finite structures remains locally finite. These results reveal a fundamental asymmetry between intersection and union operations, contributing to geometric stability theory by delineating the preservation boundaries of pregeometries in Boolean families of structures.

Victor M. Pergamenshchik, Taras Bryk, Andrij Trokhymchuk

We turn the long time puzzle of the free volume, known for its highly irregular form, into exact analytical formulae and develop statistical mechanics of the hard disk model. The free volume is exactly expressed in terms of the intersection areas of up to five exclusion circles, which can be computed analytically as functions of disk coordinates. In turn, the free volume determines the partition function and entropy. The partition function is shown to factorize into a product of free volumes and admits two exact limiting forms corresponding to gaslike and liquidlike regimes. From this construction, using Monte Carlo-generated disk coordinates, the entropy and pressure are obtained analytically and recover the known equation of state of hard disks in almost entire density range up to the close packing. At intermediate densities, the theory reveals a mixed liquid regime associated with defect formation preceding the hexagonal ordering. The intersection area of five disks emerges as a scalar measure of the local hexagonal order. The theory can be directly adopted for the hard sphere model.

Aryaman Jha, Kurt Wiesenfeld, Jorge Laval

Motivated by earlier numerical evidence for a percolation-like transition in space-time jamming, we present an analytic description of the transient dynamics of the deterministic traffic model elementary cellular automaton rule 184 (ECA184). By exploiting the deterministic structure of the dynamics, we reformulate the problem in terms of a height function constructed directly from the initial condition, and obtain an equilibrium statistical mechanics-like description over the lattice configurations. This formulation allows macroscopic observables in space-time, such as the total jam delay and jam relaxation time, as well as microscopic jam statistics, to be expressed in terms of geometric properties of the height function. We thereby derive the associated scaling forms and recover the critical exponents previously observed in numerical studies. We discuss the physical implications of this space-time geometric approach.

N.S. Tokmagambetov, B.K. Tolegen

In this article, new q-analogues of Lyapunov-type inequalities are presented for two-point fractional boundary value problems involving the Riemann–Liouville fractional q-derivative with well-posed q-boundary conditions. The study relies on the properties of the q-Green’s function, which is constructed to solve such problems and allows for the analytical derivation of the inequalities. These inequalities find application in two directions: establishing precise lower bounds for the eigenvalues of corresponding q-fractional spectral problems and formulating criteria for the absence of real zeros in q-analogues of Mittag-Leffler functions. The obtained results generalize classical and fractional Lyapunov inequalities, offering new perspectives for the analysis of stability and spectral properties of q-fractional differential systems. The relevance of the work is driven by the growing interest in q-calculus in discrete models, such as viscoelastic systems or quantum circuits, where discrete dynamics play a key role. The convenience of closed-form analytical expressions makes the results practically applicable. The research lays the foundation for further generalizations, including Caputo derivatives or multidimensional q-systems, which may stimulate new discoveries in discrete fractional analysis.

D. Swarnakar, R. Kumar, V. Ganesh Kumar et al.

A proposed numerical approximation method is presented for solving a singularly perturbed second-order differential-difference equation with both the delay and advance shifts. The algorithm utilises a nonpolynomial spline with a fitting factor finite difference scheme. The application of finite difference approximations for higher order derivatives leads to the derivation of a tri-diagonal system. To efficiently solve this system of equations, an algorithm based on discrete invariant imbedding is employed and the stability of the method is analysed. An assessment of the applicability and efficiency of the proposed scheme is conducted by performing three numerical experiments and comparing the results with other methods. The maximum absolute errors are used as the basis for comparison. The impact of minor shifts on the boundary layer behaviour of the solution is illustrated using plotted graphs featuring different degrees of shifts. The method is theoretically and numerically analysed using uniformly convergent solutions with quadric convergence rate.

A.T. Assanova, Zh.M. Kadirbayeva, R.A. Medetbekova et al.

In this article, the problem for a differential-algebraic equation with a significant loads is studied. Unlike previously studied problems for differential equations with a significant loads, in the considered equation, there is a matrix in the left part with a derivative that is not invertible. Therefore, the system of equations includes both differential and algebraic equations. To solve the problem, we propose a modification of the Dzhumabaev’s parametrization method. The considered problem is reduced to a parametric problem for the differential-algebraic equation with significant loads. We apply the Weierstrass canonical form to this problem. We obtain parametric initial value problem for a differential equations and an algebraic equations with a significant loads. The solvability conditions for the considered problem are established.

Leonardo Di G. Sigalotti, Otto Rendón, José-Rubén Luévano

We show that a physical correspondence between Brownian motion and quantum mechanics can be established by formal analytic continuation if Wick rotation (t→it) is replaced by the mirror symmetric, complex conjugate time transformation t→±it. Invariance of the square modulus of the wave function under this transformation reveals a tight connection between Born's interpretation of the probability density and the proper time t2=(−it)(+it), as being both the result of a time symmetry breaking in the informational content of the microscopic (quantum) world, which takes place as we move to a macroscopic level. We find that under the transformation t→±it, the Schrödinger equation and its complex conjugate conforms with a stochastic-mechanical model of the Brownian motion of a particle. From the present analysis, Nelson's osmotic velocity arises naturally and the quantum phase function admits a classical interpretation in terms of a continuous (scalar) potential field that influences the motion of the particle. The maximum phase variation defines the direction of the osmotic and current velocity. Whereas the paper focuses on the correlation between quantum mechanics and Ficks's second law for a constant diffusion coefficient, a discussion is also provided on the possibility of a quantum mechanical formulation for anomalous diffusion, where the diffusion coefficient is a function of space and/or time.

M. L. Kerr, G. De Rosi, K. V. Kheruntsyan

We present a comprehensive review on the state-of-the-art of the approximate analytic approaches describing the finite-temperature thermodynamic quantities of the Lieb-Liniger model of the one-dimensional (1D) Bose gas with contact repulsive interactions. This paradigmatic model of quantum many-body-theory plays an important role in many areas of physics -- thanks to its integrability and possible experimental realization using, e.g., ensembles of ultracold bosonic atoms confined to quasi-1D geometries. The thermodynamics of the uniform Lieb-Liniger gas can be obtained numerically using the exact thermal Bethe ansatz (TBA) method, first derived in 1969 by Yang and Yang. However, the TBA numerical calculations do not allow for the in-depth understanding of the underlying physical mechanisms that govern the thermodynamic behavior of the Lieb-Liniger gas at finite temperature. Our work is then motivated by the insights that emerge naturally from the transparency of closed-form analytic results, which are derived here in six different regimes of the gas and which exhibit an excellent agreement with the TBA numerics. Our findings can be further adopted for characterising the equilibrium properties of inhomogeneous (e.g., harmonically trapped) 1D Bose gases within the local density approximation and for the development of improved hydrodynamic theories, allowing for the calculation of breathing mode frequencies which depend on the underlying thermodynamic equation of state. Our analytic approaches can be applied to other systems including impurities in a quantum bath, liquid helium-4, and ultracold Bose gas mixtures.

Hong Yuan, Chang-Pu Sun

To address the observation of Max Born (M. Born 1969) that the Newton's second law can emerge from a purely statistical perspective, we derive the evolution equation about the statistical distribution for dilute gas based solely on statistical principles, without invoking Newtonian mechanics, and then obtain the equations of motion for individual particles. Newton's second law for a single particle naturally emerges when the distribution reaches equilibrium. We demonstrate that the magnitude of an external force, traditionally measured by particle acceleration, can be understood as a measure of distribution inhomogeneity. We further show that the entropic force (utilized in current gravity studies) is equivalent to the statistical force and under non-equilibrium conditions, a deviation arises between the entropic force and the Newtonian force. This framework offers a novel perspective distinct from classical Newtonian mechanics and broadens the potential scope of its application.

Kristian Uldall Kristiansen, Peter Szmolyan

Any attracting, hyperbolic and proper node of a two-dimensional analytic vector-field has a unique strong-stable manifold. This manifold is analytic. The corresponding weak-stable manifolds are, on the other hand, not unique, but in the nonresonant case there is a unique weak-stable manifold that is analytic. As the system approaches a saddle-node (under parameter variation), a sequence of resonances (of increasing order) occur. In this paper, we give a detailed description of the analytic weak-stable manifolds during this process. In particular, we relate a ``flapping-mechanism'', corresponding to a dramatic change of the position of the analytic weak-stable manifold as the parameter passes through the infinitely many resonances, to the lack of analyticity of the center manifold at the saddle-node. Our work is motivated and inspired by the work of Merle, Raphaël, Rodnianski, and Szeftel, where this flapping mechanism is the crucial ingredient in the construction of $C^\infty$-smooth self-similar solutions of the compressible Euler equations.

A.R. Yeshkeyev, A.R. Yarullina, S.M. Amanbekov et al.

Given article is devoted to the study of semantic Jonsson quasivariety of universal unars of signature containing only unary functional symbol. The first section of the article consists of basic necessary concepts. There were defined new notions of semantic Jonsson quasivariety of Robinson unars JCU, its elementary theory and semantic model. In order to prove the main result of the article, there were considered Robinson spectrum RSp(JCU) and its partition onto equivalence classes [∆] by cosemanticness relation. The characteristic features of such equivalence classes [∆]∈RSp(JCU) were analysed. The main result is the following theorem of the existence of: characteristic for every class [∆] the meaning of which is Robinson theories of unars; class [∆] for any arbitrary characteristic; criteria of equivalence of two classes [∆]1, [∆]2. The obtained results can be useful for continuation of the various Jonsson algebras’ research, particularly semantic Jonsson quasivariety of S-acts over cyclic monoid.

Hasse N.J. Dekker, Woutijn J. Baars, Fulvio Scarano et al.

The unsteady flow behaviour of two side-by-side rotors in ground proximity is experimentally investigated. The rotors induce a velocity distribution interacting with the ground causing the radial expansion of the rotor wakes. In between the rotors, an interaction of the two wakes takes place, resulting in an upward flow similar to a fountain. Two types of flow topologies are examined and correspond to two different stand-off heights between the rotors and the ground: the first one where the height of the fountain remains below the rotor disks, and a second one where it emerges above, being re-ingested. The fountain unsteadiness is shown to increase when re-ingestion takes place, determining a location switch from one rotor disk to the other, multiple times during acquisition. Consequently, variable inflow conditions are imposed on each of the two rotors. The fountain dynamics is observed at a frequency that is about two orders of magnitude lower than the blade passing frequency. The dominant characteristic time scale is linked to the flow recirculation path, relating this to system parameters of thrust and ground stand-off height. The flow field is analysed using proper orthogonal decomposition, in which coupled modes are identified. Results from the modal analysis are used to formulate a simple dynamic flow model of the re-ingestion switching cycle.

M.I. Ramazanov

On August 22, 2023, the talented mathematician, specialist in the field of functional analysis and its applications, Doctor of Physical and Mathematical Sciences, Professor Kanguzhin Baltabek Esmatovich turned 70 years old.

S. Malek, L. Gibson

Abstract We investigate the elastic behavior of periodic hexagonal honeycombs over a wide range of relative densities and cell geometries, using both analytical and numerical approaches. Previous modeling approaches are reviewed and their limitations identified. More accurate estimates of all nine elastic constants are obtained by modifying the analysis of Gibson and Ashby (1997) to account for the nodes at the intersection of the vertical and inclined members. The effect of the nodes becomes significant at high relative densities. We then compare the new analytical equations with previous analytical models, with a numerical analysis based on a computational homogenization technique and with data for rubber honeycombs over a wide range of relative densities and cell geometries. The comparisons show that both the new analytical equations and numerical solutions give a remarkably good description of the data. The results provide new insights into understanding the mechanics of honeycombs and designing new cellular materials in the future.

G. Teichert, A. Natarajan, Anton Van der Ven et al.

Abstract The free energy of a system is central to many material models. Although free energy data is not generally found directly, its derivatives can be observed or calculated. In this work, we present an Integrable Deep Neural Network (IDNN) that can be trained to derivative data obtained from atomic scale models and statistical mechanics, then analytically integrated to recover an accurate representation of the free energy. The IDNN is demonstrated by training to the chemical potential data of a binary alloy with B2 ordering. The resulting DNN representation of the free energy is used in a mesoscopic, phase field simulation and found to predict the appropriate formation of antiphase boundaries in the material. In contrast, a B-spline representation of the same data failed to resolve the physics of the system with sufficient fidelity to resolve the antiphase boundaries. Since the fine scale physics harbors complexity that emerges through the free energy in coarser-grained descriptions, the IDNN represents a framework for scale bridging in materials systems.