This work presents a unified perspective on thermal equilibrium and quantum dynamics by examining the simplest quantum system, a qubit, as a minimal model. We show that both the thermal partition function and the Loschmidt amplitude can be understood as extensions of a single analytic function along different paths in the complex plane. The zeros of Loschmidt amplitude encode dynamical features such as orthogonality, rate function singularities, and quantum speed limits, in analogy with the role of partition function zeros in equilibrium statistical mechanics. We further establish, through the Cauchy-Riemann equations, that the high-temperature specific heat corresponds to early-time evolution. The discussion follows a pedagogical progression from a single qubit to an interacting spin chain, all with finite dimensional Hilbert spaces.
Quantum mechanics broadly classifies the particles into two categories: $(1)$ fermions and $(2)$ bosons. Fermions are half-integer spin particles, obeying Pauli's exclusion principle and Fermi-Dirac statistics. Whereas bosons are integer spin particles, not obeying Pauli's exclusion principle, and obeying Bose-Einstein statistics. However, there are two exceptions to this standard case: first, anyons, which exist in 2-dimensional systems, and secondly, paraparticles, which can exist in any dimension. Paraparticles follow their non-trivial parastatistics, obeying their generalised exclusion principle. In this paper, we provide a detailed review of the foundations of paraparticle statistics established in \cite{wang2025particle}. We extend this work further and then derive an important expression for the heat capacity of paraparticles for a specific case, which would provide a handle for the experimental detection of paraparticles in appropriate systems.
Abstract Time-varying mesh stiffness is considered the primary excitation in gear vibrations. An accurate evaluation of mesh stiffness is of great importance when studying gear vibrations. Based on contact mechanics and gear engagement principle, we propose an improved analytical model to determine gear mesh stiffness. Gears with addendum modification and varying-load conditions are considered in the presented model. It is shown that the gear mesh stiffness is load-dependent, and the contribution of each component to the gear mesh stiffness is discussed. Experimental investigations based on strain gauges are presented to measure the mesh stiffness of an internal spur gear under different loading torques. In addition, existing methods are compared and discussed with the presented model. The results indicate that the proposed model can calculate time-varying mesh stiffness accurately under non-stationary conditions.
Nur Hasan Mahmud Shahen, Md. Al Amin, Foyjonnesa
et al.
Abstract This article delves into the dynamic constructions of distinctive traveling wave solutions for wave circulation in shallow water mechanics, specifically addressing the time-fractional couple Drinfel’d–Sokolov–Wilson (DSW) equation. Introducing the previously untapped $$exp(-\phi (\xi ))$$ e x p ( - ϕ ( ξ ) ) -expansion method, we successfully generate a diverse set of analytic solutions expressed in hyperbolic, trigonometric, and rational functions, each with permitted parameters. Visualization through three-dimensional (3D) as well two-dimensional (2D) plots, including contour plots, reveals inherent wave phenomena in the DSW equation. These newly obtained wave solutions serve as a catalyst for refining theories in applied science, offering a means to validate mathematical simulations for the proliferation of waves in shallow water as well as other nonlinear scenarios. Obtained wave solutions demonstrate the bright soliton, periodic, multiple soliton, and kink soliton shape. The simplicity and efficacy of the implemented methods are demonstrated, providing a valuable tool for approximating the considered equation. All figures are devoted to demonstrate the complete wave futures of the attained solutions to the studied equation with the collaboration of specific selections of the chosen parameters. Moreover, it may have summarized that the attained wave solutions and their physical phenomena might be useful to comprehend the various kind of wave propagation in mathematical physics and engineering.
In the study the existence of solutions of a class of fractional integro-differential equations with boundary conditions was considered. The main tool, we employ, is the conventional monotone iterative technique, which is highly effective method to examine the quantitative and qualitative characteristics of various nonlinear problems. This technique produces monotone sequences whose iterations are unique solutions of the certain linear problems. These bounds converge uniformly to the maximal solutions of the given problems. Some types of coupled solutions are considered to obtain the claim of the main results under suitable conditions.
Diego A. Huyke, Alexandre S. Avaro, Thomas Kroll
et al.
Microfluidic mixers offer new possibilities for the study of fast reaction kinetics down to the microsecond time scale, and methods such as soft X-ray absorption spectroscopy are powerful analysis techniques. These systems impose challenging constraints on mixing time scales, sample volume, detection region size and component materials. The current work presents a novel micromixer and jet device which aims to address these limitations. The system uses a so-called ‘theta’ mixer consisting of two sintered and fused glass capillaries. Sample and carrier fluids are injected separately into the inlets of the adjacent capillaries. At the downstream end, the two streams exit two micron-scale adjoining nozzles and form a single free-standing jet. The flow-rate difference between the two streams results in the rapid acceleration and lamination of the sample stream. This creates a small transverse dimension and induces diffusive mixing of the sample and carrier stream solutions within a time scale of 0.9 microseconds. The reaction occurs at or very near a free surface so that reactants and products are more directly accessible to interrogation using soft X-ray. We use a simple diffusion model and quantitative measurements of fluorescence quenching (of fluorescein with potassium iodide) to characterize the mixing dynamics across flow-rate ratios.
Physical mechanisms of phase separation in living systems can play key physiological roles and have recently been the focus of intensive studies. The strongly heterogeneous and disordered nature of such phenomena in the biological domain poses difficult modeling challenges that require going beyond mean field approaches based on postulating a free energy landscape. The alternative pathway we take in this work is to tackle the full statistical mechanics problem of calculating the partition function in these systems, starting from microscopic interactions, by means of cavity methods. We illustrate the procedure first on the simple binary case, and we then apply it successfully to ternary systems, in which the naive mean field approximations are proved inadequate. We then demonstrate the agreement with lattice model simulations, to finally contrast our theory also with experiments of coacervate formation by associative de-mixing of nucleotides and poly-lysine in aqueous solution. In this way, different types of evidence are provided to support cavity methods as ideal tools for quantitative modeling of biomolecular condensation, giving an optimal balance between the accurate consideration of spatial aspects of the microscopic dynamics and the fast computational results rooted in their analytical tractability.
A boundary value problem for a fractionally loaded heat equation is considered in the first quadrant. The loaded term has the form of the Riemann-Liouville’s fractional derivative with respect to the time variable, and the order of the derivative in the loaded term is less than the order of the differential part. The study is based on reducing the boundary value problem to a Volterra integral equation. The kernel of the obtained integral equation contains a special function, namely, the Wright function. The kernel is estimated, and the conditions for the unique solvability of the integral equation are obtained.
Md. Tarikul Islam, Md. Ali Akbar, J.F. Gómez-Aguilar
et al.
Fractional order nonlinear evolution equations have emerged in recent times as being very important model for depicting the interior behavior of nonlinear phenomena that exist in the real world. In particular, Schrodinger-type fractional nonlinear evolution equations constitute an aspect of the field of quantum mechanics. In this study, the (2 + 1)-dimensional time-fractional nonlinear Schrodinger equation and (1 + 1)-dimensional time-space fractional nonlinear Schrodinger equation are revealed as having different and novel wave structures. This is shown by constructing appropriate analytic wave solutions. A successful implementation of the advised rational (1/ϕ′(ξ))-expansion method generates new outcomes of the considered equations, by comparing them with those already noted in the literature. On the basis of the conformable fractional derivative, a composite wave variable conversion has been used to adapt the suggested equations into the differential equations with a single independent variable before applying the scheme. Finally, the well-furnished outcomes are plotted in different 3D and 2D profiles for the purpose of illustrating various physical characteristics of wave structures. The employed technique is competent, productive and concise enough, making it feasible for future studies.
In this paper, based on the splitting method scheme, the existence and uniqueness theorem on the whole time interval t ∈ [0 ,T ) ,T ≤∞ for the full nonlinear Boltzmann equation in the nonequilibrium case is proved where the intermolecular interactions are hard-sphere molecule and central forces. Considering the existence of a bounded solution in the space C, the strict positivity of the solution to the full nonlinear Boltzmann equation is proved when the initial function is positive. On the basis of this some mathematical justification of the H -theorem of Boltzmann is shown.
T. K. Yuldashev, B. J. Kadirkulov, A. R. Marakhimov
In this article we consider the questions of one-valued solvability and numerical realization of initial value problem for a nonlinear Hilfer type fractional differential equation with maxima. By the aid of uncomplicated integral transformation based on Dirichlet formula, this initial value problem is reduced to the nonlinear Volterra type fractional integral equation. The theorem of existence and uniqueness of the solution of given initial value problem in the segment under consideration is proved. For numerical realization of solution the generalized Jacobi–Galerkin method is applied. Illustrative examples are provided.
A simple procedure is presented for the study of the conservation of energy equation with dissipation in continuum mechanics in 1D. This procedure is used to transform this nonlinear evolution-diffusion equation into a hyperbolic PDE; specifically, a second-order quasi-linear wave equation. An immediate implication of this procedure is the formation of a least action principle for the balance of energy with dissipation. The corresponding action functional enables us to establish a complete analytic mechanics for thermomechanical systems: a Lagrangian–Hamiltonian theory, integrals of motion, bracket formalism, and Noether’s theorem. Furthermore, we apply our procedure iteratively and produce an infinite sequence of interlocked variational principles, a variational hierarchy, where at each level or iteration the full implication of the least action principle can be shown again.
In this paper a mathematical model is proposed, which incorporates quarantine and hospitalization to assess the community impact of social distancing and face mask among the susceptible population. The model parameters are estimated and fitted to the model with the use of laboratory confirmed COVID-19 cases in Turkey from March 11 to October 10, 2020. The partial rank correlation coefficient is employed to perform sensitivity analysis of the model, with basic reproduction number and infection attack rate as response functions. Results from the sensitivity analysis reveal that the most essential parameters for effective control of COVID-19 infection are recovery rate from quarantine individuals (δ1), recovery rate from hospitalized individuals (δ4), and transmission rate (β). Some simulation results are obtained with the aid of mesh plots with respect to the basic reproductive number as a function of two different biological parameters randomly chosen from the model. Finally, numerical simulations on the dynamics of the model highlighted that infections from the compartments of each state variables decreases with time which causes an increase in susceptible individuals. This implies that avoiding contact with infected individuals by means of adequate awareness of social distancing and wearing face mask are vital to prevent or reduce the spread of COVID-19 infection.
In this work the partial differential equations with involutions are considered. The mixed problems for the parabolic type equation, with constant and variable constants, corresponding to the Dirichlet type boundary conditions is investigated. The involution is contained by the second derivative with respect to the variable x, which is the difficult case for investigations. One-dimensional differential operators with involution have an infinite number of positive and negative eigenvalues. This means that the operator on the right-hand side of the equation under study is not semi-bounded. In the case of classical problems, ordinary differential operators usually appear on the right-hand side of the equations, which are semibounded. Therefore, the incorrectness of mixed problems for a parabolic equation with an involution is discussed in this paper. Examples are given. Sufficient conditions for the initial data are found when the problem under study has a unique solution. The representation of the solution in the form of partial sums of the Fourier series in eigenfunctions is found. The density in the space L2 (−1, 1) of the set of initial functions is proved everywhere, when the problem has a unique solution.
Random quantum circuits yield minimally structured models for chaotic quantum dynamics, which are able to capture, for example, universal properties of entanglement growth. We provide exact results and coarse-grained models for the spreading of operators by quantum circuits made of Haar-random unitaries. We study both 1+1D and higher dimensions and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems. In 1+1D, we demonstrate that the out-of-time-order correlator (OTOC) satisfies a biased diffusion equation, which gives exact results for the spatial profile of the OTOC and determines the butterfly speed v_{B}. We find that in 1+1D, the “front” of the OTOC broadens diffusively, with a width scaling in time as t^{1/2}. We address fluctuations in the OTOC between different realizations of the random circuit, arguing that they are negligible in comparison to the broadening of the front within a realization. Turning to higher dimensions, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a purely classical droplet growth problem. This implies that the width of the front of the averaged OTOC scales as t^{1/3} in 2+1D and as t^{0.240} in 3+1D (exponents of the Kardar-Parisi-Zhang universality class). We support our analytic argument with simulations in 2+1D. We point out that, in two or higher spatial dimensions, the shape of the spreading operator at late times is affected by underlying lattice symmetries and, in general, is not spherical. However, when full spatial rotational symmetry is present in 2+1D, our mapping implies an exact asymptotic form for the OTOC, in terms of the Tracy-Widom distribution. For an alternative perspective on the OTOC in 1+1D, we map it to the partition function of an Ising-like statistical mechanics model. As a result of special structure arising from unitarity, this partition function reduces to a random walk calculation which can be performed exactly. We also use this mapping to give exact results for entanglement growth in 1+1D circuits.
A family multipoint - integral boundary value problems for a third order differential equation with variable coefficients is considered. The questions of a existence unique solution of the considered problem and ways of its construction are investigated. The family multipoint - integral boundary value problems for the differential equation of third order with variable coefficients is reduced to a family multipoint - integral boundary value problems for a system of three differential equations by introducing new functions. For solve of resulting family of multipoint - integral boundary value problems is applied a parametrization method. An algorithms of finding the approximate solution to the family multipoint - integral boundary value problems for the system of three differential equations are proposed and their convergence is proved. The conditions of the unique solvability of the family multipoint - integral boundary value problems for the system of three differential equations are obtained in the terms of initial data. The results also formulated relative to the original of the family multipoint - integral boundary value problems for the differential equation of third order with variable coefficients.
В статье рассмотрены задачи Коши, Гурса и Дарбу для нагруженного уравнения колебания струны uxx - uyy = λu(x0,y). Построено явное представление решения задачи Коши, которое при λ = 0 совпадает с известным представлением решения задачи Коши для уравнения колебания струны. Описаны области зависимости, влияния и определения данных Коши. Показано их существенное отличие от аналогичных областей в случае задачи Коши для уравнения колебания струны. Сформулированы задачи Дарбу и Гурса в нелокальной постановке и предложен алгоритм построения их решений.
Важным направлением в исследовании закономерностей динамики экономических процессов является изучение общей тенденции развития. В статье рассмотрены факторы, формирующие тенденции (тренды) социально - экономического развития РК, и представлены результаты проведенного исследования по анализу и прогнозированию перспектив развития основных факторов экономики страны. Рассмотрены категории экономического роста и развития в аспекте их устойчивости и стабильности. Подчеркнута диалектическая взаимосвязь факторов между категориями на уровне социума, макро - микроэкономики. Кроме того, созданы базисные ориентиры для построения прогнозных моделей, формирующих парадигму социально - экономического развития.