Abstract Deriving macroscopic phenomenological laws of irreversible thermodynamics from simple microscopic models is one of the tasks of non-equilibrium statistical mechanics. We consider stationary energy transport in crystals with reference to simple mathematical models consisting of coupled oscillators on a lattice. The role of lattice dimensionality on the breakdown of the Fourier's law is discussed and some universal quantitative aspects are emphasized: the divergence of the finite-size thermal conductivity is characterized by universal laws in one and two dimensions. Equilibrium and non-equilibrium molecular dynamics methods are presented along with a critical survey of previous numerical results. Analytical results for the non-equilibrium dynamics can be obtained in the harmonic chain where the role of disorder and localization can be also understood. The traditional kinetic approach, based on the Boltzmann–Peierls equation is also briefly sketched with reference to one-dimensional chains. Simple toy models can be defined in which the conductivity is finite. Anomalous transport in integrable non-linear systems is briefly discussed. Finally, possible future research themes are outlined.
This article is written in honor of the 8th Kazakh–French Logical Colloquium. We expand on an unpublished research note of the second author. We record some results concerning local Keisler measures with respect to a formula which is stable in a model. We prove that in this context, every local Keisler measure on the associated local type space is a weighted sum of (at most countably many) local types. Using this observation, we give an elementary proof of the commutativity of the Morley product in this context. We then give a functional analytic proof that the double limit property lifts to the appropriate evaluation map on pairs of local measures. We conclude with observations regarding the NOP and local Keisler measures in the (properly) stable context. Finally, we provide two proofs that the evaluation map on pairs of local Keisler measures is stable (in continuous logic). The first follows almost immediately from the work of Ben Yaacov and Keisler on the randomization; the other proof follows from the VC theorem.
The optimal selection of favorable reservoir is a fundamental and necessary task after logging and before fracturing operations, which can help improve the success rate of coalbed methane exploration. The target coalbed reservoir of the first coalbed methane exploration well, Kexin1H, in the Turpan-Hami Basin is located at a depth of 3 163~3 350 meters. The geological characteristics analysis and optimal selection of a favorable reservoir of coalbed methane are of great significance for subsequent exploration and development. This study conducted more than 270 tests on coal quality, coal structure, in-suit gas content, as well as isothermal adsorption, porosity, permeability, stress sensitivity, rock mechanics, and other parameters, providing a comprehensive understanding on the deep coalbed methane reservoir in the Turpan-Hami Basin and a deep analysis on optimal selection of favorable reservoir. The results show that: ① coals in the Xishanyao Formation drilled by Kexin1H well have a vitrinite reflectance ranging from 0.64 % to 1.02 %, with vitrinite content ranging from 63.40 % to 89.57 %. Based on the industrial analysis results of coal-rock, it is believed that the main coal seam is a long-flame coal or non-caking coal reservoir; ② based on the measurements of closed-core and isothermal adsorption, the gas saturation of coal rock reservoir ranges from 1.47 to 1.98. The three-axis elastic modulus and Poisson's ratio of coal rock are between 3.594 GPa to 7.795 GPa and 0.17 to 0.27, respectively. The coal structure is mainly fractured coal and granular coal. The coal rock reservoir belongs to the over-saturated, medium-low rank, fractured-soft, low-permeability, and strong stress-sensitive reservoir; ③ an evaluation index system for the optimal selection of a favorable reservoir in a single well based on fuzzy analytic hierarchy process was developed, which includes three main categories of primary indicators and twelve secondary indicators. It is believed that the producibility of the coalbed reservoir in the third section of Well Kexin 1H is high in production potential. However, attention should be paid to the impact of coal fines during engineering operations.
In Ref. arXiv:2502.08816, Hawking radiation was analyzed through a statistical mechanics framework, revealing a structured microstate description of black hole horizons and information transfer into the radiation background. This study extends that approach by formulating Hawking radiation and black hole evaporation in the language of stochastic mechanics, employing an analytical Langevin framework and a numerical Euler iteration scheme. Both methods confirm that small black holes behave as thermal systems with Gaussian noise, while larger black holes develop a structured noise spectrum that aligns with the gradual contraction of the horizon. This suggests an alternative interpretation of Hawking radiation as an effective surface fuzziness, encoding horizon-scale fluctuations. The appendix provides a Wolfram Mathematica blueprint for numerical simulations, open to heuristic modifications for further exploration of black hole noise spectra.
Unilateral transmission refers to the scenario in which the waves transmitted through a system remain in pure tension or pure compression. This transmission phenomenon may occur in systems that exhibit different effective elasticity in compression and tension; i.e. bilinear elasticity. We present a computational investigation of unilateral transmission in the steady-state response of harmonically driven mechanical systems with bilinear coupling. Starting with two bilinearly coupled oscillators, we find that breaking the mirror symmetry of the system, in either elastic or inertial properties, facilitates unilateral transmission by allowing it to occur near a primary resonance. This asymmetry also enables nonreciprocal transmission to occur. We then investigate the nonreciprocal dynamics of the system, including linear stability analysis, with a focus on unilateral transmission. We also extend our discussion to a bilinear periodic structure, for which we investigate the influence of the number of units and energy dissipation on unilateral transmission. We report on the existence of stable nonreciprocal unilateral transmission near primary and internal resonances of the system, as well as other nonreciprocal features such as period-doubled and quasiperiodic response characteristics.
This article is devoted to the study of source identification problems for reverse parabolic partial differential equations with nonlocal boundary conditions. The principal aim of the work is to construct and analyze stable difference schemes that can be effectively employed for obtaining approximate solutions of such inverse problems. In particular, attention is focused on the Rothe difference scheme, and stability estimates for the corresponding discrete solutions are rigorously derived. These estimates guarantee the reliability and convergence of the proposed numerical method. A stability theorem for the solution of the difference scheme related to the source identification problem is proved. To establish the well-posedness of the underlying differential problem, the operator-theoretic approach is employed, ensuring a solid analytical foundation for the numerical method. Furthermore, the investigation is extended to an abstract setting for difference schemes, which is then applied to the numerical solution of reverse parabolic equations under boundary conditions of the first kind. This unified framework emphasizes both the theoretical justification and the computational effectiveness of the proposed approach. Finally, the efficiency of the developed method is demonstrated through a numerical illustration with a test example.
Mahesh S. Nagargoje, Eneko Lazpita, Jesús Garicano-Mena
et al.
Many cardiovascular diseases occur due to an abnormal functioning of the heart. A diseased heart leads to severe complications and in some cases death of an individual. The medical community believes that early diagnosis and treatment of heart diseases can be controlled by referring to numerical simulations of image-based heart models. Computational fluid dynamics (CFD) is a commonly used tool for patient-specific simulations in cardiac flows, and it can be equipped to allow a better understanding of flow patterns. In this paper, we review the progress of CFD tools to understand the flow patterns in healthy and dilated cardiomyopathic (DCM) left ventricles (LVs). The formation of an asymmetric vortex in a healthy LV shows an efficient means of blood transport. The vortex pattern changes before any change in the geometry of LVs is noticeable. This flow change can be used as a marker of DCM progression. We can conclude that the vortex dynamics in LVs can be understood using the widely used vortex index, the vortex formation number (VFN). The VFN coupled with data-driven approaches can be used as an early diagnosis tool and leads to improvement in DCM treatment.
Philipp P. Vieweg, Theo Käufer, Christian Cierpka
et al.
Albeit laboratory experiments and numerical simulations have proven themselves successful in enhancing our understanding of long-living large-scale flow structures in horizontally extended Rayleigh–Bénard convection, some discrepancies with respect to their size and induced heat transfer remain. This study traces these discrepancies back to their origins. We start by generating a digital twin of one standard experimental set-up. This twin is subsequently simplified in steps to understand the effect of non-ideal thermal boundary conditions, and the experimental measurement procedure is mimicked using numerical data. Although this allows for explaining the increased observed size of the flow structures in the experiment relative to past numerical simulations, our data suggests that the vertical velocity magnitude has been underestimated in the experiments. A subsequent reassessment of the latter's original data reveals an incorrect calibration model. The reprocessed data show a relative increase in $u_{z}$ of roughly $24\,\%$, resolving the previously observed discrepancies. This digital twin of a laboratory experiment for thermal convection at Rayleigh numbers $Ra = \{ 2, 4, 7 \} \times 10^{5}$, a Prandtl number $Pr = 7.1$ and an aspect ratio $\varGamma = 25$ highlights the role of different thermal boundary conditions as well as a reliable calibration and measurement procedure.
A trigonometric spline based computational technique is suggested for the numerical solution of layer behavior differential-difference equations with a fixed large delay. The continuity of the first order derivative of the trigonometric spline at the interior mesh point is used to develop the system of difference equations. With the help of singular perturbation theory, a fitting parameter is inserted into the difference scheme to minimize the error in the solution. The method is examined for convergence. We have also discussed the impact of shift or delay on the boundary layer. The maximum absolute errors in comparison to other approaches in the literature are tallied, and layer behavior is displayed in graphs, to demonstrate the feasibility of the suggested numerical method.
Markus Sudmanns, Athanasios P. Iliopoulos, Andrew J. Birnbaum
et al.
Mesoscale simulations of discrete defects in metals provide an ideal framework to investigate the micro-scale mechanisms governing the plastic deformation under high thermal and mechanical loading conditions. To bridge size and time-scale while limiting computational effort, typically the concept of representative volume elements (RVEs) is employed. This approach considers the microstructure evolution in a volume that is representative of the overall material behavior. However, in settings with complex thermal and mechanical loading histories careful consideration of the impact of modeling constraints in terms of time scale and simulation domain on predicted results is required. We address the representation of heterogeneous dislocation structure formation in simulation volumes using the example of residual stress formation during cool-down of laser powder-bed fusion (LPBF) of AISI 316L stainless steel. This is achieved by a series of large-scale three-dimensional discrete dislocation dynamics (DDD) simulations assisted by thermo-mechanical finite element modeling of the LPBF process. Our results show that insufficient size of periodic simulation domains can result in dislocation patterns that reflect the boundaries of the primary cell. More pronounced dislocation interaction observed for larger domains highlight the significance of simulation domain constraints for predicting mechanical properties. We formulate criteria that characterize representative volume elements by capturing the conformity of the dislocation structure to the bulk material. This work provides a basis for future investigations of heterogeneous microstructure formation in mesoscale simulations of bulk material behavior.
Carolanne V.M. Vouriot, Thomas D. Higton, P.F. Linden
et al.
Displacement ventilation, where cool external air enters a room through low-level vents and warmer air leaves through high-level vents, is characterised by vertical gradients in pressure arising from the warmer indoor temperatures. Models usually assume that horizontal variations of temperature difference are small in comparison and are, therefore, unimportant. Small-scale laboratory experiments and computational fluid dynamics were used to examine these flows, driven by a uniformly heated floor. These experiments and simulations show that the horizontal variations of temperature difference can be neglected for predictions of the bulk ventilation rate; however, they also evidence that these horizontal variations can be significant and play a critical role in establishing the pattern of flow within the room – this renders the horizontal position of the low- and high-level vents (relative to one another) important. We consider two cases: single-ended (where inlet and outlet are at the same end of the room) and opposite-ended. In both cases the ventilation flow rate is the same. However, in the opposite-ended case a dead zone is established in the upper part of the room which results in significant horizontal variations. We consider the formation of this dead zone by examining the streamline patterns and the age of air within the room. We discuss the implications for occupant exposure to pollutants and airborne disease.
<p>Polar sea ice is a critical component of Earth’s climate system. As a material, it is a multiscale composite of pure ice with temperature-dependent millimeter-scale brine inclusions, and centimeter-scale polycrystalline microstructure which is largely determined by how the ice was formed. The surface layer of the polar oceans can be viewed as a granular composite of ice floes in a sea water host, with floe sizes ranging from centimeters to tens of kilometers. A principal challenge in modeling sea ice and its role in climate is how to use information on smaller-scale structures to find the effective or homogenized properties on larger scales relevant to process studies and coarse-grained climate models. That is, how do you predict macroscopic behavior from microscopic laws, like in statistical mechanics and solid state physics? Also of great interest in climate science is the inverse problem of recovering parameters controlling small-scale processes from large-scale observations. Motivated by sea ice remote sensing, the analytic continuation method for obtaining rigorous bounds on the homogenized coefficients of two-phase composites was applied to the complex permittivity of sea ice, which is a Stieltjes function of the ratio of the permittivities of ice and brine. Integral representations for the effective parameters distill the complexities of the composite microgeometry into the spectral properties of a self-adjoint operator like the Hamiltonian in quantum physics. These techniques have been extended to polycrystalline materials, advection diffusion processes, and ocean waves in the sea ice cover. Here we discuss this powerful approach in homogenization, highlighting the spectral representations and resolvent structure of the fields that are shared by the two-component theory and its extensions. Spectral analysis of sea ice structures leads to a random matrix theory picture of percolation processes in composites, establishing parallels to Anderson localization and semiconductor physics and providing new insights into the physics of sea ice.</p>
The knuckleball is considered to be one of the hardest pitches to hit in baseball due to its seemingly unpredictable motion. It has gained popularity in cricket in recent times. It is shown that the delivery referred to as knuckleball in cricket, at present, does not exhibit a zigzag motion and is, therefore, a misnomer. We propose a delivery in cricket that is associated with an erratic trajectory similar to the knuckleball pitch in baseball. Force measurement experiments in a wind tunnel on a new cricket ball in various orientations of the seam to the incoming flow and at different Reynolds number are carried out. The results are utilized to estimate the trajectory of knuckleball deliveries. The key parameters are the seam angle, speed and spin rate of the ball at the time of its release. Their effect on the trajectory is studied in detail. The optimal combination of these parameters that result in a knuckleball, which is likely hard for the batter to play, is identified.
We show that the conductance $G$ of molecular tunnel junctions wherein the charge transport is dominated by a single energy level can be expressed in closed analytic form which is exact and valid at arbitrary temperature $T$ and model parameter values. On this basis, we show that the single-step tunneling mechanism is compatible with a continuous thermal transition from a weakly $T$-dependent $G$ at low $T$ (Sommerfeld regime) to a nearly exponential $1/T$-dependent $G$ at high $T$ (Arrhenius-like regime). We predict that this Sommerfeld-Arrhenius transition can be observed in real molecular junctions % (e.g., based on perylene diimide) and can be continuously tuned, e.g., via mechanical stretching.
This article is devoted to the study of the forcing companions of the Jonsson AP-theories in the enriched signature. It is proved that the forcing companion of the theory does not change when expanding the theories under consideration, which have some properties, by adding new predicate and constant symbols to the language. The model-theoretic results obtained in this paper in general form are supported by examples from differential algebra. An approach in combining a Jonsson and non-Jonsson theories is demonstrated. In this paper, for the first time in the history of Model Theory. This will allow us to further develop the methods of research of Jonsson theories and expand the apparatus for studying incomplete theories.
In this study, improved Bernoulli sub-equation function method (IBSEFM) is presented to construct the exact solutions of the nonlinear conformable fractional Schr¨odinger-Hirota equation (FSHE). By using the traveling wave transformation FSHE turns into the ordinary differential equation (ODE) and by the aid of symbolic calculation software, new exact solutions are obtained. 2D, 3D figures and contour surfaces acquired from the values of the solutions are plotted. The results show that the proposed method is powerful, effective and straightforward for formulating new solutions to various types of nonlinear fractional partial differential equations in applied sciences.
In this paper, we study formal characters of simple modules with singular highest weights over classical Lie algebras of type B over an algebraically closed field of characteristic p ≥ h, where h is the Coxeter number. Assume that the highest weights of these simple modules are restricted. We have given a description of their formal characters. In particular, we have obtained some new examples of simple Weyl modules. In the restricted region, the representation theory of algebraic groups and its Lie algebras are equivalent. Therefore, we can use the tools of the representation theory of semisimple and simply-connected algebraic groups in positive characteristic. To describe the formal characters of simple modules, we construct Jantzen filtrations of Weyl modules of the corresponding highest weights.
In this paper, we consider the non-homogeneous wave equation generated by the Bessel operator. We prove the existence and uniqueness of the solution of the non-homogeneous wave equation generated by the Bessel operator. The representation of the solution is given. We obtained a priori estimates in Sobolev type space. This problem was firstly considered in the work of M. Assal [1] in the setting of Bessel-Kingman hypergroups. The technique used in [1] is based on the convolution theorem and related estimates. Here, we use a different approach. We study the problem from the point of the Sobolev spaces. Namely, the conventional Hankel transform and Parseval formula are widely applied by taking into account that between the Hankel transformation and the Bessel differential operator there is a commutation formula [2].