Notes

May be the hereditary rule enhanced with regard to reference conservation?
g., the lineal-path function). We examine a wide spectrum of representative random systems with distinct geometrical and topological features. We find that, generally, successively incorporating higher-order P_n functions and, thus, the higher-order morphological information encoded in these descriptors leads to superior accuracy of the reconstructions. However, incorporating more P_n functions into the reconstruction also significantly increases the complexity and roughness of the associated energy landscape for the underlying stochastic optimization, making it difficult to convergence numerically.Recently, a number of sufficiency conditions have been shown for the occurrence of a Z_2-symmetry breaking phase transition (Z_2-SBPT) starting from geometric-topological concepts of potential energy landscapes. In particular, a Z_2-SBPT can be triggered by double-well potentials, or equivalently by dumbbell-shaped equipotential surfaces. In this paper, we introduce two models with a Z_2-SBPT that, due to their essential feature, show in the clearest way the generating mechanism of a Z_2-SBPT. Although they cannot be considered physical models, they all have the features of such models with the same kind of SBPT. At the end of the paper, the ϕ^4 model is revisited in light of this approach. In particular, the landscape of one of the models introduced here turned out to be equivalent to that of the mean-field ϕ^4 model in a simplified version.The putative generalization of the thermodynamic uncertainty relation (TUR) to underdamped dynamics is still an open problem. So far, bounds that have been derived for such a dynamics are not particularly transparent and they do not converge to the known TUR in the overdamped limit. Furthermore, it was found that there are restrictions for a TUR to hold such as the absence of a magnetic field. In this article we first analyze the properties of driven free diffusion in the underdamped regime and show that it inherently violates the overdamped TUR for finite times. Based on numerical evidence, we then conjecture a bound for one-dimensional driven diffusion in a potential which is based on the result for free diffusion. This bound converges to the known overdamped TUR in the corresponding limit. Moreover, the conjectured bound holds for observables that involve higher powers of the velocity as long as the observable is odd under time reversal. mTOR inhibition Finally, we address the applicability of this bound to underdamped dynamics in higher dimensions.We derive generalized Fokker-Planck equations (FPEs) based on two nonextensive entropy measures S_± that depend exclusively on the probability. These entropies have been originally obtained from the superstatistics framework, therefore they regard nonequilibrium systems outlined by a long-term stationary state in view of a spatiotemporally fluctuating intensive quantity. Moreover, entropies S_± as well as Boltzmann-Gibbs (BG) entropy S_B both pertain to the same asymptotical equivalence class, thus suggesting that S_± could depict a consistent thermodynamic generalization of BG. For these reasons, we assert that transport phenomena to be accounted for by our models shall coincide with the portrait given by the conventional FPEs for systems comprehending short-range interactions or a high number of accessible microstates, whereas, for systems composed of a small number of microstates, or those with long-range interactions, the governing equations of motion are to be the FPEs here derived, as long as the system fulfills the attributes mentioned above. We discuss the anomalous diffusion exhibited by the two generalized FPEs and also present some numerical applications. In particular, we find that there are models regarding biological sciences, for the study of congregation and aggregation behavior, the structure of which coincides with the one of our models.Understanding of multiphase flow in porous media is important for a wide range of applications such as soil science, environmental remediation, energy resources, and CO_2 sequestration. This phenomenon depends on the complex interplay between the fluid and solid forces such as gravitational, capillary, and viscous forces, as well as wettability of the solid phase. Such interactions along with the geometry of the medium give rise to a variety of complex flow regimes. Although much research has been done in the area of wettability, its mechanical effect is not well understood, and it continues to challenge our understanding of the phenomena on macroscopic and microscopic scales. In this paper, therefore, the effect of wettability on the deformation of porous media and fluid-fluid patterns is studied through a series of three-dimensional (3D) simulations. To this end, the discrete element method (DEM) and volume of fluid (VOF) are coupled to accurately model free-surface flow interaction in a cylindrical pack of spheres. The fluid-particle interactions are modeled by exchanging information between DEM and VOF, while the effect of wettability is considered to study how it controls fluid displacement. The results indicate that the drag force and deformation in the pack vary with the change in wettability and capillary number. To demonstrate the effect of both wettability and capillary number, a series of numerical experiments were conducted with two capillary numbers and three wettability conditions. Our results show that the drag force was greatest for near extreme wettability conditions, which resulted in a larger deformation.Individual fire ants are inherently active as they are living organisms that convert stored chemical energy into motion. However, each individual ant is not equally disposed to motion at any given time. In an active aggregation, most of the constituent ants are active, and vice versa for an inactive aggregation. Here we look at the role activity plays on the nonlinear mechanical behavior of the aggregation through large amplitude oscillatory shear measurements. We find that the level of viscous nonlinearity can be decreased by increasing the activity or by increasing the volume fraction. In contrast, the level of elastic nonlinearity is not affected by either activity or volume fraction. We interpret this in terms of a transient network with equal rates of linking and unlinking but with varying number of linking and unlinking events.A fundamental theory is presented for the mechanical response of polymer networks undergoing large deformation which seamlessly integrates statistical mechanical principles with macroscopic thermodynamic constitutive theory. Our formulation permits the consideration of arbitrary polymer chain behaviors when interactions among chains may be neglected. This careful treatment highlights the naturally occurring correspondence between single-chain mechanical behavior and the equilibrium distribution of chains in the network, as well as the correspondences between different single-chain thermodynamic ensembles. We demonstrate these important distinctions with the extensible freely jointed chain model. This statistical mechanical theory is then extended to the continuum scale, where we utilize traditional macroscopic constitutive theory to ultimately retrieve the Cauchy stress in terms of the deformation and polymer network statistics. Once again using the extensible freely jointed chain model, we illustrate the importance of the naturally occurring statistical correspondences through their effects on the stress-stretch response of the network. We additionally show that these differences vanish when the number of links in the chain becomes sufficiently large enough, and discuss why certain methods perform better than others before this limit is reached.We consider the Adlam-Allen (AA) system of partial differential equations, which, arguably, is the first model that was introduced to describe solitary waves in the context of propagation of hydrodynamic disturbances in collisionless plasmas. Here, we identify the solitary waves of the model by implementing a dynamical systems approach. The latter suggests that the model also possesses periodic wave solutions-which reduce to the solitary wave in the limiting case of an infinite period-as well as rational solutions that are obtained herein. In addition, employing a long-wave approximation via a relevant multiscale expansion method, we establish the asymptotic reduction of the AA system to the Korteweg-de Vries equation. Such a reduction is not only another justification for the above solitary wave dynamics, but may also offer additional insights for the emergence of other possible plasma waves. Direct numerical simulations are performed for the study of multiple solitary waves and their pairwise interactions. The stability of solitary waves is discussed in terms of potentially relevant criteria, while the robustness of spatially periodic wave solutions is touched upon via numerical experiments.We report here an alternative kind of fingering instability observed during fracture of an unconfined gel consisting of two cuboids joined by a thin gel disk, and all prepared monolithically. When the blocks are sheared across the joint, fracture ensues with the appearance of fingers at the fracture front. The spacing between the fingers remains independent of the shearing speed, planar shape of the joint, and the shear modulus of gel. Importantly this instability appears without any effect of confinement of the gel block, and its wavelength remains dependent on the lateral size of the disk, in contrast to all known instances of fingering phenomena in confined viscous, elastic, and viscoelastic systems.Balance theory proposed by Heider for the first time modeled triplet interaction in a signed network, stating that relationships between two people, friendship or enmity, is dependent on a third person. The Hamiltonian of this model has an implicit assumption that all triads are independent, meaning that the type of each triad, being balanced or imbalanced, determined apart from the state of other triads. This independence forces the network to have completely balanced final states. However, there exists evidence indicating that real networks are partially balanced, raising the question of what is the mechanism preventing the system to be perfectly balanced. Our suggestion is to consider a quartic interaction which dissolves the triad's independence. We use the mean-field method to study the thermal behavior of such systems where the temperature is a parameter that allows the stochastic behavior of agents. We show that under a certain temperature, the symmetry between balanced and imbalanced triads will spontaneously break and we have a discrete phase transition. As a consequence, stability arises where either similar balanced or imbalanced triads dominate, hence the system obtains two new imbalanced stable states. In this model, the critical temperature depends on the second power of the number of nodes, which was a linear dependence in thermal balance theory. Our simulations are in good agreement with the results obtained by the mean-field method.We show the asymptotic equivalence of two forcing schemes in the lattice Boltzmann method (LBM) within second-order accuracy through the asymptotic analysis instead of the Chapman-Enskog analysis. We consider the single relaxation time LBM with the following two forcing schemes the simplest scheme by He et al. [J. Stat. Phys. 87, 115 (1997)10.1007/BF02181482] (referred to as He forcing); the most popular scheme by Guo et al. [Phys. Rev. E 65, 046308 (2002)10.1103/PhysRevE.65.046308] (referred to as Guo forcing). It has been shown by using the Chapman-Enskog analysis that the He forcing leads the unphysical terms in the macroscopic equations due to the spatial and time derivatives of the body force, whereas the Guo forcing does not lead such terms. However, we find by using the asymptotic analysis that the order of the unphysical terms is comparable to or less than (Δx)^3 for the continuity equation and (Δx)^4 for the Navier-Stokes equations (where Δx is the lattice spacing). Therefore, not only the Guo forcing but also the He forcing give the macroscopic flow velocity and pressure for incompressible viscous fluid with relative errors of O[(Δx)^2].
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