Hookean Force Research Articles

On black hole thermodynamics, singularity, and gravitational entropy

Many fundamental issues remain for black hole thermodynamics after almost half a century of its conception. For example, what are the underlying degrees of freedom of a black hole horizon that give rise to said thermodynamical properties? Furthermore, classical black holes also harbor a spacetime singularity. Although it is often believed that quantum gravity would "cure" the singularity, as emphasized by Penrose, this viewpoint requires a deeper examination. In this review, I will examine the possibility that singularities remain in quantum gravity, the roles they may play, and the possible links between singularity and black hole thermodynamics. I will also discuss how -- inspired by Penrose's Weyl curvature hypothesis -- gravitational entropy for a black hole can be defined using curvature invariants, and the surprising implication that the entropy of black holes in different theories of gravity are different manifestations of spacetime curvature, i.e., their underlying microstructures could be different. Finally, I review the "Hookean law" recently established for singly rotating Myers-Perry black holes (including Kerr black holes) that connect black hole fragmentation -- a consequence of the second law of black hole thermodynamics -- with the maximum "Hookean force", as well as with the thermodynamic geometry of Ruppeiner. This also suggests a new way to study black hole microstructures, and hints at the possibility that some black holes are beyond the Hookean regime (and thus have different microstructures). While examining the remarkable connections between black hole thermodynamics, spacetime singularities and cosmic censorship, as well as gravitational entropy, I shall point out some subtleties, provide some new thoughts, and raise some hard but fundamental questions, including whether black hole thermodynamics is really just "ordinary thermodynamics" or something quite different.

Stochastic harmonic trapping of a Lévy walk: transport and first-passage dynamics under soft resetting strategies

We introduce and study a Lévy walk (LW) model of particle spreading with a finite propagation speed combined with soft resets, stochastically occurring periods in which an harmonic external potential is switched on and forces the particle towards a specific position. Soft resets avoid instantaneous relocation of particles that in certain physical settings may be considered unphysical. Moreover, soft resets do not have a specific resetting point but lead the particle towards a resetting point by a restoring Hookean force. Depending on the exact choice for the LW waiting time density and the probability density of the periods when the harmonic potential is switched on, we demonstrate a rich emerging response behaviour including ballistic motion and superdiffusion. When the confinement periods of the soft-reset events are dominant, we observe a particle localisation with an associated non-equilibrium steady state. In this case the stationary particle probability density function turns out to acquire multimodal states. Our derivations are based on Markov chain ideas and LWs with multiple internal states, an approach that may be useful and flexible for the investigation of other generalised random walks with soft and hard resets. The spreading efficiency of soft-rest LWs is characterised by the first-passage time statistic.

Resonance behavior for a generalized Mittag-Leffler fractional Langevin equation with hydrodynamic interactions

The phenomenological model for the heavy tracers in viscoelastic media modeled by a generalized Mittag-Leffler fractional Langevin equation with the generalized Stokes force, the Basset force, the Hookean force, and the thermal force has been revisited. Under the fluctuation-dissipation relation, the generalized Stokes force describes the viscoelastic media by a Mittag-Leffler (ML) memory kernel. Furthermore, based on the background of ML function, the generalized Mittag-Leffler fractional derivative is introduced. Moreover, the exact expression of stationary first moment and the expression of spectral amplification (SPA) of a tracer model have been deserved by the generalized form of Shapiro-Loginov formula. The generalized stochastic resonance (GSR) phenomena has been systematically studied. Moreover, the GSR, reverse stochastic resonance (SR) phenomenon, bona fide SR, stochastic multi-resonance (SMR) phenomena, increasing multi-resonance and decreasing multi-resonance have been found. Especially, the periodic resonance phenomenon could be induced by the generalized Mittag-Leffler (GML) noise, which has been few observed in the previous literatures.

Global Optimization of Adsorbate–Surface Structures While Preserving Molecular Identity

As the complexity of atomistic simulations in catalysis and surface science increases, the challenge of manually finding the lowest-energy adsorbate–surface geometries grows significantly. In the current work, a global optimization approach that preserves adsorbate identity is applied to enable the automated search for optimized binding geometries. This technique is based on the minima hopping method developed by Goedecker, but is modified to preserve the molecular identity of adsorbates by the application of a new class of Hookean constraints. These constraints are completely inactive when the adsorbate identity is preserved, but act to restore the adsorbate structure via a Hookean force when the bond length exceeds a threshold distance. Additionally, a related Hookean constraint has been developed to prevent adsorbates (particularly such adsorbates as CO and CH2O that have stable gas-phase forms) from volatilizing during the molecular dynamics portion of the minima hopping technique. This combination, referred to herein as the constrained minima hopping method, was tested for its suitability in finding the minimum-energy binding configuration for a set of 17 C x H y O z adsorbates on a stepped Cu fcc(211) surface and in all cases found the global minima in comparable or fewer steps than the previous brute force methodologies. It is expected that methods such as this will be crucial to finding low-energy states in more complex systems, such as those with high coverages of adsorbed species or in the presence of explicit solvent molecules.

Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions

We consider Levy flights subject to external force fields. This anomalous transport process is described by two approaches, a Langevin equation with Levy noise and the corresponding generalized Fokker-Planck equation containing a fractional derivative in space. The cases of free flights, constant force and linear Hookean force are analyzed in detail, and we corroborate our findings with results from numerical simulations. We discuss the non-Gibbsian character of the stationary solution for the case of the Hookean force, i.e. the deviation from Boltzmann equilibrium for long times. The possible connection to Tsallis's q-statistics is studied.

High-Speed Video Cinematographic Demonstration of Stalk and Zooid Contraction of Vorticella convallaria

Stalk contraction and zooid contraction of living Vorticella convallaria were studied by high-speed video cinematography. Contraction was monitored at a speed of 9000 frames per second to study the contractile process in detail. Complete stalk contraction required approximately 9 ms. The maximal contraction velocity, 8.8 cm/s, was observed 2 ms after the start of contraction. We found that a twist appeared in the zooid during contraction. As this twist unwound, the zooid began to rotate like a right-handed screw. The subsequent stalk contraction steps, the behavior of which was similar to that of a damped harmonic oscillator, were analyzed by means of the equation of motion. From the beginning of stalk contraction, the Hookean force constant increased, and reached an upper limit of 2.23 × 10 −4 N/m 2–3 ms after the start of contraction. Thus, within 2 ms, the contraction signal spread to the entire stalk, allowing the stalk to generate the full force of contraction. The tension of an extended stalk was estimated to be 5.58 × 10 −8 N from the Hookean force constant of a stalk. This value coincides with that of the isometric tension of a glycerol-treated V. convallaria, confirming that the contractile system of V. convallaria is well preserved despite glycerol treatment.

A finitely extensible network strand model with nonlinear backbone forces and entanglement kinetics

In an earlier paper, a nonaffine network model of polymer melts was presented in which the rotation caused by shearing as well as the extension of the test strand are hindered by interactions with the network itself. In that work, it was shown that such a strand motion leads to qualitatively correct steady shear and elongational material properties, even though the strand disentanglement rate was constant and the strand force law was linear. These simplifications were accepted in order to emphasize the effects of the strand motion on material properties. In this paper, however, we show that these idealizations cause the model to fail in the start-up of shearing flow because no overshoot is seen in the shear stress growth function. To address this failure, the finitely extensible nonlinear elastic (FENE) network model is introduced in which the FENE force law replaces the Hookean force law used in the earlier finitely extensible network strand (FENS) model. Also, a nonlinear expression for the kinetics of strand disentanglement replaces the assumption of a constant rate of disentanglement. Material properties for the FENE network model are generated by stochastic simulation. The simulation results show that these modifications produce overshoot in the shear stress growth function and result in a more consistent description of finite network strand extensibility.

Nonlinear dumbbell model for flexible polymers: dynamical phenomena

We study the stress growth upon the inception of steady shear and steady elongational flow for a nonlinear dumbbell model for flexible macromolecules suspended in a Newtonian fluid. The internal force law is made nonlinear by adding a cubic term to the Hookean force law. The generalization to a nonlinear chainlike bead—spring model is outlined. In order to consider higher concentration effects also the configuration-dependent tensorial mobility is included. We use the mean-field approximation and test this approximation with a Brownian dynamics simulation. The mean-field results for the material functions are compared with experimental results for both flow situations.