Infinitesimal Change Research Articles

Local fibrations of topological entropy for fibred systems

Given a fibred dynamical system, we introduce the notions of entropy fiber of a fibre for topological entropy, Bowen entropy and packing entropy, which quantifies the “infinitesimal change” in the dynamics of a fibre with respect to its neighboring fibres, this gives rise to an (upper semicontinuous) fibre function. Besides, we show that the topological entropy (Bowen entropy and packing entropy, resp.) of the system is the supremum of the topological entropy fiber (Bowen entropy and packing entropy, resp.) of its fibres, which provides a new perspective on the study of entropy in fibred systems.

Heat engines with finite reservoirs

Typical textbook analyses of heat engines assume that the temperatures of thermal reservoirs do not change; that is, the reservoirs have infinite heat capacity. Several authors have investigated the performance of reversible heat engines for which reservoirs have finite heat capacities and thus the reservoir temperatures do change with time. We find that previous studies have been too restrictive in that they assumed that the reservoir temperature change per cycle is infinitesimal and that the engine ceases operation when the hot reservoir cools to match the temperature of the cold reservoir. We model a Carnot-like engine and show that (1) the problem can be solved exactly with no requirement for infinitesimal temperature changes and (2) there is nothing special about the instant when the temperatures converge, with the device transitioning smoothly from a heat engine to a refrigerator. We find explicit expressions for the limiting reservoir temperature and the total efficiency.

Quantum–Classical Hybrid Dynamics: Coupling Mechanisms and Diffusive Approximation

In this paper we demonstrate that any Markovian master equation defining a completely positive evolution for a quantum–classical hybrid state can always be written in terms of four basic coupling mechanisms. Each of them is characterized by a different “backaction” on each subsystem. On this basis, for each case, we find the conditions under which a diffusive limit is approached, that is, the time evolution can be approximated in terms of the first and second derivatives of the hybrid state with respect to a classical coordinate. In this limit, the restricted class of evolutions that guaranty the positivity of the hybrid state at all times (quantum Fokker-Planck master equations) emerges when the coupling mechanisms lead to infinitesimal (non-finite) changes in both the quantum and classical subsystems. A broader class of diffusive evolutions is obtained when positivity is only granted after a transient time or alternatively is granted after imposing an initial finite width on the state of the classical subsystem. A set of representative examples support these results.

Direction of Spontaneous Processes in Non-Equilibrium Systems with Movable/Permeable Internal Walls.

We consider three different systems in a heat flow: an ideal gas, a van der Waals gas, and a binary mixture of ideal gases. We divide each system internally into two subsystems by a movable wall. We show that the direction of the motion of the wall, after release, under constant boundary conditions, is determined by the same inequality as in equilibrium thermodynamics dU-đQ≤0. The only difference between the equilibrium and non-equilibrium laws is the dependence of the net heat change, đQ, on the state parameters of the system. We show that the same inequality is valid when introducing the gravitational field in the case of both the ideal gas and the van der Waals gas in the heat flow. It remains true when we consider a thick wall permeable to gas particles and derive Archimedes' principle in the heat flow. Finally, we consider the Couette (shear) flow of the ideal gas. In this system, the direction of the motion of the internal wall follows from the inequality dE-đQ-đWs≤0, where dE is the infinitesimal change in total energy (internal plus kinetic) and đWs is the infinitesimal work exchanged with the environment due to the shear force imposed on the flowing gas. Ultimately, we synthesize all these cases within a general framework of the second law of non-equilibrium thermodynamics.

Gradients of properties increase the morphing and stiffening performance of bioinspired synthetic fin rays

State-of-the-art morphing materials are either very compliant to achieve large shape changes (flexible metamaterials, compliant mechanisms, hydrogels), or very stiff but with infinitesimal changes in shape that require large actuation forces (metallic or composite panels with piezoelectric actuation). Morphing efficiency and structural stiffness are therefore mutually exclusive properties in current engineering morphing materials, which limits the range of their applicability. Interestingly, natural fish fins do not contain muscles, yet they can morph to large amplitudes with minimal muscular actuation forces from the base while producing large hydrodynamic forces without collapsing. This sophisticated mechanical response has already inspired several synthetic fin rays with various applications. However, most ‘synthetic’ fin rays have only considered uniform properties and structures along the rays while in natural fin rays, gradients of properties are prominent. In this study, we designed, modeled, fabricated and tested synthetic fin rays with bioinspired gradients of properties. The rays were composed of two hemitrichs made of a stiff polymer, joined by a much softer core region made of elastomeric ligaments. Using combinations of experiments and nonlinear mechanical models, we found that gradients in both the core region and hemitrichs can increase the morphing and stiffening response of individual rays. Introducing a positive gradient of ligament density in the core region (the density of ligament increases towards the tip of the ray) decreased the actuation force required for morphing and increased overall flexural stiffness. Introducing a gradient of property in the hemitrichs, by tapering them, produced morphing deformations that were distributed over long distances along the length of the ray. These new insights on the interplay between material architecture and properties in nonlinear regimes of deformation can improve the designs of morphing structures that combine high morphing efficiency and high stiffness from external forces, with potential applications in aerospace or robotics.

Gradualism, bifurcation and fading qualia

Abstract When reasoning about dependence relations, philosophers often rely on gradualist assumptions, according to which abrupt changes in a phenomenon of interest can result only from abrupt changes in the low-level phenomena on which it depends. These assumptions, while strictly correct if the dependence relation in question can be expressed by continuous dynamical equations, should be handled with care: very often the descriptively relevant property of a dynamical system connecting high- and low-level phenomena is not its instantaneous behaviour but its stable fixed points (those in the vicinity of which it spends most of the time, after comparatively short transitory periods), and stable fixed points can change abruptly as a result of infinitesimal changes of the low-level phenomenon. We illustrate this potential gradualist trap by showing that Chalmers’ fading qualia argument falls into it.

Design guidelines for the morphing of stiff lattice materials

Materials and structures with shape morphing capabilities are attractive in aerospace, maritime vehicles and robotics: such systems offer lighter weight, better distributed stresses, simpler kinematics, smaller numbers of actuators and higher reliability. Current morphing materials are either very compliant to achieve large shape changes (metamaterials, compliant mechanisms, hydrogels), or very stiff but with infinitesimal shape changes that require large actuation forces (metallic or composite sandwich panels with piezoelectric actuation). Morphing efficiency and structural stiffness are therefore mutually exclusive properties in current engineering morphing materials, which limit the range of their application. Here we propose a design for a stiff morphing beam made of “meta-elements” with unusual combinations of elastic properties, and which can be actuated from a simple push/pull at the base. To capture the performance of the beam we propose and use three metrics: The “morphing amplitude”, the “morphing compliance” and “structural stiffness”. We develop nonlinear finite elements to explore a wide range of designs, and to show some inevitable tradeoffs of stiffness and morphing. To aid the design of the structure we propose ternary diagrams to select optimum combinations of properties for the meta-elements. Finally, we fabricated and tested three prototypes of stiff morphing beams to validate the approach.

Complexity in the Lipkin-Meshkov-Glick model.

We study complexity in a spin system with infinite-range interaction, via the paradigmatic Lipkin-Meshkov-Glick (LMG) model, in the thermodynamic limit. Exact expressions for the Nielsen complexity (NC) and the Fubini-Study complexity (FSC) are derived, which helps us to establish several distinguishing features compared to complexity in other known spin models. In a time-independent LMG model, close to phase transition, the NC diverges logarithmically, much like the entanglement entropy. Remarkably, however, in a time-dependent scenario, this divergence is replaced by a finite discontinuity, as we show by using the Lewis-Riesenfeld theory of time-dependent invariant operators. The FSC of a variant of the LMG model shows different behavior compared to quasifree spin models. Namely, it diverges logarithmically when the target (or reference) state is near the separatrix. Numerical analysis indicates that this is due to the fact that geodesics starting with arbitrary boundary conditions are "attracted" toward the separatrix and that near this line, a finite change in the affine parameter of the geodesic produces an infinitesimal change of the geodesic length. The same divergence is shared by the NC of this model as well.

Irreversible thermodynamics of surfaces and interfaces: Special reference to the strained thin solid films on the substrates: Theory and practice

The realization of nanoscale devices largely depends on our ability to control and manipulate interfacial interactions and, thus, understanding of the mechanisms of surface/interface instabilities. In this work, theoretically as well as technologically important and distinct two thermodynamic systems, which are exposed to (isobaric) and isolated from (isochoric) external body forces and surface tractions, are formulated by using irreversible thermodynamics in combination with the generalized variational method. The starting point for the present formulation closely follows up the Fowler and Guggenheim [Statistical Thermodynamics (University Press, Cambridge, 1952)] interpretation of the Planck inequality [Über Prinzip Vermehrung Entropie: Ann. Phys. Series 2(32), 462 (1887)] for isothermal reversible and irreversible (natural) infinitesimal changes in heterogeneous systems (multi-phase and multi-component). By combining this fundamental principle with the interlink between the dissipation function and global internal entropy production postulates, two distinct sets of governing equations for the surface drift-diffusion flux as well as the rate of evaporation/condensation and/or the growth/recrystallization of amorphous solid thin films are obtained for isochoric and isobaric systems. The role of Eshelby's energy-momentum tensor in the generalized potential for the interface displacement is found to differ (opposite in sign) for isochoric and isobaric systems. To demonstrate the importance of these sign conflicts, two sets of computer experiments are performed on isochoric and isobaric systems. They showed us that the elastic strain energy density contribution to the generalized driving force for surface drift-diffusion alone favoring flat and smooth surfaces in isobaric systems regardless of the sign of the uniaxial stress (healing), rather than causing the surface roughness and even catastrophic crack initiation as the case in internally strained isochoric systems. Computer simulations allowed us to track down the dynamical behavior of test modules by furnishing surface and strain energy variations, combined with the Global Helmholtz free change, which indicates the existence of two regimes: initial smooth surface undulations followed up by the rather chaotic crack formation and propagation stage at the middle of the thin film supported by the stiff substrate. In this study, we mainly focused on the development kinetics of “Stranski–Krastanow” island-type morphology, initiated by the nucleation route rather than the surface roughening scheme. The physicomathematical model, which is based on the irreversible thermodynamics treatment of surfaces and interfaces with singularities [T. O. Ogurtani, J. Chem. Phys. 124, 144706 (2006)], furnishes us to have autocontrol on otherwise free-motion of the triple junction contour line between the substrate and the droplet without presuming any equilibrium dihedral contact (wetting) angles at edges. We have also demonstrated the formation of the Stranski–Krastanow (SK)-type doublet islanding (quantum dots) as a stationary nonequilibrium state in an epitaxially strained thin flat droplet on a rigid substrate by introducing the wetting potential—invoked by the quantum confinement—into the scenario and carefully selecting the system parameters (size and shape) for the isochoric system represented by [Ge/Si (100)]. It has been also shown that on the contrary to common perceptions, the Stranski–Krastanow islands are in genuine stationary nonequilibrium states in the sense of Prigogine if one invokes proper free-moving boundary conditions at triple junctions deduced from the irreversible thermodynamics rather than ad hoc periodic or reflecting constrains at the edges.

On elastic gaps in strain gradient plasticity: 3D discrete dislocation dynamics investigation

Although presenting attractive features in dealing with small-scale size effects, strain gradient plasticity (SGP) theories can lead to uncommon phenomena for some boundary value problems. Almost all non-incremental (Gurtin-type) SGP theories including thermodynamically-consistent higher-order dissipation predict elastic gaps under certain non-proportional loading conditions. An elastic gap is defined as a finite change in the equivalent yield stress after an infinitesimal change in the strain conditions, at the occurrence of the non-proportional loading source. The existence of such gaps in reality is largely questioned and represents a major source of uncertainty preventing the development of robust SGP theories for real small-scale applications. Using 3D discrete dislocation dynamics (3D-DDD), the present paper aims at investigating size effects within micron-scale single crystal structures under various non-proportional loading conditions, including tension–compression–passivation, bending–passivation and tension–bending. An in-depth investigation of the occurrence of elastic gaps under these conditions, which are known to entail such gaps when using classical non-incremental SGP theories, is conducted. The obtained 3D-DDD results reproduce well known experimentally confirmed size effects like Hall–Petch effect, Asaro’s type III kinematic hardening and reversible plasticity. However, no evidence of the phenomenon of elastic gaps is found, which constitutes a first indication that this phenomenon may not exist in reality. The simulations are performed on face-centered cubic (FCC) Nickel single grains with cuboid shapes ranging from 2μm to 15μm and different orientations.

Heartbeat and Respiration Rate Prediction Using Combined Photoplethysmography and Ballisto Cardiography

Owing to the recent trends in remote health monitoring, real-time applications for measuring Heartbeat Rate and Respiration Rate (HARR) from video signals are growing rapidly. Photo Plethysmo Graphy (PPG) is a method that is operated by estimating the infinitesimal change in color of the human face, rigid motion of facial skin and head parts, etc. Ballisto Cardiography (BCG) is a nonsurgical tool for obtaining a graphical depiction of the human body’s heartbeat by inducing repetitive movements found in the heart pulses. The resilience against motion artifacts induced by luminance fluctuation and the patient’s mobility variation is the major difficulty faced while processing the real-time video signals. In this research, a video-based HARR measuring framework is proposed based on combined PPG and BCG. Here, the noise from the input video signals is removed by using an Adaptive Kalman filter (AKF). Three different algorithms are used for estimating the HARR from the noise-free input signals. Initially, the noise-free signals are subjected to Modified Adaptive Fourier Decomposition (MAFD) and then to Enhanced Hilbert vibration Decomposition (EHVD) and finally to Improved Variation mode Decomposition (IVMD) for attaining three various results of HARR. The obtained values are compared with each other and found that the EHVD is showing better results when compared with all the other methods.

Synergistic effect of h-BN on thermal conductivity of polymer composites

The conductivity characteristics of polymers and polymer composites have become more significant recently. Good heat dissipation is required in many applications, such as circuit boards and heat exchangers, so it is essential to develop the thermal conductivity characteristics of the materials. The micro-fillers have been replaced with nano or hybrid fillers to increase the low thermal conductivity of the polymer. Hexagonal boron nitride (h-BN) and multi-walled carbon nanotubes (MW-CNT), both of which have good conductivity properties, are two popular filling materials. The presence of hydroxyl and amino active groups at the corners of the hexagonal structure of BN improves the thermal conductivity properties of the polymer composite. In addition, it shows high thermal conductivity behavior in polymer composite structures with BN and MW-CNT. It is essential to demonstrate the effects of the volume fraction of additives on the thermal properties of composites with various approaches. In this study, the thermal conductivity behaviors of h-BN/high-density polyethylene and h-BN/MW-CNT/high-density polyethylene composites are demonstrated using the theoretical Bruggeman model, which is based on the assumption that there are constant infinitesimal changes in the material so that there is an interaction between particles. The coefficient of determination (R²) between the thermal conductivity values of the composites and the predictions of the Bruggeman theoretical model is greater than 0.98. This way, the synergetic effect of h-BN and MW-CNT/h-BN additives on thermal conductivity has been theoretically proven.

Access Convergence for Heavy Load Markov Ethernet Bursty Traffic Using Two-level Statistical Multiplexing

A method for modeling aggregated heavy Markov bursty Ethernet traffic from different sources is proposed in this paper, particularly that prevailing between gateway services and internet routing devices, with the aim of achieving rate accommodation. In other words, to accommodate different rates while filtering out delays in the queue, to achieve access network convergence. Although gateway functions can be used to achieve this by adapting service rates, as many gateways as services are required. Instead of considering the distributed gateway services method, statistical multiplexing is chosen for this study for cost efficiency in network resources. Unfortunately, statistical multiplexing exhibits greater packet variation (jitter) and transfer delay. These delays, basically resulting from positive correlations or time dependency in the queue system, are addressed through infinitesimal queue modeling, based on the diffusion process approximated by Ornstein-Uhlenbeck, which deals with infinitesimal changes in the Markov queue. The related analysis has resulted in an exponential queueing model for univariate and/or multivariate servers obtained through Markov Gaussian approximation. An experiment based on two different voice algorithms shows rate accommodation, and a fluid solution, which is dynamically outputted according to the transmission link availability during each transition time, without any significant delay. Hence, better transfer delay and rate control is obtained through the proposed two multiplexing levels within an Ethernet LAN

Adsorption of Methylene Blue on the Surface of Polymer Membrane; Dependence on the Isotopic Composition of Liquid Matrix.

As was found in our previous works, when Nafion swells in water, polymer fibers unwind into the bulk of the surrounding liquid. This effect is controlled by the content of deuterium in water. Here, we present the results of studying the dynamics of methylene blue (MB) adsorption on the Nafion surface for MB solutions based on natural water (deuterium content is 157 ppm, the unwinding effect occurs) and based on deuterium-depleted water (DDW; deuterium content is 3 ppm, there is no unwinding). In addition, we studied the dynamics of water desorption during drying of the Nafion polymer membrane after soaking in MB solution based on natural water and DDW. It turned out that in the case of natural water, the rate of MB adsorption and water desorption is higher than in the case of DDW. It also turned out that the amount of MB adsorbed on the membrane in the case of natural water is greater than in the case of DDW. Finally, it was found that the desorption of water during drying is accompanied by a rearrangement of the absorption spectrum of Nafion. This rearrangement occurs earlier in the case of DDW. Thus, by infinitesimal changes in the deuterium content (from 3 to 157 ppm) in an aqueous solution, in which a polymer membrane swells, we can control the dynamics of adsorption and desorption processes. A qualitative model, which connects the observed effects with the slowing down of diffusion processes inside the layer of unwound fibers, is proposed.

Effective bilipschitz bounds on drilling and filling

This paper proves explicit bilipschitz bounds on the change in metric between the thick part of a cusped hyperbolic 3-manifold N and the thick part of any of its long Dehn fillings. Given a bilipschitz constant J > 1 and a thickness constant epsilon > 0, we quantify how long a Dehn filling suffices to guarantee a J-bilipschitz map on epsilon-thick parts. A similar theorem without quantitative control was previously proved by Brock and Bromberg, applying Hodgson and Kerckhoff's theory of cone deformations. We achieve quantitative control by bounding the analytic quantities that control the infinitesimal change in metric during the cone deformation. Our quantitative results have two immediate applications. First, we relate the Margulis number of N to the Margulis numbers of its Dehn fillings. In particular, we give a lower bound on the systole of any closed 3-manifold M whose Margulis number is less than 0.29. Combined with Shalen's upper bound on the volume of such a manifold, this gives a procedure to compute the finite list of 3-manifolds whose Margulis numbers are below 0.29. Our second application is to the cosmetic surgery conjecture. Given the systole of a one-cusped hyperbolic manifold N, we produce an explicit upper bound on the length of a slope involved in a cosmetic surgery on N. This reduces the cosmetic surgery conjecture on N to an explicit finite search.

The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization

This article is about the quantitative homogenization theory of linear elliptic equations in divergence form with random coefficients. We derive gradient estimates on the homogenization error, i. e. on the difference between the actual solution and the two-scale expansion of the homogenized solution, both in terms of strong norms (oscillation) and weak norms (fluctuation). These estimates are optimal in terms of scaling in the ratio between the microscopic and the macroscopic scale.The purpose of this article is to highlight the usage of the recently introduced annealed Calderón-Zygmund (CZ) estimates in obtaining the above, previously known, error estimates. Moreover, the article provides a novel proof of these annealed CZ estimate that completely avoids quenched regularity theory, but rather relies on functional analysis. It is based on the observation that even on the level of operator norms, the Helmholtz projection is close to the one for the homogenized coefficient (for which annealed CZ estimates are easily obtained).In this article, we strive for simple proofs, and thus restrict ourselves to ensembles of coefficient fields that are local transformations of Gaussian random fields with integrable correlations and Hölder continuous realizations. As in earlier work, we use the natural objects from the general theory of homogenization, like the (potential and flux) correctors and the homogenization commutator. Both oscillation and fluctuation estimates rely on a sensitivity calculus, i.e. on estimating how sensitively the quantity of interest does depend on an infinitesimal change in the coefficient field, which is fed into the Spectral Gap inequality. In this article, the annealed CZ estimate is the only form in which elliptic regularity theory enters.

Hessian of Hausdorff dimension on purely imaginary directions

Bridgeman–Taylor (Math. Ann. 341 (2008), 927–943) and McMullen (Invent. Math. 173 (2008), 365–425) showed that the Weil–Petersson metric on Teichmüller space can be realized by looking at the infinitesimal change of the Hausdorff dimension of certain quasi-Fuchsian deformations. In this article, we give a similar geometric interpretation of the spectral gap pressure metric introduced by Bridgeman–Canary–Labourie–Sambarino (Geom. Dedicata 192 (2018), 57–86) on the Hitchin component for PSL d ( R ) ${{\mathsf {PSL}}}_d(\mathbb {R})$ . More generally, we investigate the Hessian of the Hausdorff dimension as a function on the space of (1,1,2)-hyperconvex representations, a class introduced in (J. reine angew. Math. 774 (2021), 1–51) which includes small complex deformations of Hitchin representations and of Θ $\Theta$ -positive representations. As another application, we prove that the Hessian of the Hausdorff dimension of the limit set at the inclusion Γ → PO ( n , 1 ) → PU ( n , 1 ) ${{\Gamma }}\rightarrow {{\mathsf {PO}}}(n,1)\rightarrow {{\mathsf {PU}}}(n,1)$ is positive definite when Γ ${{\Gamma }}$ is co-compact in PO ( n , 1 ) ${{\mathsf {PO}}}(n,1)$ (unless n = 2 $n=2$ and the deformation is tangent to X ( Γ , PO ( 2 , 1 ) ) $\mathfrak {X}({{\Gamma }}, {{\mathsf {PO}}}(2,1))$ ).

Phase equilibria and stability boundaries in a two[sbnd]component Lennard-Jones mixture

The mechanical and diffusion stability conditions of binary Lennard-Jones mixtures with respect to infinitesimal changes in state parameters are considered. In stable and metastable states, by molecular dynamics method, the pressure is determined as a function of temperature, density and composition of a Lennard-Jones mixture simulating the methane–nitrogen system. The phase equilibrium parameters are calculated. Based on the data obtained, a unified mixture state equation for liquid and vapor was created. The diffusion and mechanical spinodals of the mixture were determined.

Trigonal cluster-based ultra-sensitive surface plasmon resonance sensor for multipurpose sensing

We present an ultra-sensitive Surface Plasmon Resonance-based Photonic Crystal Fiber (SPR-PCF) sensor that can be employed for multipurpose sensing of analyte, temperature, and magnetic field. Our prototype has a trigonal cluster-based strategic pattern of circular air holes inside the fiber and can be easily fabricated using the standard Stack-and-Draw method. Thin layers of gold (Au) and titanium dioxide (TiO2) are used as the plasmonic materials surrounding the PCF. The Finite Element Method (FEM) of the commercial software COMSOL Multiphysics 5.3a is used to estimate the sensor properties. After optimization of different parameters, we recorded a maximum Amplitude Sensitivity (AS) of 7223.62 RIU−1, a maximum wavelength sensitivity (WS) of 28,500 nm/RIU, and a leading figure of merit (FOM) of 914 RIU−1. Our sensor is capable of detecting unknown analytes within the refractive index (RI) range of 1.33 to 1.42 with sensor resolutions of 1.38 × 10−6 (amplitude) and 3.51 × 10−6 RIU (wavelength). Furthermore, it can also detect temperature and magnetic field variations with the corresponding maximum sensitivity of 1.25 nm/°C (1250 pm/°C) and 0.16 nm/Oe (160 pm/Oe). An infinitesimal change in the sensor performance was observed for the ±10% fabrication tolerance analysis. So, it can be concluded that our sensor can significantly contribute to scientific, biomedical, and industrial fields due to its supreme sensing capabilities and versatile features.

Using the Carnot cycle to determine changes of the phase transition temperature

The Clausius–Clapeyron relation and its analogs in other first-order phase transitions, such as type-I superconductors, are derived using very elementary methods without appealing to the more advanced concepts of entropy or Gibbs free energy. The reasoning is based on Kelvin's formulation of the second law of thermodynamics and should be accessible to high school students. After recalling some basic facts about the Carnot cycle, we present two very different systems that undergo discontinuous phase transitions (ice/water and normal/superconductor) and construct engines that exploit the properties of these systems to produce work. In each case, we show that if the transition temperature Ttr was independent of other parameters, such as pressure or magnetic field, it would be possible to violate Kelvin's principle, i.e., to construct a perpetuum mobile of the second kind. Since the proposed cyclic processes can be realized reversibly in the limit of infinitesimal changes in temperature, their efficiencies must be equal to that of an ordinary Carnot cycle. We immediately obtain an equation of the form dT/dX=f(T,X), which governs how the transition temperature changes with the parameter X.