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We define and study the Dirichlet fractional Gaussian fields on the Sierpinski gasket and show that they are limits of fractional discrete Gaussian fields defined on the sequence of canonical approximating graphs., v2: Accepted version, to appear in Stoch. Proc. Appl

One of the main difficulties in proving convergence of discrete models of surface growth to the Kardar-Parisi-Zhang (KPZ) equation in dimensions higher than one is that the correct way to take a scaling limit, so that the limit is nontrivial, is not known in a rigorous sense. To understand KPZ growth without being hindered by this issue, this article introduces a notion of "local KPZ behavior", which roughly means that the instantaneous growth of the surface at a point decomposes into the sum of a Laplacian term, a gradient squared term, a noise term that behaves like white noise, and a remainder term that is negligible compared to the other three terms and their sum. The main result is that for a general class of surfaces, which contains the model of directed polymers in a random environment as a special case, local KPZ behavior occurs under arbitrary scaling limits, in any dimension., Comment: 32 pages. Minor revisions in this update. To appear in Comm. Math. Phys

Recent investigations have observed superdiffusion in integrable classical and quantum spin chains. An intriguing connection between these spin chains and Kardar-Parisi-Zhang (KPZ) universality class has emerged. Theoretical developments (e.g. generalized hydrodynamics) have highlighted the role of integrability as well as spin-symmetry in KPZ behaviour. However understanding their precise role on superdiffusive transport still remains a challenging task. The widely used quantum spin chain platform comes with severe numerical limitations. To circumvent this barrier, we focus on a classical integrable spin chain which was shown to have deep analogy with the quantum spin-$\frac{1}{2}$ Heisenberg chain. Remarkably, we find that KPZ behaviour prevails even when one considers integrability-breaking but spin-symmetry preserving terms, strongly indicating that spin-symmetry plays a central role even in the non-perturbative regime. On the other hand, in the non-perturbative regime, we find that energy correlations exhibit clear diffusive behaviour. We also study the classical analog of out-of-time-ordered correlator (OTOC) and Lyapunov exponents. We find significant presence of chaos for the integrability-broken cases even though KPZ behaviour remains robust. The robustness of KPZ behaviour is demonstrated for a wide class of spin-symmetry preserving integrability-breaking terms., 10 pages, 9 figures (including supplementary material)

The thermodynamic relations in the Tsallis statistics were studied with physical quantities. An additive entropic variable related to the Tsallis entropy was introduced by assuming the form of the first law of the thermodynamics. The fluctuations in the Tsallis statistics were derived with physical quantities with the help of the introduced entropic variable. It was shown that the mean squares of the fluctuations of the physical quantities in the Tsallis statistics are the same as those in the conventional statistics. The mean square of the fluctuation of the Tsallis entropy and the mean square of the fluctuation of the Tsallis temperature were also derived. The mean square of the relative fluctuation of the Tsallis entropy and the mean square of the relative fluctuation of the Tsallis temperature are represented with heat capacities. It was shown that these fluctuations of the Tsallis quantities have the $q$-dependent terms in the Tsallis statistics of the entropic parameter $q$., Comment: 10 pages

The asymptotics is found for eigenvalues of polynomially compact pseudodifferential operators of the zeroth order.

Schramm's Locality Conjecture asserts that the value of the critical parameter pc of a graph satisfying pc < 1 depends only on its local structure. In this paper, we prove this conjecture in the particular case of transitive graphs with polynomial growth. Our proof relies on two recent works about such graphs, namely supercritical sharpness of percolation by the same authors and a finitary structure theorem by Tessera and Tointon., Electronic Communications in Probability, 28, ISSN:1083-589X

The aim of this paper is to unify the extended Mittag-Leffler function and generalized Q function and define a unified Mittag-Leffler function. Both the extended Mittag-Leffler function and generalized Q function can be obtained from the unified Mittag-Leffler function. The Laplace, Euler beta, and Whittaker transformations are applied for this function, and generalized formulas are obtained. These formulas reproduce integral transformations of various deduced Mittag-Leffler functions and Q function. Also, the convergence of this unified Mittag-Leffler function is proved, and an associated fractional integral operator is constructed.

We study the antiferromagnetic (AF) spin-1 Blume-Capel (BC) model on a recursive lattice called twofold Cayley tree. Both sublattice magnetizations of the Ising spins are exactly calculated with the aim to obtain phase diagrams and thermal variations of the sublattice magnetizations. The finite-temperature phase diagrams exhibit a small reentrant region if the twofold Cayley tree with a sufficiently high coordination number $$q>5$$ is considered. The results are also compared to obtained by Bethe lattice consideration in recursive approach.

For a real biquadratic field, we denote by $$\lambda$$ , $$\mu$$ and $$\nu$$ the Iwasawa invariants of cyclotomic $$\mathbb{Z}_{2}$$ -extension of $$k$$ . We give certain families of real biquadratic fields $$k$$ such that $$\mu=0$$ .

The $S=1$ bilinear-biquadratic Heisenberg exchange model on the triangular lattice with a single-ion anisotropy has previously been shown to host a number of exotic magnetic and nematic orders [Moreno-Cardoner $\textit{et al.}$, Phys. Rev. B $\textbf{90}$, 144409 (2014)], including an extensive region of "supersolid" order. In this work, we amend the model by an XXZ anisotropy in the exchange interactions. Tuning to the limit of an exactly solvable $S=1$ generalized Ising-/Blume-Capel-type model provides a controlled limit to access phases at finite transverse exchange. Notably, we find an additional macroscopically degenerate region in the phase diagram and study its fate under perturbation theory. We further map out phase diagrams as a function of the XXZ anisotropy parameter, ratio of bilinear and biquadratic interactions and single-ion anisotropy, and compute corrections to the total ordered moment in various phases using systematically constructed linear flavor-wave theory. We also present linear flavor-wave spectra of various states, finding that the lowest-energy band in three-sublattice generalized (i.e. with $S^z=\pm1,0$) Ising/Blume-Capel states, stabilized by strong exchange anisotropies, is remarkably flat, opening up the way to flat-band engineering of magnetic excitation via stabilizing non-trivial Ising-ordered ground states., 23 pages, 13 figures, 2 tables

We study the transient behavior of damage propagation in the two-dimensional spin-$1$ Blume-Capel model using Monte Carlo simulations with Metropolis dynamics. We find that, for a particular region in the second-order transition regime of the crystal field--temperature phase diagram of the model, the average Hamming distance decreases exponentially with time in the weakly damaged system. Additionally, its rate of decay appears to depend linearly on a number of Hamiltonian parameters, namely the crystal field, temperature, applied magnetic field, but also on the amount of damage. Finally, a comparative study using Metropolis and Glauber dynamics indicates a slower decay rate of the average Hamming distance for the Glauber protocol., Comment: 17 pages Latex including 8 captioned pdf figures, J. Stat. Phys. (in press) 2022

Tsallis entropy is a flexible extension of Shanon (logarithm) entropy. Since entropy measures indeterminacy of an uncertain random variable, this paper proposes the concept of partial Tsallis entropy for uncertain random variables as a flexible devise in chance theory. An approach for calculating partial Tsallis entropy for uncertain random variables, based on Monte Carlo simulation, is provided. As an application in finance, partial Tsallis entropy is invoked to optimize portfolio selection of uncertain random returns via crow search algorithm.

Oriented short hemp fibre mats were produced using dynamic sheet forming (DSF) incorporating cellulose nanocrystals (CNCs) to improve their integrity. The CNCs were found to act as a binder and improve mechanical strength of the mats as well as the strength of polypropylene matrix composites produced with the mats. Improved thermal stability was also obtained for composites by using CNC treatment of fibre mats.

We establish a nonequilibrium dynamic and thermodynamic formalism in the landscape-flux framework for open Hamiltonian systems in contact with multiple heat baths governed by stochastic dynamics. To systematically characterize nonequilibrium steady states, the nonequilibrium trinity construct is developed, which consists of detailed balance breaking, nonequilibrium potential landscape, and irreversible probability flux. We demonstrate that the temperature difference of the heat baths is the physical origin of detailed balance breaking, which generates the nonequilibrium potential landscape characterizing the nonequilibrium statistics and creates the irreversible probability flux signifying time irreversibility, with the latter two aspects closely connected. It is shown that the stochastic dynamics of the system can be formulated in the landscape-flux form, where the reversible force drives the conservative Hamiltonian dynamics, the irreversible force consisting of a landscape gradient force and an irreversible flux force drives the dissipative dynamics, and the stochastic force adds random fluctuations to the dynamics. The possible connection of the nonequilibrium trinity construct to nonequilibrium phase transitions is also suggested. A set of nonequilibrium thermodynamic equations, applicable to both nonequilibrium steady states and transient relaxation processes, is constructed. We find that an additional thermodynamic quantity, named the mixing entropy production rate, enters the nonequilibrium thermodynamic equations. It arises from the interplay between detailed balance breaking and transient relaxation, and it also relies on the conservative dynamics. At the nonequilibrium steady state, the heat flow, entropy flow, and entropy production are demonstrated to be thermodynamic manifestations of the nonequilibrium trinity construct. The general nonequilibrium formalism is applied to a class of solvable systems consisting of coupled harmonic oscillators. A more specific example of two harmonic oscillators coupled to two heat baths is worked out in detail. The example may facilitate connection with experiments.

Shear viscosity of a nonequilibrium quantum field φ4 is considered in the framework of the Classical statistical approximation. The dependence on the initial energy distribution is investigated. The generalisation of the Green–Kubo relation to the case of the stationary nonequilibrium field is obtained.

It has been argued that non-Gaussian statistics provide a natural framework to investigate semiclassical effects in the context of Planck-scale deformations of the Heisenberg uncertainty relation. Here we substantiate this point by considering the Unruh effect as a specific playground. By working in the realm of quantum field theory, we reformulate the derivation of the modified Unruh effect from the generalized uncertainty principle (GUP) in the language of the nonextensive Tsallis thermostatistics. We find a nontrivial monotonic relation between the nonextensivity index q and the GUP deformation parameter $$\beta $$ β , which generalizes an earlier result obtained in quantum mechanics. We then extend our analysis to black hole thermodynamics. We preliminarily discuss our outcome in the broader context of an effective description of Planck-scale gravitational physics based on Tsallis theory.

A simplified method for calculating the spectral emission of nonequilibrium air plasmas is developed. In order to obtain the nonequilibrium energy level populations, the nonequilibrium coefficients are introduced into the Saha-Boltzmann equation. These nonequilibrium coefficients are calculated by using several significant radiative processes. An approach to the determination of nonequilibrium electronic energy level populations of diatomic molecules is also presented. Based on the method, spectral emission of atoms and molecules in a typical air plasma cell is investigated. The results reveal that there is a significant difference between the nonequilibrium and equilibrium emission. We apply the method to the nonequilibrium AVCO R-156 experiment. Good agreement with the NEQAIR code and the measured data is shown, indicating that the method is reasonable and has good accuracy.

R-package for percolation analysis

The one-dimensional Kardar-Parisi-Zhang (KPZ) equation is becoming an overarching paradigm for the scaling of nonequilibrium, spatially extended, classical and quantum systems with strong correlations. Recent analytical solutions have uncovered a rich structure regarding its scaling exponents and fluctuation statistics. However, the zero surface tension or zero viscosity case eludes such analytical solutions and has remained ill-understood. Using numerical simulations, we elucidate a well-defined universality class for this case that differs from that of the viscous case, featuring intrinsically anomalous kinetic roughening, despite previous expectations for systems with local interactions and time-dependent noise and ballistic dynamics. The latter may be relevant to recent quantum spin chain experiments which measure KPZ and ballistic relaxation under different conditions. We identify the ensuing set of scaling exponents in previous discrete interface growth models related with isotropic percolation, and show it to describe the fluctuations of additional continuum systems related with the noisy Korteweg-de Vries equation. Along this process, we additionally elucidate the universality class of the related inviscid stochastic Burgers equation., 9 pages, 8 figures, Physical Review E (in press, 2022)

Depinning of elastic systems advancing on disordered media can usually be described by the quenched Edwards-Wilkinson equation (qEW). However, additional ingredients such as anharmonicity and forces that can not be derived from a potential energy may generate a different scaling behavior at depinning. The most experimentally relevant is the Kardar-Parisi-Zhang (KPZ) term, proportional to the square of the slope at each site, which drives the critical behavior into the so-called quenched KPZ (qKPZ) universality class. We study this universality class both numerically and analytically: by using exact mappings we show that at least for $d=1,2$ this class encompasses not only the qKPZ equation itself, but also anharmonic depinning and a well-known class of cellular automata introduced by Tang and Leschhorn. We develop scaling arguments for all critical exponents, including size and duration of avalanches. The scale is set by the confining potential strength $m^2$. This allows us to estimate numerically these exponents as well as the $m$-dependent effective force correlator $\Delta(w)$, and its correlation length $\rho:= \Delta(0)/|\Delta'(0^+)|$. Finally we present a new algorithm to numerically estimate the effective ($m$-dependent) elasticity $c$, and the effective KPZ non-linearity $\lambda$. This allows us to define a dimensionless universal KPZ amplitude ${\cal A}:=\rho \lambda /c$, which takes the value ${\cal A}=1.10(2)$ in all systems considered in $d=1$. This proves that qKPZ is the effective field theory for all these models. Our work paves the way for a deeper understanding of depinning in the qKPZ class, and in particular, for the construction of a field theory that we describe in a companion paper., Comment: companion paper to arXiv:2207.09037. 19 pages, 17 figures. v2: accepted version

We introduce a field-theoretic approach for describing the critical behavior of nonextensive systems, systems displaying global correlations among their degrees of freedom, encoded by the nonextensive parameter $q$. As some applications, we report, to our knowledge, the first analytical computation of both universal static and dynamic $q$-dependent nonextensive critical exponents for O($N$) vector models, valid for all loop orders and $|q - 1| < 1$. Then emerges the new nonextensive O($N$)$_{q}$ universality class. We employ six independent methods which furnish identical results. Particularly, the results for nonextensive 2d Ising systems, exact within the referred approximation, agree with that obtained from computer simulations, within the margin of error, as better as $q$ is closer to $1$. We argue that the present approach can be applied to all models described by extensive statistical field theory as well. The results show an interplay between global correlations and fluctuations., Submitted to Journal on November-23-2021, 5 pages, IX Tables

We present a purely elementary proof that the Rauzy gasket has Hausdorff dimension strictly smaller than two., 10 pages, 2 figures

We study a quantum spin-1/2 chain that is dual to the canonical problem of non-equilibrium Kawasaki dynamics of a classical Ising chain coupled to a thermal bath. The Hamiltonian is obtained for the general disordered case with non-uniform Ising couplings. The quantum spin chain (dubbed Ising-Kawasaki) is stoquastic, and depends on the Ising couplings normalized by the bath's temperature. We give its exact ground states. Proceeding with uniform couplings, we study the one- and two-magnon excitations. Solutions for the latter are derived via a Bethe Ansatz scheme. In the antiferromagnetic regime, the two-magnon branch states show intricate behavior, especially regarding their hybridization with the continuum. We find that that the gapless chain hosts multiple dynamics at low energy as seen through the presence of multiple dynamical critical exponents. Finally, we analyze the full energy level spacing distribution as a function of the Ising coupling. We conclude that the system is non-integrable for generic parameters, or equivalently, that the corresponding non-equilibrium classical dynamics are ergodic., Comment: 11+3 pages, 7+2 figures. v3: Minor changes including more detailed description of low energy excited states

We generalize time-evolving matrix product operators method to nonequilibrium quantum transport problems. The nonequilibrium current is obtained via numerical differentiation of the generating functional which is represented as a tensor network. The approach is numerically exact and the non-Markovian effects are fully taken into account. In the transport process, a part of the heat that flows out from a bath flows into the system and other baths, and the rest is stored in the system-bath coupling part. We take the spin-boson model as a demonstration to show the details of this heat flowing and the establishment of a steady current between two baths., 9 pages, 9 figures

Gibbs-Boltzmann entropy leads to systems that have a strong dependence on initial conditions. In reality, most materials behave quite independently of initial conditions. Nonextensive entropy or Tsallis entropy leads to nonextensive statistical mechanics. In this paper, we calculate the Tsallis form of Clausius inequality and then determine the upper bound for extracting work from the small system in the nonextensive statistical mechanics with mutual information. In the following, we extract mutual information and adjust Maxwell's demon with quantum feedback control., 13 pages

Matlab code for analyzing developmental data, and mat files.

How does the percolation transition behave in the absence of quenched randomness? To address this question, we study two nonrandom self-dual quasiperiodic models of square-lattice bond percolation. In both models, the critical point has emergent discrete scale invariance, but none of the additional emergent conformal symmetry of critical random percolation. From the discrete sequences of critical clusters, we find fractal dimensions of $D_f=1.911943(1)$ and $D_f=1.707234(40)$ for the two models, significantly different from $D_f=91/48=1.89583...$ of random percolation. The critical exponents $\nu$, determined through a numerical study of cluster sizes and wrapping probabilities on a torus, are also well below the $\nu=4/3$ of random percolation. While these new models do not appear to belong to a universality class, they demonstrate how the removal of randomness can fundamentally change the critical behavior., Comment: 17 pages (17 figures) + 3 appendices (6 figures, 3 tables). v3 revisions: integrated supplement into main text; added analysis of a second quasiperiodic model, determination of fractal dimension of hulls, and investigation of universality; revised estimates of $D_f$ and $\nu$ for original model using nominally infinite methods; removed discussion of energy correlator and red bonds

More detailed and accurate modeling is very important for analyzing and optimizing the tribological performance of the piston-ring-cylinder liner system. However, due to the difficulty of modeling and solving, theoretical studies on the three-dimensional (3D) tribodynamics of piston rings are limited. The tribodynamic model which couples the dynamics, mixed lubrication, and blow-by of piston-ring pack assemblies has not been found yet. Therefore, in this study, a 3D tribodynamic model of the piston-ring pack is developed considering the influence of piston secondary motion and the interaction forces and moments between piston ring and cylinder liner as well as between piston ring and ring groove. In addition to the ring end gaps, the influence of ring dynamics is also contained in the blow-by model. Coupled with gas flows and piston rotation, ring motions in the ring groove are investigated. It is found that ring dynamics has significant effects on the tribological performance, the axial reversing movement of piston ring is the main cause of gas pressure oscillation, piston motion has an obvious influence on the ring dynamics, the interaction forces and moments between piston and rings increase the secondary motion amplitude of piston, especially near the fire top dead center.

The theory of superstatistics, originally proposed for the study of complex nonequilibrium systems, has recently been extended to studies of small systems interacting with a finite environment, because such systems display interestingly similar statistical behavior. In both situations there are several applicable definitions of inverse temperature, either intrinsic or dependent of the statistical ensemble. In this work we develop these concepts focusing our attention on a region of an isolated, one-dimensional Ising chain as an example of a subsystem that does not follow the canonical Gibbs distribution. For this example, we explicitly show that superstatistics cannot describe the behavior of the subsystem, and verify a recently reported relation between the fundamental and microcanonical inverse temperatures. Our results hint at a new framework for dealing with regions of microcanonical systems with positive heat capacity, which should be described by some new class of statistical ensembles outside superstatistics but still preserving the notion of temperature fluctuations.

In the realm of Boltzmann-Gibbs statistical mechanics there are three well known isomorphic connections with random geometry, namely (i) the Kasteleyn-Fortuin theorem which connects the $\lambda \to 1$ limit of the $\lambda$-state Potts ferromagnet with bond percolation, (ii) the isomorphism which connects the $\lambda \to 0$ limit of the $\lambda$-state Potts ferromagnet with random resistor networks, and (iii) the de Gennes isomorphism which connects the $n \to 0$ limit of the $n$-vector ferromagnet with self-avoiding random walk in linear polymers. We provide here strong numerical evidence that a similar isomorphism appears to emerge connecting the energy $q$-exponential distribution $\propto e_q^{-\beta_q \varepsilon}$ (with $q=4/3$ and $\beta_q \omega_0 =10/3$) optimizing, under simple constraints, the nonadditive entropy $S_q$ with a specific geographic growth random model based on preferential attachment through exponentially-distributed weighted links, $\omega_0$ being the characteristic weight., Comment: 5 pages and 2 figures

In momentum-based impact models used in accident reconstructions, the restitution coefficient is a key role to determine impulses and delta-V accurately in car-to-car collisions. The restitution co...

An analytical method for programming piston displacements for constant flow rate piston pumps is presented. A total of two trigonometric transition functions are introduced to express the piston velocities during the transition processes, which can guarantee both constant flow rates and the continuity of piston accelerations. A kind of displacement function of pistons, for two-piston pumps, and two other kinds, for three-piston pumps, are presented, and the physical meaning of their parameters is also discussed. The results show that, with the given transition functions, cam profiles can be designed analytically with parameterized forms, and the maximum accelerations of the pistons are determined by the width of the transition domain and the rotational velocities of the cams, which will affect contact forces between cams and followers.

Applying the (infinite) density-matrix renormalisation group technique, we explore the effect of an explicit dimerisation on the ground-state phase diagram of the spin-1 $XXZ$ chain with single-ion anisotropy $D$. We demonstrate that the Haldane phase between large-$D$ and antiferromagnetic phases survives up to a critical dimerisation only. As a further new characteristic the dimerisation induces a direct continuous Ising quantum phase transition between the large-$D$ and antiferromagnetic phases with central charge $c=1/2$, which terminates at a critical end-point where $c=7/10$. Calculating the critical exponents of the order parameter, neutral gap and spin-spin-correlation function, we find $\beta=1/8$ (1/24), $\nu=1$ (5/9), and $\eta=1/4$ (3/20), respectively, which proves the Ising (tricritical Ising) universality class in accordance with field-theoretical predictions., Comment: 5 pages, 4 figures

In a fluidized bed evaporator, heat transfer between walls of heated tubes and solutions belongs to a vapor-liquid-solid flow boiling, and solid particles are fluidized by a vapor-liquid flow. In t...

Multivariate Mittag-Leffler functions are a strong generalisation of the univariate and bivariate Mittag-Leffler functions which are known to be important in fractional calculus. We consider the ge...

Within the framework of Tsallis nonextensive statistics, the criteria for the Jeans gravitational instability are derived for a self-gravitating protoplanetary disk, whose substance consists of a mixture of a conducting ideal q-gas and modified radiation of a photon gas. The instability criteria are derived from the corresponding dispersion relations written for both neutral disk matter and magnetized plasma with modified blackbody radiation. The thermodynamics of a photon gas are constructed based on the nonextensive Tsallis quantum entropy, which depends on the deformation parameter. It is shown that blackbody q-radiation can stabilize the state of a nonextensive medium for a purely gaseous disk, and for an electrically conducting disk, the Jeans instability criterion is modified by the magnetic field and radiation pressure only in the transverse propagation mode of the disturbance wave.

This review gives a survey of some results about systems of interacting particles and the laws characterizing their behavior on large scales, which are common for a number of phenomena unified under the notion of the Kardar–Parisi–Zhang universality class.

This paper is contributed to the 1-D piston problem for the compressible relativistic full Euler system. Given some small BV perturbations of both the piston velocity and the initial state, we establish the structural stability of a contact discontinuity when the piston moves relatively still to the gas. We also consider the non-relativistic limits of such entropy solutions to piston problem as the light speed c → + ∞ . Moreover, we study the long time behavior of these solutions as t → + ∞ .

Fredholm determinants associated to deformations of the Airy kernel are closely connected to the solution to the Kardar‐Parisi‐Zhang (KPZ) equation with narrow wedge initial data, and they also appear as largest particle distributions in models of positive‐temperature free fermions. We show that logarithmic derivatives of the Fredholm determinants can be expressed in terms of a 2 × 2 Riemann‐Hilbert problem, and we use this to derive asymptotics for the Fredholm determinants. As an application of our result, we derive precise lower tail asymptotics for the solution of the KPZ equation with narrow wedge initial data, refining recent results by Corwin and Ghosal.

Many models with long-range interactions, including the Nagel-Kardar (NK) model, have been proposed to demonstrate and explore the differences between microcanonical ensembles and canonical ensembles. In the NK model, the first-order phase transition always accompanied by the temperature jump in the microcanonical ensemble, while this anomalous temperature jump can not be observed in the canonical ensemble. Moreover, the tricritical points of this model are different from each other in these two different ensembles. In this paper, the microcanonical-canonical crossover behavior of the NK model is studied by placing the NK model in contact with a two-level reservoir, and transition of the tricritical point is presented.

Biquadratic tensors play a central role in many areas of science. Examples include elastic tensor and Eshelby tensor in solid mechanics, and Riemannian curvature tensor in relativity theory. The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor, respectively. The tensor product operation is closed for biquadratic tensors. All of these motivate us to study biquadratic tensors, biquadratic decomposition, and norms of biquadratic tensors. We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure. Then, either the number of variables is reduced, or the feasible region can be reduced. We show constructively that for a biquadratic tensor, a biquadratic rank-one decomposition always exists, and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition. We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor. Finally, we define invertible biquadratic tensors, and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse, and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor, and the spectral norm of its inverse.

A series of conventional studies on free surface roughening have shown empirically that the surface roughness increases linearly with the equivalent strain. However, the effect of the deformation mode on the rate of surface roughening has not been clarified. We continuously observed the evolution of surface roughening under the deformation modes of uniaxial tension, plane strain tension, and equal biaxial tension in the metal forming. The deformation modes affect surface roughening, especially as the equivalent strain increases. It is found that this is caused by the change in surface area with plastic deformation. By removing the effect, a new deformation-mode-independent material constant of free surface roughening is derived.

The evidence here provided shows that the thermodynamics of the second law, as currently understood, originated in a correction of the flaws affecting Clausius original work on this matter. The body of knowledge emerging from this correction has been here called post-Clausius’ thermodynamics. The said corrections, carried on with the intended goal of preserving the validity of Clausius’ main result, namely the law of increasing entropy, made use of a number of counterintuitive or logically at fault notions. A joint revision of Clausius’ and post-Clausius’ work on the second law, carried on retaining some of Clausius original notions, and disregarding others introduced by post-Clausius thermodynamics, led this author to results in direct contradiction to the law of increasing entropy. Among the key results coming out of this work we find the one stating that the total-entropy change for spontaneous thermodynamic processes is the result of the summation of the opposite-sign contributions coming from the entropic (energy degrading) and negentropic (energy upgrading) changes subsumed by any such process. These results also show, via the total-entropy change for a non-reversible heat engine, that negentropic thermodynamics subsumes post-Clausius thermodynamics as a special case.

Dynamical properties of dust-ion-acoustic waves (DIAWs) are analysed in a collisional nonextensive dusty plasma composing of mobile ions, q-nonextensive electrons and stationary dust grains with slight collisions between dusts and ions. Reductive perturbation technique (RPT) is practiced to acquire the damped Korteweg-de Vries (DKdV) equation. In absence of collision between dusts and ions, the damping term vanishes and the DKdV equation reduces to the KdV equation. Bifurcation analysis of the dynamical system obtained from the KdV equation is carried out. Analytical solitary wave solution and numerical periodic wave solution of the KdV equation are presented. Approximate analytical DIAW solutions for the DKdV equation are furnished using the novel $ (G'/G) $ -expansion technique. As an outcome, novel approximate analytical solutions of DIAWs are obtained for the DKdV equation. The application of our work is significant to explore nonextensive plasma environment, such as, ionosphere of Earth.

In this article, we review fractal calculus ( $ F^{\alpha } $ -calculus) and define generalized Tsallis entropy on the fractal sets which is called fractal Tsallis entropy. We define q-fractal calculus to obtain q-Gaussian or generalized stable and Lévy distribution on fractal sets. The conditions for the nonlinear coupling of the statistical states are given. The relationship of fractal dimension which is defined in $ F^{\alpha } $ -calculus with q-parameter of the Tsallis entropy of the Hadron system is proposed.

We study numerical methods for dissipative particle dynamics (DPD), which is a system of stochastic differential equations and a popular stochastic momentum-conserving thermostat for simulating complex hydrodynamic behavior at mesoscales. We propose a new splitting method that is able to substantially improve the accuracy and efficiency of DPD simulations in a wide range of the friction coefficients, particularly in the extremely large friction limit that corresponds to a fluid-like Schmidt number, a key issue in DPD. Various numerical experiments on both equilibrium and transport properties are performed to demonstrate the superiority of the newly proposed method over popular alternative schemes in the literature.

The difference between homogeneous and bubbling fluidization behaviors has been studied for the past 70 years, where several researchers have reported on the influence of interparticle forces in fluidization. Although interparticle forces such as van der Waals forces are evident in a real system, these forces are not the determinant in homogeneous fluidization, which can be simulated without any interparticle forces. In our previous study, the difference in fundamental mechanisms of the two fluidization states was analytically determined with a dimensionless gravity term, comprising the Reynolds number, Archimedes number, and density ratio. Nevertheless, some researchers insist that interparticle forces are dominant and a hydrodynamic force is not dominant. In this study, a dimensional analysis was applied to obtain a dominant parameter for distinguishing two fluidizations. Furthermore, some parameters were examined by comparing the experimental data in previous studies. The results indicated that hydrodynamic force is the dominant factor and the dimensionless gravity term is the dominant parameter in differentiating the two fluidized states.

In this paper we consider an Ising model with nearest-neighbour interactions with spin space [0, 1] on a Cayley tree. We present a sufficient condition under which the Ising model has a unique splitting Gibbs measure.

We present an overview of a proposal in relativistic proton-proton (pp) collisions emphasizing the thermal or kinetic freeze-out stage in the framework of the Tsallis distribution. In this paper we take into account the chemical potential present in the Tsallis distribution by following a two step procedure. In the first step we used the redudancy present in the variables such as the system temperature, T, volume, V, Tsallis exponent, q, chemical potential, &mu, and performed all fits by effectively setting to zero the chemical potential. In the second step the value q is kept fixed at the value determined in the first step. This way the complete set of variables T,q,V and &mu, can be determined. The final results show almost no (or at best a very weak) energy dependence in pp collisions at the centre-of-mass energy s=20 TeV to 13 TeV. The chemical potential &mu, at kinetic freeze-out shows a steep increase with beam energy. This considerably simplifies the description of the thermal freeze-out stage in pp collisions as the values of T and of the freeze-out radius R remain constant to a good approximation over a wide range of beam energies.

Cumulative Tsallis entropy and its residual have developed by many others, and they proposed alternative (second) forms of them. In this article, we illustrate the differences and the relation between the proposed model and its alternative form. Furthermore, we study some properties and features for the concept of cumulative Tsallis entropy and its residual. In addition, we propose a dynamic past form of cumulative Tsallis entropy and obtain some of its properties.