Reply to the Comments of Sharipov on “Symmetry of the Linearized Boltzmann Equation”

This is a reply to the comments by Sharipov on the papers published in J. Stat. Phys. 136:751 (2009) and in J. Stat. Phys. 136:945 (2009). The present reply will show that the comments do not apply to the papers.     

Probability Description and Entropy of Classical and Quantum Systems

Tomographic approach to describing both the states in classical statistical mechanics and the states in quantum mechanics using the fair probability distributions is reviewed. The entropy associated with the probability distribution (tomographic entropy) for classical and quantum systems is studied. The experimental possibility to check the inequalities like the position–momentum uncertainty relations and entropic uncertainty relations are considered.     

Statistical-Mechanical Entropy of Garfinkle-Horowitz-Strominger Black Hole

Taking WKB approximation to solve the scalar field equation in the Garfinkle-Horowitz-Strominger (GHS) black hole spacetime, we can get the classical momenta. Substituting the classical momenta into state density equation corrected by the generalized uncertainty principle, we will obtain the number of quantum states with energy less than ω. It is convergent in the neighborhood of the horizon. Then, it is used to calculate the statistical-mechanical entropy of the scalar field in the GHS black hole spacetime. The calculation shows that the entropy is proportional to the horizon area.     

Trial Equation Method Based on Symmetry and Applications to Nonlinear Equations Arising in Mathematical Physics

To find exact traveling wave solutions to nonlinear evolution equations, we propose a method combining symmetry properties with trial polynomial solution to nonlinear ordinary differential equations. By the method, we obtain some exact traveling wave solutions to the Burgers-KdV equations and a kind of reaction-diffusion equations with high order nonlinear terms. As a result, we prove that the Burgers-KdV equation does not have the real solution in the form a ₀+a ₁tan ξ+a ₂tan ² ξ, which indicates that some types of the solutions to the Burgers-KdV equation are very limited, that is, there exists no new solution to the Burgers-KdV equation if the degree of the corresponding polynomial increases. For the second equation, we obtain some new solutions. In particular, some interesting structures in those solutions maybe imply some physical meanings. Finally, we discuss some classifications of the reaction-diffusion equations which can be solved by trial equation method.     

Long-Range Order, “Tower” of States,and Symmetry Breaking in Lattice Quantum Systems

Journal of Statistical Physics - In a quantum many-body system where the Hamiltonian and the order operator do not commute, it often happens that the unique ground state of a finite system exhibits...     

Micro-chemical and micro-structural investigation of the corrosion products on “The Dancing Satyr” (Mazara del Vallo, Sicily, Italy)

The “Dancing Satyr”, a bronze statue measuring more than 2 metres in height and weighting 108 kg, represents one of the most important recent archaeological finds in Italy. The statue was discovered on the floor of the Sicilian channel (the portion of the Mediterranean sea between Sicily and Tunisia), not far from the south-western Sicilian coast, under 500 metres of seawater in 1998.     

Cross-sectional morphological profiles of ripples on Si, SiC, and HOPG

The cross-sectional profiles of ripple structures on silicon (Si), silicon carbide (SiC), and highly oriented pyrolytic graphite (HOPG) were studied by direct observation. The ripple structures were cut by an ion beam, and their cross sections were observed by scanning electron microscopy. The results showed that the cross sections of coarse ripples on Si and SiC have a convex shape with narrower valleys, whereas those of HOPG have sharp ridges and wide wings with a poorer aspect ratio. This difference may arise from the difference in material phase conversion processes induced by femtosecond laser irradiation. The cross-sectional profiles of fine ripples on SiC and HOPG, which give useful information on the ripple formation process, are also discussed.     

A Remark on the Residual Entropy of the Antiferromagnetic Ising Model in the Maximal Critical Field

By means of a transfer matrix method, we show that the residual entropy S of the two-dimensional square lattice antiferromagnetic Ising model in the maximal critical field satisfies (ln λ _n )/(n+1)≤S≤(ln λ _n )/n, where λ _n is the largest eigenvalue of the transfer matrix F _n on a strip of width n. Using these bounds, we numerically calculate the value of S, with precise estimates on the errors, namely, S=0.394198±0.020747.     

Hawking Radiation Spectrum and Entropy Correction of Apparent Horizon in a FRW Universe

Taking the reaction of the radiation to the spacetime into consideration, we discuss Hawking radiation spectrum and Bekenstein-Hawking entropy correction in Friedmann-Robertson-Walker (FRW) universe by the analytical continuation method. We derive the radiation spectrum that satisfies the unitary principle and the logarithmic correction term of entropy in FRW universe.     

Remarks on Undecidability,Incompleteness and the Integrability Problem

In their seminal paper Undecidability and incompleteness in classical mechanics (Int. J. Theor. Phys. 30:1041–1073, 1991), N.C.A. da Costa and F.A. Doria introduced a powerful method for studying the appearance of undecidability and incompleteness in mathematics and theoretical physics. In this work their results are applied to integrability theory. Specifically, it is pointed out that it is not possible to expect the existence of an algorithm able to decide whether a given partial differential equation is integrable or not.     

Effect of annealing temperature on hardness, thickness and phase structure of carbonitrided 304 stainless steel

Carbonitriding of AISI 304 austenitic stainless steel was performed at a plasma-processing power of 450 W using inductively coupled radio frequency (rf) plasma in a gas mixture of 50% N₂ and 50% C₂H₂. The rate of carbonitriding, microhardness, phase structure of the compound layer, surface microstructure and cross-section morphology were studied before and after the annealing process. At the annealing temperature up to 800°C, the microhardness values of the compound zones decrease, while the associated values of the diffused zones increase. Little change was found in the thickness of the compound and diffused zones when the carbonitrided samples were annealed up to 400°C. However, at a higher annealing temperature, the thicknesses of both zones increase. The γ-Fe austenite is the main crystalline phase that can be detected by X-ray diffraction. As the annealing temperature increases up to 500°C, X-ray spectra show α-Fe and Fe₅C₂ phases. Nitrogen diffuses more deeply from the near surface to the interior of the treated sample as the annealing temperature increases up to 800°C and this might explain the extent of carbonitrided thickness and the enhanced microhardness of the diffused zone.     

Entropy of Kerr-Newman Black Hole to All Orders in the Planck Length

Using the quantum statistical method, the difficulty of solving the wave equation on the background of the black hole is avoided. We directly solve the partition functions of Bose and Fermi field on the background of an axisymmetric Kerr-Newman black hole using the new equation of state density motivated by the generalized uncertainty principle in the quantum gravity. Then near the black hole horizon, we calculate entropies of Bose and Fermi field between the black hole horizon surface and the hypersurface with the same inherent radiation temperature measured by an observer at an infinite distance. In our results there are not cutoffs and little mass approximation introduced in the conventional brick-wall method. The series expansion of the black hole entropy is obtained. And this series is convergent. It provides a way for studying the quantum statistical entropy of a black hole in a non-spherical symmetric spacetime.     

“Conserved charges” of the Bondi-Metzner-Sachs algebra in the Brans-Dicke theory

The asymptotic symmetries in the Brans-Dicke theory are analyzed using Penrose's conformal completion method,which is independent of the coordinate system used.These symmetries,indeed,include supertranslations and Lorentz transformations for an asymptotically flat spacetime.With the Wald-Zoupas formalism,“conserved charges”and fluxes of the Bondi-Metzner-Sachs algebra are computed.The scalar degree of freedom contributes only to the Lorentz boost charge,even though it plays a role in various fluxes.The flux-balance laws are further applied to constrain the displacement memory,spin memory,and center-of-mass memory effects.     

Entropy and Entanglement of a Single-mode Vacuum Field Interacting with a Ξ-type Three-level Atom with Detuning

The evolution of the field entropy and the entanglement between the atom and the field for the system of a single-mode vacuum field interacting with a Ξ-type three-level atom have been studied by using the reduced quantum entropy. The influences of the detuning of the light field and the setting of the initial state of the atom on the field entropy and entanglement of the system under consideration are discussed emphatically. It is showed that the detuning of the light field and the setting of the initial state of the atom play an important role for the evolution of the field entropy and the entanglement between the atom and the field. The general conclusions reached are illustrated by numerical results.     

Construction of AFLT States for W_NH Symmetry,Analytic Continuation and Integrability on AGT Relation

We propose a Hamiltonian to the construction of the AFLT states for WNsymmetry. We generalize the AGT relation to generic(extended) conformal field theory with 1 ≤ c ∞. We analyze the triangular structure hidden in the AGT relation with WNsymmetry in detail and the triangular structure implies the integrability.     

Entropy of Stationary Nonequilibrium Measures of Boundary Driven Symmetric Simple Exclusion Processes

We examine the entropy of stationary nonequilibrium measures of boundary driven symmetric simple exclusion processes. In contrast with the Gibbs–Shannon entropy (Bahadoran in J. Stat. Phys. 126(4–5):1069–1082, 2007; Derrida et al. in J. Stat. Phys. 126(4–5):1083–1108, 2007), the entropy of nonequilibrium stationary states differs from the entropy of local equilibrium states.     

On the Entropy Production of Time Series with Unidirectional Linearity

There are non-Gaussian time series that admit a causal linear autoregressive moving average (ARMA) model when regressing the future on the past, but not when regressing the past on the future. The reason is that, in the latter case, the regression residuals are not statistically independent of the regressor. In previous work, we have experimentally verified that many empirical time series indeed show such a time inversion asymmetry. For various physical systems, it is known that time-inversion asymmetries are linked to the thermodynamic entropy production in non-equilibrium states. Here we argue that unidirectional linearity is also accompanied by entropy generation. To this end, we study the dynamical evolution of a physical toy system with linear coupling to an infinite environment and show that the linearity of the dynamics is inherited by the forward-time conditional probabilities, but not by the backward-time conditionals. The reason is that the environment permanently provides particles that are in a product state before they interact with the system, but show statistical dependence afterwards. From a coarse-grained perspective, the interaction thus generates entropy. We quantitatively relate the strength of the non-linearity of the backward process to the minimal amount of entropy generation. The paper thus shows that unidirectional linearity is an indirect implication of the thermodynamic arrow of time, given that the joint dynamics of the system and its environment is linear.     

Effect of Symmetry and H‐bond Strength of Hard Segments on the Structure‐Property Relationships of Segmented,Nonchain Extended Polyurethanes and Polyureas

Segmented, nonchain extended polyurethanes and polyureas based on PTMO soft segments (SS) and hard segments (HSs) based on only single molecules of a diisocyanate were synthesized. Type and nature of the diisocyanate was systematically varied in order to analyze the effect of HS symmetry and type of linkage between the HS and SS on the structure‐property relationship of these segmented copolymers. Results showed that the increased symmetry of the diisocyanates allows a more efficient packing of the HSs which leads to a microphase‐separated structure with the crystalline hard ribbon or thread‐like domains percolated throughout the SS matrix, even with a low HS content (ca. 13 wt.%). The service window of these segmented copolymers was significantly influenced by the symmetry and type of linkage between the HS and SS. Most copolymers also showed evidence of strain hardening accented by the strain induced crystallization of the PTMO SS.     

Quantum Entropy of a Nonlinear Two-level Atom with Atomic Motion

The wave function of a system governed by the time-dependent nonlinear Jaynes-Cummings (JC) model is obtained. We compute analytically the eigenvalues of the reduced field density operator by which the dynamics of the entropy of entanglement of the cavity field are analyzed. The influences of the atomic motion, the field-mode structure and the Kerr-like medium on this phenomenon are illustrated. The population dynamics of an excited atom is also discussed for the same set of parameters. The cavity field is assumed to be initially excited in either a Fock or a coherent states. The cavity excitation in a Fock state generates a class of an entanglement without death with fixed amplitude by adjusting the parameters of the atomic motion as well as the Kerr and the field-mode structure. In case of a coherent cavity, the only phenomenon to be noted is the periodical behavior of the dynamics under study when the atomic motion is considered. Although the Kerr medium affects the strength of the entanglement negatively, the entropy of entanglement loses its zeros where the Kerr is taken into consideration.