Ordinary Entropy Research Articles

Entropy-Based Parameter Estimation for Extended Three-Parameter Burr III Distribution for Low-Flow Frequency Analysis

Low-flow analysis is important for management of water supply and droughts, determining the health of ecosystems, and for sustained agriculture. Two entropy-based methods, the ordinary entropy (ENT) method and the parameter space expansion method (PSEM), are employed for estimating parameters of the extended Burr III distribution, which is useful for low-flow frequency analysis. With parameters estimated in this manner, the extended Burr III distribution is applied to six low-flow datasets, and quantiles (discharges) corresponding to different return periods are computed. These return periods are then compared with those when the extended Burr III distribution parameters are estimated using the methods of moments (MOM), probability weighted moments (PWM), and maximum likelihood estimation (MLE). It is shown that PSEM yields the same quantiles as does MLE for discrete cases, while ENT is found comparable to MOM and PWM. The quantile (discharge) values estimated by the four methods are close to each other for smaller return periods, but large differences occur for long return periods.

The second entropy: a general theory for non-equilibrium thermodynamics and statistical mechanics

The second entropy is introduced, which is a new type of entropy that provides a basis for the non-equilibrium thermodynamics of time-dependent systems. Whereas the first or ordinary entropy counts the molecular configurations associated with a given structure, the second entropy counts the molecular configurations associated with a transition between two given structures in a specified time. Maximization of the second entropy gives the optimum rate of change or flux, and as such it provides a quantitative principle for non-equilibrium systems. In contrast, the second law of thermodynamics only provides a direction for change, not a rate of change. The probability distribution function for time-dependent systems is also given, which is the focal point of non-equilibrium statistical mechanics. This and the second entropy are used, for example, to derive the Langevin equation, the Green-Kubo relations, the transition and path probability, the fluctuation and work theorems, and a generalised fluctuation-dissipation theorem. They are also used to develop computer simulation algorithms suited for time-dependent systems, specifically non-equilibrium Monte Carlo and stochastic Molecular Dynamics. The analysis is illustrated and quantitatively tested for the case of steady heat flow, and for the case of time-varying, driven, Brownian motion.

Entanglement through conformal interfaces

We consider entanglement through permeable interfaces in the c=1 (1+1)-dimensional conformal field theory. We compute the partition functions with the interfaces inserted. By the replica trick, the entanglement entropy is obtained analytically. The entropy scales logarithmically with respect to the size of the system, similarly to the universal scaling of the ordinary entanglement entropy in (1+1)-dimensional conformal field theory. Its coefficient, however, is not constant but controlled by the permeability, the dependence on which is expressed through the dilogarithm function. The sub-leading term of the entropy counts the winding numbers, showing an analogy to the topological entanglement entropy which characterizes the topological order in (2+1)-dimensional systems.

Entropy-based parameter estimation for extended Burr XII distribution

Two entropy-based methods, called ordinary entropy (ENT) method and parameter space expansion method (PSEM), both based on the principle of maximum entropy, are applied for estimating parameters of the extended Burr XII distribution. With the parameters so estimated, the Burr XII distribution is applied to six peak flow datasets and quantiles (discharges) corresponding to different return periods are computed. These two entropy methods are compared with the methods of moments (MOM), probability weighted moments (PWM) and maximum likelihood estimation (MLE). It is shown that PSEM yields the same quantiles as does MLE for discrete cases, while ENT is found comparable to the MOM and PWM. For shorter return periods (<10–30 years), quantiles (discharges) estimated by these four methods are in close agreement, but the differences amongst them grow as the return period increases. The error in quantiles computed using the four methods becomes larger for return periods greater than 10–30 years.

Improvement of goodness-of-fit test for normal distribution based on entropy and power comparison

Vasicek's entropy test for normality is based on sample entropy and a parametric entropy estimator. These estimators are known to have bias in small samples. The use of Vasicek's test could affect the capability of detecting non-normality to some extent. This paper presents an improved entropy test, which uses bias-corrected entropy estimators. A Monte Carlo simulation study is performed to compare the power of the proposed test under several alternative distributions with some other tests. The results report that as anticipated, the improved entropy test has consistently higher power than the ordinary entropy test in nearly all sample sizes and alternatives considered, and compares favorably with other tests.

Fluctuation theorem for the renormalized entropy change in the strongly nonlinear nonequilibrium regime

A nonlinear relaxation process is considered for a macroscopic thermodynamic quantity, generalizing recent work by Taniguchi and Cohen [J. Stat. Phys. 126, 1 (2006)] that was based on the Onsager-Machlup theory. It is found that the fluctuation theorem holds in the nonlinear nonequilibrium regime if the change of entropy characterized by local equilibria is appropriately renormalized. The fluctuation theorem for the ordinary entropy change is recovered in the linear near-equilibrium case.

Tsallis ensemble as an exact orthode

We show that Tsallis ensemble of power-law distributions provides a mechanical model of nonextensive equilibrium thermodynamics for small interacting Hamiltonian systems, i.e., using Boltzmann's original nomenclature, we prove that it is an exact orthode. This means that the heat differential admits the inverse average kinetic energy as an integrating factor. One immediate consequence is that the logarithm of the normalization function can be identified with the entropy, instead of the q-deformed logarithm. It has been noted that such entropy coincides with Renyi entropy rather than Tsallis entropy, it is non-additive, tends to the standard canonical entropy as the power index tends to infinity and is consistent with the free energy formula proposed in [S. Abe et. al. Phys. Lett. A 281, 126 (2001)]. It is also shown that the heat differential admits the Lagrange multiplier used in non-extensive thermodynamics as an integrating factor too, and that the associated entropy is given by ordinary nonextensive entropy. The mechanical approach proposed in this work is fully consistent with an information-theoretic approach based on the maximization of Renyi entropy.

Nonunitary higher derivative gravity classically equivalent to Einstein gravity and its Newtonian limit

Runaway solutions can be avoided in fourth order gravity by a doubling of the matter operator algebra with a symmetry constraint with respect to the exchange of observable and hidden degrees of freedom together with the change in sign of the ghost and the dilaton fields. The theory is classically equivalent to Einstein gravity, while its non-unitary Newtonian limit is shown to lead to a sharp transition, around $10^{11}$ proton masses, from the wavelike properties of microscopic particles to the classical behavior of macroscopic bodies, as well as to a trans-Planckian regularization of collapse singularities. A unified reading of ordinary and black hole entropy emerges as entanglement entropy with hidden degrees of freedom. The emergent picture gives a substantial agreement with B-H entropy and Hawking temperature.

Axioms and uniqueness theorem for Tsallis entropy

The Shannon-Khinchin axioms for the ordinary information entropy are generalized in a natural way to the nonextensive systems based on the concept of nonextensive conditional entropy and a complete proof of the uniqueness theorem for the Tsallis entropy is presented. This improves the discussion of dos Santos [J. Math Phys. 38 (1997) 4104].

Regularization of Quantum Relative Entropy in Finite Dimensions and Application to Entropy Production

The fundamental concept of relative entropy is extended to a functional that is regular-valued also on arbitrary pairs of nonfaithful states of open quantum systems. This regularized version preserves almost all important properties of ordinary relative entropy such as joint convexity and contractivity under completely positive quantum dynamical semigroup time evolution. On this basis a generalized formula for entropy production is proposed, the applicability of which is tested in models of irreversible processes. The dynamics of the latter is determined by either Markovian or non-Markovian master equations and involves all types of states.

Additivity of the entropies of black holes and matter in equilibrium.

We consider static spherical spacetimes in which a black hole is in equilibrium with surrounding matter. The grand canonical ensemble is defined by the geometry of the boundary of a region of spacetime and the chemical potential evaluated on this boundary. Additivity of the actions of gravity and matter and the principle of equivalence then jointly imply that the total entropy is the simple sum of the accepted black-hole entropy and the ordinary matter entropy. However, energy and thermodynamic potentials exhibit interaction terms and are not simply additive..

On the entropy of a quantum field

Recently, Kandrup and Hu have proposed a new definition of nonequilibrium field. This letter shows that the definition is consistent with conventional wisdom by demonstrating that, for a massless Klein-Gordon field at equilibrium in Minkowski space, one recovers the ordinary entropy associated with a Wigner function.

A comparison of the entropies of collapsing stars and black holes

We considered three modes of black hole formation: (I) a black hole kernel first forms at the centre of a collapsing star and as the outer matter falls, the kernel grows until the whole star becomes a black hole; (II) all the layers of a collapsing simultaneously satisfy the Schwarzschild condition; (III) the outermost layer first satisfies the Schwarzschild condition. For each mode, we calculated the entropy carried by the collapsing matter, S m , and the entropy of the black hole so formed, S BH . We found S BH to be 10 19 times S m and the lower limit of the mass capable of becoming a black hole to be the Planck mass, M p = 10 −5g. A discussion on the nature of S BH led us to think that S BH possibly contains things other than the ordinary thermodynamical entropy.

Wave propagation in thermo-viscous magnetofluiddynamics

Wave propagation in thermo-viscous fluids, acted on by a magnetic field, is investigated through a model accounting for viscosity and heat conduction via hidden variables. On adopting the approximations of standard magnetifluiddynamics the speeds of the resulting waves are derived explicity. The close connection between such waves and the ordinary entropy waves, Alfvén waves, and magnetoacoustic waves is outlined.

Does black-hole entropy make sense?

Bekenstein and Hawking saved the second law of thermodynamics near a black hole by assigning to the hole an entropyS h proportional to the area of its event horizon. It is tempting to assume thatS h possesses all the features commonly associated with the physical entropy. Kundt has shown, however, thatS h violates several reasonable physical expectations. We review his criticism, augmenting it as follows: (a)S h is a badly behaved state function requiring knowledge of the hole's future history; and (b) close analogs of event horizons in other space-times do not possess an “entropy.” We also discuss these questions: (c) IsS h suitable for all regions of a black-hole space-time? And (b) shouldS h be attributed to the exterior of a white hole? One can retainS h for the interior (respectively, exterior) of a black (respectively, white) hole, but we reject this as contrary to the information-theoretic derivation of horizon entropy given by Bekenstein. The total entropy defined by Kundt (all ordinary entropy on space-section cutting through the hole, no horizon term) and that of Bekenstein-Hawking (ordinary entropy outside horizon plus horizon term) appear to be complementary concepts with separate domains of validity. In the most natural choice, an observer inside a black hole will use Kundt's entropy, and one remaining outside that of Bekenstein-Hawking.

On the notions of entropy and information

In this review article we look for some of the historical reasons for the “identification” of the information-theoretical and thermodynamic entropy concepts. We also discuss the Baron-Jauch entropy concept and explicitly show that, for classical systems in thermodynamic equilibrium, there exists a very simple connection between this general definition and the ordinary experimental entropy.

A note on the thermodynamic theory of nonhydrostatically stressed solids

In a recent paper printed in this journal, Kamb [1961] reviewed the thermodynamic theory of nonhydrostatically stressed solids. I agree with his view cited as follows: ‘Since the Gibbs free energy is only a secondarily defined quantity in ordinary hydrostatic thermodynamics, energy and entropy being the primary quantities of physical significance, it is of course also permissible in nonhydrostatic thermodynamics to attempt to define a Gibbs free energy in a suitable way’ (p. 265). However, I should like to comment on his discussion. Kamb presented the theoretical basis both for the preferred orientation of minerals and for phase transformation under the nonhydrostatic stress (Appendix 2, p. 269). His derivation of the equilibrium condition does not seem to be explicit. His equation [49] (hereafter equations that appeared in Kamb's [1961] paper are denoted by [ ]) must be: