Transformation Entropy Research Articles

Erratum to: “On entropy and generators of measure-preserving transformations”

Lemma (4.2) of this paper that appeared in Trans. Amer. Math. Soc. 149 (1970), 453-464, was stated incorrectly. To obtain the correct statement replace p(A0) >p(A1) + 2p(A2) by p(A0)>2p(A1)+2p(A2). (Cf. [2]. The lemma follows from Zaslavskil's work by using the coding system 221, 211, and by leaving symbols other than 0, 1, 2 unchanged in the coding.) The lemma together with an argument of the kind as indicated on p. 463 does not yield Theorem (4.3). Only a weaker estimate can be obtained in this way. Theorem (4.3) is valid, however, as can be seen from [1].

The Addition Theorem for the Entropy of Transformations of G-Spaces

The Addition Theorem for the Entropy of Transformations of G-Spaces

Topological entropy of distal affine transformations on compact abelian groups

Topological entropy of distal affine transformations on compact abelian groups

The addition theorem for the entropy of transformations of 𝐺-spaces

For a measure-preserving transformation T T which is a skew-product of a measure-preserving transformation S S and a topological group endomorphism σ \sigma , it is shown that the entropy h h satisfies the following “addition theorem": h ( T ) = h ( S ) + h ( σ ) h(T) = h(S) + h(\sigma ) .

On entropy and generators of measure-preserving transformations

Let T be an ergodic measure-preserving transformation of a Lebesgue measure space with entropy h(T). We prove that T has a generator of size k where eh(T) _ k < eh(T)+ 1

Interstitial Order-Disorder Transformation in the Ti-O Solid Solution. III. A Statistical Theory

A theory of the interstitial order-disorder transformation in the h.c.p. lattice has been developed on the basis of the Bragg-Williams approximation. The calculated results are compared with the experiments on the Ti-O system reported in Part I and Part II. Oxygen atoms situated on the octahedral interstices are considered to repel each other. As the interstitial disordering proceeds in the two successive steps of the intra- and inter-plane disordering, analytical calculations are made by treating the two processes independently. Then numerical computations are carried out taking account of mutual correlations of these processes. It is revealed that the intra-plane disordering is considerably modified by simultaneous occurrence of the inter-plane disordering. The transformation energy and entropy are deduced as a function of oxygen contents. A tentative phase diagram is constructed by evaluating the repulsive energies for three pairs of the oxygen atoms. These results are consistent with the characteristi...

Interstitial Order-Disorder Transformation in the Ti-O Solid Solution. II. A Calorimetric Study

Heat capacity of the solid solution of alpha-titanium with 8∼28 at% oxygen has been measured to investigate calorimetric effects associated with the interstitial order-disorder transformation of oxygen atoms. Anomalous heat absorption is observed in the temperature range from 300 to 600°C varying with oxygen contents. The transformation at oxygen contents higher than about 15 at% is characterized by the appearance of double peaks in the specific heat curve. In the light of the results of diffraction studies, the peaks are interpreted to be due to the two successive processes from the ordered phase α'' to the disordered solid solution α through the intermediate phase α'. The transformation energy and entropy as well as the critical temperature are determined as functions of composition. It is found that two maxima appear in the transition energy-composition curve around 15 and 24 at% O with nearly the same value of about 390 cal/mol, and that a similar behavior is also observed in the transition entropy-co...

Effect of a magnetic field on phase transformations in austenitic stainless steels

A differential equation is obtained and solved for the entropy of transformation in first-order phase transformations with allowance for the effect of an external magnetic field. The effect of a constant magnetic field on phase transformations in an austenitic stainless steel is investigated. It is shown that the theoretical values for the magnetic δT effect are of the same order as the experimental values.

On the entropy of uniquely ergodic transformations

0. Introduction. Ergodic theory involves itself with the study of transformations of a measure space. Topological dynamics is involved with homeomorphisms of a topological space. The entropy of a measure preserving transformation is a gauge of its randomness. There are certain qualitative properties in topological dynamics which seem to play a similar role. For example, equicontinuity, distality, and minimality restrict the randomness of a transformation group. The question arises: If a measure preserving transformation is also a homeomorphism is there a strict relation between these notions? For instance it is known that equicontinuity and distality imply zero entropy. It was not known whether minimality implied zero entropy. W. Parry raised the following question: Let X be a compact metric space and T a continuous map of X into itself. Suppose there is a unique T invariant probability measure and suppose (X, T) is minimal (every orbit is dense). Can (X, T) have positive entropy? The difficulty in answering such a question is the scarcity of a wide class of minimal, uniquely ergodic transformation groups (X, T). We give here a method of constructing minimal uniquely ergodic transformation groups (X, T). Furthermore our method is designed to show that there exist transformation groups (X, T), X a compact metric space, which are minimal and uniquely ergodic but have arbitrarily large finite entropy. Thus minimality and unique ergodicity do not bound the entropy of (X, T) and the answer to Parry's question is no. We carry out our program by constructing closed shift invariant subspaces of a sequence space. In ?1 we establish some notation and terminology. In ??2 and 3 we either modify or translate some known theorems into combinatorial terms which are to be used in the succeeding sections. In ?4 we define the sequences which will be used to build the uniquely ergodic minimal pairs (X, T) of large but finite entropy. ??5, 6, 7, 8 are used to carry out the detailed examination of the properties of our sequence. ?9 indicates the modification necessary in ?4 to show that the same construction can be done on bisequences and thus Tmay also be a homeomorphism. The results are summarized in Corollaries 8.14 and 9.7. The principal idea which makes our result possible is the weak law of large numbers. Lemmas 1.1 and 1.2 are merely forms of this law. The remainder is

Entropy of conservative transformations

The definition of entropy of a measure-preserving transformation (called: endomorphism) of a finite measure space into itself makes no sense for σ-finite measure spaces. Using induced transformations (introduced by Kakutani [1]) we give a definition which applies to conservative endomorphisms in σ-finite measure spaces. (This covers all cases of interest, since dissipative endomorphisms have a rather simple structure.) A theorem of Abramov [2] implies that for finite measure spaces the new definition is equivalent to the old one. Entropy as a metric invariant of conservative transformations has many, but not all of the properties discovered by Kolmogorov, Sinai, Rokhlin and others in the finite case. Major differences between the finite and the σ-finite case occur in the investigation of transformations with entropy 0. After giving the basic definitions in section 1 we first prove a theorem on antiperiodic transformations, which will be needed in all other sections, unless the reader is willing to assume that all transformations are ergodic. In section 3 we define entropy and prove a theorem which permits its computation. As an example the entropy of the Markov shift for null-recurrent Markov chains is computed in section 4. We then investigate simple properties such as h(T n )=nh(T) (section 5) and give the ergodic decomposition of h(T) in section 6. Section 7 is devoted to the investigation of transformations with entropy zero, especially an example is given which shows that a known necessary and sufficient condition for a transformation with finite invariant measure to have entropy zero is not sufficient for transformations with a σ-finite invariant measure unless they satisfy an additional assumption. Finally section 8 is devoted to the proof of category statements about the set of conservative transformations and the subset of those among them which have entropy zero.

High-Temperature Heat Contents and Related Thermodynamic Functions of Eight Rare-Earth Metals: Sc, Gd, Tb, Dy, Ho, Er, Tm, and Lu

The high-temperature heat contents of scandium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, and lutetium were measured from 0°C to at least the melting points of these metals. The specific heats, heats of transition, heats of fusion, and related thermodynamic functions were calculated. The smoothed values of H°T—H°298.15, (H°T—H°298.15)/T, S°T, and − (F°T—H°298.15)/T are tabulated at 100° intervals. An examination of the entropies of transition and fusion of all of the measured rare-earth metals shows that the entropy of the fcc(or hcp)⇄bcc transition depends upon the number of valence electrons and the entropy of fusion decreases with increasing rare-earth-metal size. The electronic dependence of the entropy of transformation has also been observed to hold for other metals undergoing similar transformations.

Thermal behaviour of germanium dioxide

The thermal behaviour of the two crystalline forms of germanium dioxide has been investigated by differential thermal analysis. Small amounts of lithium increase the rate of transformation. The enthalpy and entropy of transformation have been estimated as 5·05 ± 0·6 kcal. mole–1 and 3·9 ± 0·5 e.u., and the enthalpy of fusion as 4·1 ± 0·5 kcal. mole–1, at the respective transformation and fusion temperatures. Use of the available specific-heat data for both forms of germania to extrapolate to 298°K leads to a standard entropy of tetragonal germania of 9·7 ± 0·7 e.u.

Effect of temperature on the structure of manganites

The effect of temperature on the structure of some “normal” manganites has been studied by high temperature X-ray diffraction and by differential thermal analysis. At rcom temperature they are isomorphous with hausmannite, with c/a > 1. On heating to elevated temperatures, the c,a, and c/a parameters are found to remain constant until just before a critical transformation temperature whereupon c/a falls abruptly to unity, coinciding with the structure transforming from tetragonal to cubic. This abrupt transformation is reminiscent of a first order process. The structure of the tetragonal forms at various temperatures, and the cubic form just after transition, have been determined by X-ray diffraction. The theoretical model for the variation of distortion with temperature has been discussed, and a correlation between the transition temperature and the cell distortion has been shown. An anomalous behaviour has been observed for magnesium manganite because of its tendency to change its cation distribution and alter, on heating, from a “normal” to a “random” structure. The site preference energy for the cations in MgMn 2O 4 has been calculated from the change in cation distribution with temperature. The D.T.A. data support the transition temperatures of various manganites as obtained by X-ray and also lead to the determination of the entropy of transformation.