The near-equilibrium nature of dynamic linear response when a system is acted upon by an external force is clarified by using the linearity condition to identify a limited class of near-equilibrium density matrices that characterize the response. The analysis incorporates a distinction between undriven and driven time dependences. Irreversibility is introduced in a traditional way by means of a nonunitary component in the evolution of the undriven dynamics.

Experimental data of temporal relaxation phenomena in diverse types of complex correlated condensed matter systems seem to follow stretched exponential behavior (or its frequency space counterpart). Descriptions of such phenomena based on ‘coupling model’ schemes invoke three timescales: τ0 associated with an early linear exponential; τ∗ associated with a late stretched exponential with a stretching index n (0 ≤ n < 1); and ωc−1, the time of onset of the effect of coupling (correlation) which enters into a relation between τ0 and τ∗, namely, ωcτ∗ = [(1 − n)ωcτ0]11−n·τ0 represents the relaxation time before correlations become operative and τ∗ is the relaxation time when correlations are operative. ‘Coupling model’ schemes described here have been successful in bringing together a large body of experimental data for a significant class of complex correlated systems (CCS) and demonstrate certain widely shared behaviors. A review of the conceptual notions used in coupling model schemes in the past decade is outlined. The scope and underlying assumptions of these schemes are made explicit. In an appendix, a general phenomenology that brings out the dynamic structure of relaxation in CCS is also outlined.

It is pointed out that the use of acceleration studies implies that stress does not change the mechanisms of failure but only contracts the time to reach failure level. This implication is translated into a requirement that the parameter set for the failure distribution does not change, but values of the parameters may. The generality of the relationship between parameter values at various stress ...

本文不用“大数近似”（ＬＮＡ），而在平均值的基础上为费米子、玻色子和玻尔兹曼粒子的分布提供一种新的推导方法．这个推导只用到传统的组合方法，其物理内容的适用性并不依赖于最可几值和“大数近似”方法的运用．本方法的基础，是把密度矩阵的本征值作为香农（Ｓｈａｎｎｏｎ）熵中的几率，把密度矩阵本征值的简并度与玻尔兹曼的计数因子（ｅｎｕｍｅｒａｔｉｏｎ ｆａｃｔｏｒ）等同起来，因此避免了诸如与最速下降法、极限过渡技巧和玻尔兹曼热力学几率相关的问题．此外，本方法立足于密度矩阵的本征值，也不需用到系综的概念．

A homogeneous complex system, i.e., one with interacting modes of the same type, that is in thermodynamic equilibrium is described formally in terms of effective independent modes. This formal description causes an apparent increase in system entropy. It is argued that this apparent increase in entropy is accompanied by the introduction of an effective temperature T* that is lower than the actual temperature T of the complex system; T*/T=f, 0

In order to gain insight into the nature of the Wigner and related distribution functions, bivariate averaging functions of real unbounded variables with absolutely continuous marginals that are ordinary probabilities are considered. Accordingly variables are chosen to be phase space variables that are respectively eigenvalues of position and momentum operators. The impact of the condition that the marginals are squared magnitudes of amplitudes that are Fourier transforms of one another is emphasized by the delay of the introduction of this Fourier transform condition until after the form for a bivariate distribution with the given marginals is obtained. When the respective amplitudes are fourier transforms of one another, special cases of the bivariate averaging function correspond to generalized Wigner functions characterized by a parameterα. Such anα-Wigner function can be used as the basis of a consistent averaging procedure if an appropriate corresponding representation for underlying operators to be averaged is specified. Properties of theα-Wigner functions are summarized.

Discussion de divers modeles pour expliquer la forme exponentielle etiree de la relaxation d'enthalpie au-dessus de la transition vitreuse

Density matrix averaged uncertainty relations for general operators  and B̂ are discussed. A minimum uncertainty relation between the averages of Hermitian and anti-Hermitian combinations of †B̂ is obtained. The microscopic represented by an eigenstate, and the macroscopic represented by a mixed state density matrix, coincide at minimum uncertainty.

The least maximum entropy for the free radiation field occurs for the vacuum state of the radiation field and its unitary equivalents including the coherent state of an ideal laser. Comparisons are made with number measurement entropies obtained by projection or averaging of the number representation of the ideal laser.

In this paper we view a relaxing complex system such as entangled polymer melt to consist of three parts: (1) an individual primary species PS of interest; (2) a heat bath (HB) whose interaction with the PS provides the primary mechanism of relaxation; (3) other relaxing species whose interactions with the PS, the PS-C coupling, are for us the principal characteristic of complexity. The PS-C coupling is represented by time dependent constraints whose effect begins only after the primary relaxation process due to the PS-HB interaction is already underway. The overall process is described both physically and theoretically. The latter is described classically by means of time dependent Dirac constraint theory applied to a Liouville operator formalism. The physical and theoretical discussion leads to a time dependent relaxation rate W(t). The specific form of W(t) is adduced based on the requirement of time-temperature equivalence or thermorheological simplicity. The result is a time independent relaxation rate W0 for times short compared to the onset of the effect of the time dependent constraints at tc=ω−1c, and a time dependent rate W0(ωct)−n for times long compared to tc. The case W0tc<1 is of most interest because the relaxation process then reveals the effect of complexity empirically. The empirically observed result is then just the Kohlrausch form. Furthermore, a second relation between τ0≡W−10 and the effective relaxation time in the Kohlrausch form follows immediately. It is also noted that the present framework can be applied more generally in relaxation phenomena if thermorheological simplicity is viewed as a special case of correlation or constraint scaling in which dW/W=−ndt/t for t>tc.

Several models of relaxation based on master equation approaches have obtained the Kohlrausch fractional exponential form φ(t) = exp − (tτ∗)1−n, 0 < n < 1, or its equivalent for the relaxation function in complex systems. Representative models include (i) the Cohen-Grest free-volume theory, (ii) the work of Dhar and Barma, and Skinner based on Glauber's kinetic Ising model, (iii) the theory of De Dominicis et al. based on a random energy model for the spin glass, (iv) the Ogielski-Stein theory based on dynamics in an ultrametric space, and (v) Ngai's theory of time-dependent transition rates. In view of the different nature of these models and because of the claims that they are applicable outside of their original contexts, it is useful to make an intercomparison of these models and their consequences. A presentation of these models is here given based on a unified master equation approach. By experiment, many real systems have been shown to exhibit not only the Kohlrausch form but two additional related properties which are not encompassed in model types (i)–(iv). Only models that include time-dependent transition rates have so far been shown to be consistent with the experimental observations of the three empirical relations.

New derivations of particle occupation factors that are based on mean values and do not require large number approximations (LNA) are provided for fermions, bosons, and Boltzmann particles. The derivations are closely related to traditional combinatorial approaches, so the physical content of the latter is preserved without recourse to most probable values or LNA. The approach is based on the use of eigenvalues of the density matrix as probabilities entering into the Shannon entropy. The degeneracies of the eigenvalues of the density matrix can be identified with Boltzmann enumeration factors. Problems associated with steepest descent procedures, limit theorem techniques, and the Boltzmann thermodynamic probability are avoided. In addition, since the approach is based on the eigenvalues of the density matrix, the concept of ensembles is not required.

After a brief expository account of the Shannon-Jaynes principle of maximum entropy (POME) for discrete and continuous variables, we give here an account of some recent research work which (i) a “histogram” method to contrast the discrete and continuous modes computation and the role of histogram in actual practice when dealing with continuous probability distributions. (ii) The idea of mean logarithmic decrement associated with a probability distribution is introduced and is shown to be related to the concept of differential entropy. The mean with respect to an arbitrary probability distribution of the logarithm of the ratio of the probability density function for an exponential distribution is discussed in the context of hydrological investigations. Unlike the entropy of a continuous probability distribution introduced in (i), this quantity which is an example of the Kullback-Leibler (KL) Information, is always positive and invariant under coordinate transformation. (iii) The constraints entering into POME as well as a minimum K L information are identified as a class of sufficient statistics which determine the unknown parameters in the probability density functions that occur in the most commonly used hydrological models. (iv) An example of (iii) where only the first two moments in a semi-infinite domain are givens is discussed to shed light on the limitations of POME, recentky recognized by Wragg and coworkers, and is made relevant to the work of Sonuga on rainfall-run off relationship. Finally, (v) a method of generating probability distributions starting from one basic distribution employing coordinate transformations is given. This in conjuction with (iii) leads to the notions of “Physical constraints” in contrast to the “mathematical constraints” in examing parameter estimation.

An inequality derived by both Beckner, and Bialynicki-Birula and Mycielski is used to establish an upper bound for the sum of Kullback-Leibler information for probability densities obtained from normalized squared Fourier transform pairs relative to respective probability densities for minimum Heisenberg uncertainty wavefunctions.

Annals of the New York Academy of SciencesVolume 484, Issue 1 p. 321-323 Empirical Requirements on Models of Relaxationa A. K. RAJAGOPAL, A. K. RAJAGOPAL Naval Research Loboratory, Washington, DC 20375 On leave from the Department of Physics and Astronmoy, Louisiana State University, Baton Rouge, Louisiana 70803.Search for more papers by this authorR. W. RENDELL, R. W. RENDELL Naval Research Loboratory, Washington, DC 20375 Sachs/Freeman Associates.Search for more papers by this authorK. L. NGAI, K. L. NGAI Naval Research Loboratory, Washington, DC 20375Search for more papers by this authorS. TEITLER, S. TEITLER Naval Research Loboratory, Washington, DC 20375Search for more papers by this author A. K. RAJAGOPAL, A. K. RAJAGOPAL Naval Research Loboratory, Washington, DC 20375 On leave from the Department of Physics and Astronmoy, Louisiana State University, Baton Rouge, Louisiana 70803.Search for more papers by this authorR. W. RENDELL, R. W. RENDELL Naval Research Loboratory, Washington, DC 20375 Sachs/Freeman Associates.Search for more papers by this authorK. L. NGAI, K. L. NGAI Naval Research Loboratory, Washington, DC 20375Search for more papers by this authorS. TEITLER, S. TEITLER Naval Research Loboratory, Washington, DC 20375Search for more papers by this author First published: December 1986 https://doi.org/10.1111/j.1749-6632.1986.tb49587.xCitations: 11 † This work was supported in part by Contracts N00014-85C-2315 and N00014-86-WR-24016 with the Office of Naval Research. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article.Citing Literature Volume484, Issue1Dynamic Aspects of Structural Change in Liquids and GlassesDecember 1986Pages 321-323 RelatedInformation