Memory and Entropy

I study the physical nature of traces. Surprisingly, (i) systems separation with (ii) temperature differences and (iii) long thermalization times are sufficient conditions to produce macroscopic traces. Traces of the past are ubiquitous because these conditions are largely satisfied in our universe. I quantify these thermodynamical conditions for memory and derive an expression for the maximum amount of information stored in such memories as a function of the relevant thermodynamical parameters. This mechanism transforms low entropy into available information. I suggest that all macroscopic information has this origin in past low entropy.     

Entropy and disorder

A discussion is given of some aspects of the relation between concepts of entropy and intuitive qualitative ideas concerning ‘disorder’.

In this connection, a plea is mado for moderation in the use of such idess in teaching. In particular: (i) They ought not to be so misused as to obscure the status of S as a quantity which can be treated in a purely macroscopic theory. (ii) They do not exhibit a complete correspondence with the concepts of a quantitative treatment on the molecular level.     

Entropy and gravitation

Einstein's gravitational theory is analyzed from a thermodynamic point of view. A thermodynamic potential characterizing the sources of gravitational fields is presented. By means of this potential the entropy production density is derived. Einstein's equations with dissipative terms appear as linear phenomenological laws in the sense of irreversible thermodynamics. Some thermodynamic influences on gravitational phenomena are discussed.     

Time-Reversal and Entropy

There is a relation between the irreversibility of thermodynamic processes as expressed by the breaking of time-reversal symmetry, and the entropy production in such processes. We explain on an elementary mathematical level the relations between entropy production, phase-space contraction and time-reversal starting from a deterministic dynamics. Both closed and open systems, in the transient and in the steady regime, are considered. The main result identifies under general conditions the statistical mechanical entropy production as the source term of time-reversal breaking in the path space measure for the evolution of reduced variables. This provides a general algorithm for computing the entropy production and to understand in a unified way a number of useful (in)equalities. We also discuss the Markov approximation. Important are a number of old theoretical ideas for connecting the microscopic dynamics with thermodynamic behavior.     

Correlation between elastic anisotropy and magnetic susceptibility anisotropy of sedimentary and metamorphic rock

A correlation has been revealed between the types of acoustic anisotropy and magnetic susceptibility anisotropy for specimens from noncrystalline metamorphic and sedimentary rock has been revealed. This correlation makes it possible to assume that the anisotropy of magnetic and acoustic characteristics has the same origin, namely, the rock texture. The pronounced triaxial acoustic anisotropy for specimens of polycrystalline rock in the absence of external forces makes it possible to judge the fracturing caused by tectonic loads and the degree of this fracturing. The large body of available experimental data allows the expectation that it will be possible to obtain empirical dependences suitable for estimating the intensity of geological processes based on the values of the magnetic susceptibility anisotropy and elasticity tensor anisotropy.     

Magnetochiral anisotropy

In biology, chemistry and physics one often deals with systems that can occur in two forms that are each other's mirror image. A new magneto-optical effect called magnetochiral anisotropy (MChA) is described that establishes a link between chirality and magnetic fields and that can discriminate between the two enantiomers (mirror images) of chiral systems. It is shown that through MChA a magnetic field can lead to an enantiomeric excess in a photochemical reaction with unpolarized light. The existence of MChA can be deduced from general symmetry arguments, and the question comes to mind whether this effect can manifest itself in other magnetotransport phenomena in chiral media. Theoretical evidence is presented that indeed MChA is a general phenomenon, and its existence is demonstrated experimentally in the electrical conductivity of carbon nanotubes.     

Test of a simple relation between gap anisotropy and mass enhancement anisotropy

A simple formula due to Butler and Allen relates anisotropy of the superconducting energy gap Δ_k to the anisotropy of the mass enhancement λ_k and Coulomb pseudopotential μ_k¹. This formula is tested against the numerical calculations of Peter, Ashkenazi and Dacorogna, and agrees quite well. The discrepancies are traced to two sources. (1) In regions of the Fermi surface where | λ_k-λ | or $\text{| μ}{}_{\mathrm{k}}{}^1\text{-μ}{}^1\text{|}$ are particularly large, the linear expansion (weak-anisotropy approximation) fails. (2) In regions of the Fermi surface where the important phonon frequencies happen to be especially small, the two-square-well model becomes less valid.     

Relative Entropy and Relative Entropy of Entanglement for Infinite-Dimensional Systems

In this paper, firstly, we derive some inequalities about the relative entropy for infinite-dimensional quantum systems. Secondly, we propose a new measurement based on the relative entropy of entanglement for infinite-dimensional systems with bounded mean energy, and give a lower bound on this entanglement measure. Lastly, we generalize this measure to multi-partite quantum systems.     

Directional reconstruction and anisotropy studies

The incident directions of cosmic ray particles at the highest energies are readily measured by modern detectors. Provided that there are sufficient data, the interpretation of such directions should give strong information on the origin of cosmic rays. However, the choice of statistical methods used in such an analysis can be of paramount importance in the reliability of the interpretation. This article summarizes the conclusions which may be drawn from the present data, and proposes methods needed to interpret statistically the large quantities of data which will become available in the near future. To cite this article: R.W. Clay et al., C. R. Physique 5 (2004).     

Entropy and Vacuum Radiation

It is shown that entropy increase in thermodynamic systems can plausibly be accounted for by the random action of vacuum radiation. A recent calculation by Rueda using stochastic electrodynamics (SED) shows that vacuum radiation causes a particle to undergo a rapid Brownian motion about its average dynamical trajectory. It is shown that the magnitude of spatial drift calculated by Rueda can also be predicted by assuming that the average magnitudes of random shifts in position and momentum of a particle correspond to the lower limits of the uncertainty relation. The latter analysis yields a plausible expression for the shift in momentum caused by vacuum radiation. It is shown that when the latter shift in momentum is magnified in particle interactions, the fractional change in each momentum component is on the order of unity within a few collision times, for gases and (plausibly) for denser systems over a very broad range of physical conditions. So any system of particles in this broad range of conditions would move to maximum entropy, subject to its thermodynamic constraints, within a few collision times. It is shown that the spatial drift caused by vacuum radiation, as predicted by the above SED calculation, can be macroscopic in some circumstances, and an experimental test of this effect is proposed. Consistency of the above results with quantum mechanics is discussed, and it is shown that the diffusion constant associated with the above Brownian drift is the same as that used in stochastic interpretations of the Schrödinger equation.     

Dirichlet Polynomials and Entropy

A Dirichlet polynomial d in one variable y is a function of the form d(y)=anny+⋯+a22y+a11y+a00y for some n,a0,…,an∈N. We will show how to think of a Dirichlet polynomial as a set-theoretic bundle, and thus as an empirical distribution. We can then consider the Shannon entropy H(d) of the corresponding probability distribution, and we define its length (or, classically, its perplexity) by L(d)=2H(d). On the other hand, we will define a rig homomorphism h:Dir→Rect from the rig of Dirichlet polynomials to the so-called rectangle rig, whose underlying set is R⩾0×R⩾0 and whose additive structure involves the weighted geometric mean; we write h(d)=(A(d),W(d)), and call the two components area and width (respectively). The main result of this paper is the following: the rectangle-area formula A(d)=L(d)W(d) holds for any Dirichlet polynomial d. In other words, the entropy of an empirical distribution can be calculated entirely in terms of the homomorphism h applied to its corresponding Dirichlet polynomial. We also show that similar results hold for the cross entropy.     

熵在非晶材料合成中的作用

熵作为系统的状态函数,对于真实物质体系而言是一个极为重要的物理量.在非晶态合金的制备过程中最具代表性的指导原则有"混乱原理"和井上三原则,二者皆与熵有着紧密的联系.在过去很长一段时间内,这些经验准则指导了大量新型非晶体系的发现,但近些年的实验结果对这些理论提出了质疑.除组元数目之外,还有其他尚待研究的因素也影响着合金体系的玻璃形成能力.本文总结了玻璃转变过程中熵在热力学条件、动力学条件和结构条件中所扮演的角色,阐述了其对玻璃形成能力产生的或正或反的影响.特别是对近几年发展起来的高熵非晶体系的研究有助于开发出临界尺寸更大的非晶合金,也有助于进一步探索多组元合金和非晶形成能力之间的关系.     

Quantum Information and Entropy

Thermodynamic entropy is not an entirely satisfactory measure of information of a quantum state. This entropy for an unknown pure state is zero, although repeated measurements on copies of such a pure state do communicate information. In view of this, we propose a new measure for the informational entropy of a quantum state that includes information in the pure states and the thermodynamic entropy. The origin of information is explained in terms of an interplay between unitary and non-unitary evolution. Such complementarity is also at the basis of the so-called interaction-free measurement.     

Entropy and Random Vectors

Barron⁽¹⁾ produced a proof of the Central Limit Theorem for real-valued IID random variables, in the sense of convergence in relative entropy. Here, we establish a similar result for independent real-valued random vectors, not necessarily identically distributed. The main developments required are a generalisation of De Bruijn's identity, and various inequalities proposed in ref. 2.     

Entropy and ionic conductivity

It is known that the ionic conductivity can be obtained by using the diffusion constant and the Einstein relation. We derive it here by extracting it from the steady electric current which we calculate in three ways, using statistics analysis, an entropy method, and an entropy production approach.     

Entropy and normal states

A generalized definition of entropy for any state on aC* algebra is given and studied. We prove that the entropy characterizes uniquely the normal states.