Wave Mechanics or Wave Statistical Mechanics

By comparison between equations of motion of geometrical optics and that of classical statistical mechanics, this paper finds that there should be an analogy between geometrical optics and classical statistical mechanics instead of geometrical mechanics and classical mechanics. Furthermore, by comparison between the classical limit of quantum mechanics and classical statistical mechanics, it finds that classical limit of quantum mechanics is classical statistical mechanics not classical mechanics, hence it demonstrates that quantum mechanics is a natural generalization of classical statistical mechanics instead of classical mechanics. Thence quantum mechanics in its true appearance is a wave statistical mechanics instead of a wave mechanics.     

Wave Mechanics or Wave Statistical Mechanics

By comparison between equations of motion of geometrical optics and that of classical statistical mechanics, this paper finds that there should be an analogy between geometrical optics and classical statistical mechanics instead of geometrical mechanics and classical mechanics. Furthermore, by comparison between the classical limit of quantum mechanics and classical statistical mechanics, it finds that classical limit of quantum mechanics is classical statistical mechanics not classical mechanics, hence it demonstrates that quantum mechanics is a natural generalization of classical statistical mechanics instead of classical mechanics. Thence quantum mechanics in its true appearance is a wave statistical mechanics instead of a wave mechanics.     

Anomalous isotropic luminescence from anthracene sandwich dimers

A classical statistical probability amplitude is introduced whose square modulus is the distribution function. This enables the analogy between classical statistical mechanics and quantum mechanics to be completed. The analogy is developed until quantum statistical derivations can be used in classical statistical mechanics. Two master equations are found: the classical equivalent of the Pauli Master Equation, and a generally valid master equation. Well-known classical equations are deduced from these in a special representation. Interference terms are found and discussed.     

Foundations of the quantum statistics of systems in equilibrium: A measuretheoretic approach

The fundamental equations of equilibrium quantum statistical mechanics are derived in the context of a measure-theoretic approach to the quantum mechanical ergodic problem. The method employed is an extension, to quantum mechanical systems, of the techniques developed by R. M. Lewis for establishing the foundations of classical statistical mechanics. The existence of a complete set of commuting observables is assumed, but no reference is made a priori to probability or statistical ensembles. Expressions for infinite-time averages in the microcanonical, canonical, and grand canonical ensembles are developed which reduce to conventional quantum statistical mechanics for systems in equilibrium when the total energy is the only conserved quantity. No attempt is made to extend the formalism at this time to deal with the difficult problem of the approach to equilibrium.     

Some Observations on the Early History of Equilibrium Statistical Mechanics

The article begins with some personal comments by the author on the outstanding contributions of Michael Fisher to statistical mechanics and critical phenomena. Its major aim is to trace the contributions of a number of pioneering personalities to the early history of equilibrium statistical mechanics. Four different areas are considered: (1) Classical Statistical Mechanics, (2) Quantum Statistical Mechanics, (3) Interacting Systems, and (4) The Ising Model. The article is concerned with the development and applications of statistical mechanics when certain basic assumptions are made. It does not deal with the justification of these assumptions which is a sophisticated discipline of its own.     

Generalizing Planck’s distribution by using the Carati–Galgani model of molecular collisions

Classical systems of coupled harmonic oscillators are studied using the Carati–Galgani model. We investigate the consequences for Einstein’s conjecture by considering that the exchange of energy in molecular collisions follows the Lévy type statistics. We develop a generalization of Planck’s distribution admitting that there are analogous relations in the equilibrium quantum statistical mechanics of the relations found using the nonequilibrium classical statistical mechanics approach. The generalization of Planck’s law based on the nonextensive statistical mechanics formalism is compatible with our analysis.     

用动力学方法研究原子分子碰撞过程中的统计平均问题

动力学李代数方法在研究原子分子碰撞问题中是一种很重要的方法.在计算过程中我们用密度算子导出了物理量的统计平均值.同时我们用时间演化算子计算了振转能量的跃迁几率.作为例子我们用此方法计算了H2和He的碰撞问题.     

Reinterpretation of Quantum Mechanics Based on the Statistical Interpretation

I attempt to develop further the statistical interpretation of quantum mechanics proposed by Einstein and developed by Popper, Ballentine, etc. Two ideas are proposed in the present paper. One is to interpret momentum as a property of an ensemble of similarly prepared systems which is not satisfied by any one member of the ensemble of systems. Momentum is regarded as a statistical parameter like temperature in statistical mechanics. The other is the holistic assumption that a probability distribution is determined as a whole as most likely to be realized. This is the same as the chief assumption in statistical mechanics, and maximum likelihood in classical statistics. These ideas enable us to understand statistically (1) the formalism of quantum mechanics, (2) Heisenberg's uncertainty relations, and (3) the origin of quantum equations. They also explain violation of Bell's inequality and the interference of probabilities.     

Nonextensive entropies derived from Gauss? principle

Gauss? principle in statistical mechanics is generalized for a q-exponential distribution in nonextensive statistical mechanics. It determines the associated stochastic and statistical nonextensive entropies which satisfy Greene-Callen principle concerning on the equivalence between microcanonical and canonical ensembles.     

Formulation of statistical mechanics for chaotic systems

We formulate the statistical mechanics of chaotic system with few degrees of freedom and investigated the quartic oscillator system using microcanonical and canonical ensembles. Results of statistical mechanics are numerically verified by considering the dynamical evolution of quartic oscillator system with two degrees of freedom.      

非广延统计力学与完全开放系统的统计分布

简介了非广延统计力学的Tsallis统计,用其计算了理想气体;推导出了以含有非广延熵常数的Shannon熵为基础和以Tsallis熵为基础的非广延统计力学的完全开放系统的统计分布及计算热力学量的公式;讨论表明:Tsallis熵对应的统计分布及计算热力学量的公式在非广延参量q→1时,完全过渡到了Shannon熵对应的形式.     

Classical Mechanics Without Determinism

Classical statistical particle mechanics in the configuration space can be represented by a nonlinear Schrödinger equation. Even without assuming the existence of deterministic particle trajectories, the resulting quantum-like statistical interpretation is sufficient to predict all measurable results of classical mechanics. In the classical case, the wave function that satisfies a linear equation is positive, which is the main source of the fundamental difference between classical and quantum mechanics.     

Unification theory of classical statistical uncertainty relation and quantum uncertainty relation and its applications

General classical statistical uncertainty relation is deduced and generalized to quantum uncertainty relation. We give a general unification theory of the classical statistical and quantum uncertainty relations, and prove that the classical limit of quantum mechanics is just classical statistical mechanics. It is shown that the classical limit of the general quantum uncertainty relation is the general classical uncertainty relation. Also, some specific applications show that the obtained theory is self-consistent and coincides with those from physical experiments.     

Rigged Hilbert spaces and time asymmetry: The case of the upside-down simple harmonic oscillator

The upside-down simple harmonic oscillator system is studied in the contexts of quantum mechanics and classical statistical mechanics. It is shown that in order to study in a simple manner the creation and decay of a physical system by way of Gamow vectors we must formulate the theory in a time-asymmetric fashion, namely using two different rigged Hilbert spaces to describe states evolving toward the past and the future. The spaces defined in the contexts of quantum and classical statistical mechanics are shown to be directly related by the Wigner function.     

Statistical mechanics and financial markets: Antagony between derivatives and market self-regulation

We establish specific correspondences between notions of economics and statistical mechanics. There are several situations wherein a rather accurate correspondence has already been established, for instance in utility theory for exchange economy with quasilinear utility function, which has been mapped to analogous thermodynamics. We discuss how statistical mechanics can be applied to define the efficiency of financial markets, via a mapping of stock fluctuations to the Random Energy Model (REM) at particular temperatures. We introduce the concept of reflection in economics; the effective reflection number, in particular, is found to be crucial in understanding the self-regulation of the market. We also establish a qualitative similarity between market with derivatives and certain statistical mechanics models. Such an analogy supports a hypothesis that financial derivatives are antagonistic to the self-regulation of financial markets. As a whole, our analysis is complementary to established concepts and methods of neoclassical economics for markets without derivatives.     

On the Representation of Quantum Mechanics on a Classical Sample Space

Under the common viewpoint of statistical maps,the concept of observables in quantum mechanics and inclassical probability theory are discussed and compared.It is shown that, by means of injective statistical maps, quantum mechanics can to a certain extentbe reformulated in classical terms. Some characteristicexamples are considered.     

Probability and logical structure of statistical theories

A characterization of statistical theories is given which incorporates both classical and quantum mechanics. It is shown that each statistical theory induces an associated logic and joint probability structure, and simple conditions are given for the structure to be of a classical or quantum type. This provides an alternative for the quantum logic approach to axiomatic quantum mechanics. The Bell inequalities may be derived for those statistical theories that have a classical structure and satisfy a locality condition weaker than factorizability. The relation of these inequalities to the issue of hidden variable theories for quantum mechanics is discussed and clarified.     

Thermodynamics is more powerful than the role to it reserved by Boltzmann-Gibbs statistical mechanics

We briefly review the connection between statistical mechanics and thermodynamics. We show that, in order to satisfy thermo-dynamics and its Legendre transformation mathematical frame, the celebrated Boltzmann-Gibbs (BG) statistical mechanics is sufficient but not necessary. Indeed, the N →∞ limit of statistical mechanics is expected to be consistent with thermodynamics. For systems whose elements are generically independent or quasi-independent in the sense of the theory of probabilities, it is well known that the BG theory (based on the additive BG entropy) does satisfy this expectation. However, in complete analogy, other thermostatistical theories (e.g., q-statistics), based on nonadditive entropic functionals, also satisfy the very same expectation. We illustrate this standpoint with systems whose elements are strongly correlated in a specific manner, such that they escape the BG realm.     

The Spin-Echo Experiment and Statistical Mechanics

No Heading In this paper the spin-echo experiment is examined in the light of three different approaches to statistical mechanics: the coarse-graining Gibbsian approach, the interventionist Gibbsian approach, and the Boltzmannian approach. The conclusions of this examination are almost exactly opposite to the conclusions of Ridderbos and Redhead [1]: Firstly, it is argued that the spin-echo experiment does not tell against a coarse-graining approach to statistical mechanics. Secondly, it is argued that the interventionist approach to statistical mechanics is itself somewhat problematic as its statistical mechanical counterpart of thermodynamic entropy has a number of properties that actual thermodynamic entropy seemingly does not. In the final section of this paper a feature of coarse-grained entropies (their relativity) is noted that may enable coarse-graining approaches to reconcile conflicting intuitions about the behaviour of entropy in the spin-echo experiment, which may be considered a further advantage of such approaches.