Bulletin of the Crimean Astrophysical Observatory

Stochastic models in classical and quantum mechanics

V. Yu. Terebizh

Published Jun 1, 2010• 0 citations

Abstract

Characteristic features of the probabilistic models used in classical mechanics, statistical physics, and quantum theory are discussed. The viewpoint according to which there is no room for probabilities in Nature, as such, is consistently substantiated; the researcher is forced to introduce probabilistic concepts and the corresponding models in conditions that provide only partial predictability of the phenomena being studied. This approach allows one to achieve a consistent interpretation of some important physical phenomena, in particular, the relationship between instability and irreversibility in time, the stochastic evolution of systems in the theory of deterministic chaos, Boltzmann’s H -theorem, and paradoxes of quantum mechanics.

DOI: 10.3103/s0190271710010171

Related papers

Вероятностные модели в классической и квантовой механике: Исправления к опубликованной ранее статье

V.Yu. Terebizh, В.Ю. Теребиж

Characteristic features of the stochastic models used in the classical mechanics, statistical physics and quantum theory are discussed. The arguments are given in favor of point of view that Nature, in itself, does not include probabilities. Researcher is forced to introduce the probabilistic concepts and corresponding models in conditions providing only partial predictability of a studied system behavior. This approach allows us to achieve a consistent interpretation of some important physical phenomena, in particular, the relationship between instability and irreversibility in time, the stochastic evolution of systems in the deterministic chaos theory, the H-theorem of Boltzmann, and paradoxes of quantum mechanics.

Bulletin of the Crimean Astrophysical ObservatoryJun 1, 20110 citations

Вероятностные модели в классической и квантовой механике

V.Yu. Terebizh, В.Ю. Теребиж

Stochastic models in classical and quantum mechanics, by V.Yu.Terebizh. This approach allows us to achieve a consistent interpretation of some important physical phenomena, in particular, the relationship between instability and irreversibility in time, the stochastic evolution of systems in the deterministic chaos theory, the H-theorem of Boltzmann, and paradoxes of quantum mechanics.

Bulletin of the Crimean Astrophysical ObservatoryJun 1, 20100 citations

Interrelation and Interpretation of effects in quantum mechanics and classical physics

R.KH. RAKHIMOV

Quantum mechanics based on the probabilistic approach provides a powerful tool for accurate prediction and interpretation of quantum phenomena, allowing statistically sound predictions about the behavior of microparticles and quantum systems. This statement emphasizes the probabilistic nature of quantum mechanics, its applicability to quantum phenomena and microparticles, as well as the statistical nature of its predictions when applied to the macro effects of classical physics. In addition, the role of statistics and probability in various fields of science, such as particle physics, thermodynamics, biology, sociology, psychology, economics and finance, is discussed. The philosophical implications of the probabilistic approach and the associated limitations and challenges are also considered.

Computational nanotechnologyAug 15, 20245 citations

Quantum mechanics for classical transport equations

Christof Wetterich

Classical transport equations with probabilistic initial conditions can be viewed as quantum systems. In a discrete version they are probabilistic automata. The time-local probabilistic information is encoded in a classical wave function. Its unitary evolution obeys a Schrödinger equation. Statistical observables measure properties of the classical probability distribution. In the quantum formalism they are represented by operators which do not commute with the ones associated to classical observables. Momentum, quantum energy and angular momentum or charges yield conserved quantities which constrain the evolution of the classical probability distribution. The characteristic features of quantum mechanics, as the superposition of wave functions, interference, the importance of phases, non-commuting operators or a unitary time evolution, are realized by probabilistic classical transport equations. Stochastic transport equations are described by a quantum system with a stochastic Hamiltonian.

quant-pharXiv:2605.15969May 18, 20260 citations

3. Uncertainty, statistics, probability

Arnold Neumaier

In this chapter, we consider the way quantum theory represents deterministic, statistical, and probabilistic aspects of Nature. After a discussion in Section 3.1 of uncertainty in general, we discuss in Sections 3.2-3.4 several formal notions of classical probability, their relation to the probability concept used in applied statistics, and their dependence on the description used. Section 3.5 briefly discusses various examples, where probabilistic features emerge from fully deterministic situations-a theme to be taken up in much more detail in Chapter 11. Section 3.6 then shows how the statistical aspects of the quantum formalism naturally follow from the weak law of large numbers. The notions of c-probability in the classical case and of q-probability in the quantum case are formally defined in Section 3.7. Historically, the concept of classical probability (including its use in stochastic processes) was given an undisputed formal mathematical foundation in 1933 in terms of the measure-theoretic setting of Kolmogorov [164]. Apart from this traditional axiomatic foundation of classical probability theory, there is a less well-known equivalent axiomatic treatment by Whittle [302] in terms of expectations. Here probabilities appear as the expectations of statements, {0, 1}-valued random variables. von Plato [240] discusses the history of the concept of probability. Krüger et al. [169] discuss the history of probability in the various fields of application. Sklar [275] discusses the philosophical problems of the probability concept, with an emphasis on statistical mechanics.

De Gruyter eBooksOct 24, 20190 citations

Probabilities in Physics

Claus Beisbart, Stephan Hartmann

Abstract Many theories and models from physics are probabilistic. This observation raises several philosophical questions: What are probabilities in physics? Do they reflect objective chances which exist independently of the human mind? Or do they only express subjective credences and thus capture our own uncertainty about the world? Finally, which metaphysical lessons, if at all, can one draw from the largely probabilistic character of physics? The chapters collected in this volume address these questions and provide a detailed philosophical appraisal of the status of probabilities in all of physics. Particular emphasis is laid upon statistical physics and quantum mechanics. Many chapters reflect a desire to understand probabilities from physics as objective chances. These chances are characterized, e.g., as time-averages, as probabilities from a best system in the terms of David Lewis, or using the Boltzmannian typicality approach. Other chapters are sympathetic to a Bayesian view of probabilities in physics. The chapters about quantum mechanics elucidate the peculiar characteristics of quantum correlations and discuss strategies to justify the Born Rule. Finally, the chapters of this volume demonstrate how closely interpretive issues about probabilities are entangled with other foundational problems of physics such as the Reversibility Paradox, the ontology of the quantum world and the question whether the world is deterministic.

Sep 1, 2011130 citations