Physics
Probabilistic Mechanics
Probabilistic mechanics is a branch of mechanics that incorporates probability theory to analyze and predict the behavior of systems with inherent randomness or uncertainty. It provides a framework for understanding and modeling the probabilistic nature of physical phenomena, such as the behavior of particles at the quantum level or the random fluctuations in materials.
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7 Key excerpts on "Probabilistic Mechanics"
eBook - PDFThe Road to Maxwell's Demon
Conceptual Foundations of Statistical Mechanics
- Meir Hemmo, Orly R. Shenker(Authors)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
125 impossible. The central aim of statistical mechanics is to prove Boltz- mann’s intuition. Introducing probability into statistical mechanics is subtle, especially because classical mechanics is a deterministic theory. There are two desiderata that we believe the notion of probability in statistical mechan- ics must satisfy. (i) Probability should be characterized in purely mechan- ical terms. In particular we construct the notion of probability on the basis of the interplay between macrostates and dynamical blobs as developed so far. A notion of probability based on mechanical terms is mandatory in order to describe Boltzmann’s insight (let alone prove it) in mechanics. (ii) We shall take it here that probabilistic statements must be empirically testable to the same extent as non-probabilistic statements such as Newton’s Second Law, F ¼ ma, Maxwell’s equation of the mag- netic field, or the Schroedinger equation in quantum mechanics. 2 We call this the testability criterion. Assuming that probabilistic counterparts of the laws of thermodynamics that satisfy the above two desiderata can be formulated, we shall say that they are as physically objective as mechanics and thermodynamics are. 6.2 Probability in statistical mechanics In a nutshell, probability enters statistical mechanics at the interface between macrostates and dynamical blobs owing to the fact that as the blob evolves according to the dynamics of the system, in general it partly overlaps with different macrostates. These partial overlaps between the blob and the macrostates determine the probabilities of future macro- states, and so in order to make probabilistic predictions one can follow the evolution of the blob by solving the mechanical equations of motion for the initial microstates contained in the initial macrostate, and measure the partial overlaps of this blob at the desirable future time. In this section we will describe in detail this idea and some of its consequences.
eBook - PDFQuantum Mechanics In Phase Space: An Overview With Selected Papers
An Overview with Selected Papers
- Thomas L Curtright, David B Fairlie, Cosmas K Zachos(Authors)
- 2005(Publication Date)
- World Scientific(Publisher)
Yet there are serious difficulties in effecting such a reformulation. Classical statistical mechanics is a ‘ crypto-deter- ministic ’ theory, where each element of the probability distribution of the dynamical variables specifying a given system evolves with time according to deterministic laws of motion; the whole uncertainty is contained in the form of the initial distribu. tions. A theory based on such concepts could not give a satisfactory account of such non-deterministic effects as raclioa,ctive decay or spontaneous emission (cf. Whit- taker (2)). Classical statistical mechanics is, however, only a special case in the general theory of dynamical statistical (stochastic) processes. In the general case, there is the possibihty of ‘ dEusion’ of the probability ‘fluid’, so that the transformation with time of the probability distribution need not be deterministic in the classical sense. In this paper, we shall attempt to interpret quantum mechanics as a form of such a general statistical dynamics. I. QUA.NTUM KDTEMATICS 2. THE EXISTENCE OF PHASE-SPAUE DISTRIBUTIONS I N QUANTUM THEORY In the accepted statistical interpretation of quantum theory, the possible values of a dynamical variable s are the eigenvalues si of the corresponding operator (observable ) 7-2 168 100 J. E. MOYAL S in the Hilbert space of the state vectors. The probability of finding sc in a state @ is then equal to the square of the modulus I a$ l 2 of the projection a, of @ on the corre- sponding eigenvector 1C.i. A complete or irreducible representation for a given mechanical system is given by a set of commuting observables s such that their eigenvectors pi span the whole space, i.e. such that any ? , b = Ca,$i. Hence we obtain directly from II.
- Nikolai Sergeevich Krylov, Joel S. Migdal(Authors)
- 2014(Publication Date)
- Princeton University Press(Publisher)
This conclusion must not of course be taken to mean only that classical mechanics is incapable of providing all that is necessary for laying the foundations of statistics and must be sup- plemented with some elements of probabilistic notions. Such a conclusion would be quite obvious (cf. §§2 and 4) and is now generally accepted (cf., for instance, the Ehrenfests' review [1]). The conclusion we have made means much more than that: no logically permissible combination of proba- bilistic notions together with classical mechanics 1 can help us achieve the object of laying the foundations of physical statistics; in other words, classi- cal mechanics cannot serve as a micromechanics that can form a basis for the Physically permissible is only that form of combination which is expressed by probabilistic assumptions regarding the distribution of initial microstates, which was proved by Boltzmann in his statistical interpretation of the molecular- kinetic theory; cf. §4. FOUNDATIONS OF STATISTICAL PHYSICS 99 construction of statistical physics. Indeed, our principal argument de- scribed in §§12 and 13 (as well as the arguments of §10) is based on only one main feature of classical mechanics—on the fact that equations lack the probabilistic element. This feature can be taken as a definition of classical mechanics, and therefore the rejection of some special proper- ties of classical mechanics cannot alter the conclusion we have drawn. It is precisely because the conclusion we have made indicates the in- adequacy of classical mechanics as a micromechanics forming a basis for the construction of physical statistics that the above considerations re- garding the impossibility of interpreting probability laws on its basis (§§10-15) could hardly have arisen in the days of Boltzmann and the con- temporary discussion on the foundations of thermodynamics.
eBook - PDF- Nelson Bolívar(Author)
- 2019(Publication Date)
- Arcler Press(Publisher)
It is important to note that the external physical condition does not affect the possible states per se but only affects the numerical values of the probabilities associated with these states. We shall see presently that this is so in quantum mechanics also. PROBABILITY THEORY AND STATISTICAL PHYSICS We deal with probability in both probability theory and in statistical physics (which consists of quantum mechanics and classical statistical mechanics). The definition of probability, as well as the laws of probability theory, are common to both. But there are many fundamental differences between probability theory and statistical physics regarding probability. First we consider the differences regarding possible states and the numerical values of the primary probabilities. In statistics we deal with macroscopic objects whose possible “states” are easily identifiable (like the states of height, eye color, parenthood , etc.) and treat the primary probabilities of these states as unspecified constants. Their numerical values are estimated only by making use of some statistical data. In physics, the laws of physics determine uniquely the possible states of physical systems but there is no unique method of determining the primary probability distributions. 1) In the case of an atomic system (like a single hydrogen atom) the monadic probability distributions are determined uniquely by the laws of quantum mechanics (without any reference to probability theory) [23] . 2) In the case of some collectivistic quantum phenomena, the collectivistic probability distributions are determined uniquely by the laws of quantum mechanics (without any reference to probability theory) [23] . 3) In the case of an assembly of classical systems in statistical equilibrium, the collectivistic probability distribution is determined by a combination of the laws of classical mechanics and the laws of probability theory [24] ; this is so in the case of atomic systems also [25] .
eBook - PDFPhysical Chemistry
A Modern Introduction, Second Edition
- William M. Davis(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
343 11 Statistical Mechanics Theconnectionbetweenthequantummechanicaltreatmentofindividualatomsand moleculesandmacroscopicpropertiesandphenomenaisthegoalofstatisticalmechanical analysis�Statisticalmechanicsisthemeansforaveragingcontributionstopropertiesand toenergiesoveralargecollectionofatomsandmolecules�Thefirstpartoftheanalysisis directedtothedistributionofparticlesamongavailablequantumstates�Anoutcomeof thisanalysisisthepartitionfunction,whichprovestobeanessentialelementinthermo-dynamics,inreactionkinetics,andintheintensityinformationofmolecularspectra� 11.1 Probability Inearlierchapters,weconsideredthepressureofgasintermsofcollisionswithamov-ablewallofacontainer�Wesawthatthereisarelationbetweenathermodynamicstate property,pressureofthebulkgas,andmolecularbehavior�Inprinciple,suchconnections canbeapproachedintwoways�Amechanicalapproachistousemechanicalanalysisto followasystemofmanyparticles(atomsandmolecules)intime�Macroscopicproperties of a system at equilibrium correspond to the long-time average of the system’s behav-ior,andthus,afullmechanicalanalysisprovideseverythingneededtoobtainpressure, internalenergy,temperature,andsoon�Theotherapproachisstatisticalinnature,aswe discussedinChapter1�Itinvolvesconsideringtheprobabilitiesofthedifferentwaysa systemcanexistatanyinstantandthenaveragingpropertiesoverthosedifferentways, eachweightedbyitsprobability�Amechanicalapproachmayseemratherclear-cut,but unfortunately the task of solving mechanical (classical or quantum) problems for huge numbersofparticlesoverlongperiodsoftimeisnotusuallyfeasible�Therefore,weneed thesecondapproach—thetoolsofstatisticalmechanics�Todevelopthem,weneedtokeep inmindcertaininformationaboutprobabilities� Essentialtoanydiscussionofprobabilityisitsdefinition�Ifthereare N mutuallyexclusive andequallylikelyoccurrences,and n A andonly n A ofthemleadtosomeparticularresult, A ,
- Isaac E Elishakoff, Makoto Ohsaki(Authors)
- 2010(Publication Date)
- ICP(Publisher)
Given this, the probabilistic properties of the output quantities can be determined. As Wentzel (1980) stresses, probabilistic methods are ‘frequently regarded as a kind of magic wand which produces in-1 2 Optimization and Anti-Optimization of Structures under Uncertainty formation out of a void. This is a fallacy; the theory of probability only enables information to be transformed, and conclusions on in-accessible phenomena to be drawn from data on observable ones.’ Probabilistic methods must yield a central quantity–reliability of the structure, namely, the probability that the structure will perform its mission satisfactorily. Modern society rightfully expects extremely high reliabilities and, consequently, extremely small probabilities of failure. The deterministic mechanical theories represent a cornerstone of Probabilistic Mechanics. First, the phenomenon should be understood qualitatively; next, it should be understood sufficiently well quantitatively. Then the output quantities in their intricate dependence on their input counterparts are developed as a set of equations, algebraic expressions, or numerical codes of varying complexity. At this stage the parameters to be treated as uncertain variables or functions are identified. The deterministic relation or numerical code then serves as a transfer function determining the probabilistic characteristics of the output and the reliability of the structure. Our goal being extremely high reliabilities, it is immediately understood that the deterministic relations or numerical codes must be of supreme accuracy, in order to avoid a GIGO (‘garbage in–garbage out’) situation.
eBook - PDFEntropy for Biologists
An Introduction to Thermodynamics
- Harold J. Morowitz(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
In classical mechanics the ensemble forms a continuum over all possible coordinates and momenta of the atoms subject to the con-straints of the system. This continuum approach leads to certain inde-terminacies in evaluating thermodynamic functions. Quantum mecha-nics seems to form a more natural basis for statistical thermodynamics The Quantum Mechanical Description of Systems 77 because the theory itself generates an ensemble of discrete states. Indeed, using the wisdom of hindsight, it has been pointed out that the founders of statistical mechanics might have been alerted to the existence of quantum mechanics by the problem of being able to choose an elemen-tary volume in phase space. We will therefore carry out the actual formal analysis of statistical theory in terms of quantum states, which will necessitate the briefest review of some of the elementary notions of quantum mechanics. We will shortly make some of these notions more concrete by considering the quantum-mechanical problem of the par-ticle in the box. This particular example will later be extended to a collection of particles in a box, which is, of course, the quantum analog of a perfect gas. B. T H E Q U A N T U M M E C H A N I C A L D E S C R I P T I O N OF SYSTEMS It should be understood that we cannot here attempt to explain or justify quantum mechanics, but only to present the results so that the notions of quantum state and energy eigenvalue will be more familiar when we utilize them in setting up the fundamental equations of sta-tistical mechanics. The reader interested in more detailed quantum mechanical background should consult a book like Quantum Mechanics in Chemistry by M . W. Hanna (Benjamin, New York, 1965). The formulation of quantum mechanics we shall use was carried out by Erwin Schrödinger, a man who later produced a probing examination of the relationship between biology and the statistical aspects of thermal physics.
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