Statistical Mechanics of Assemblies of Coupled Oscillators
dc.contributor.author | Ford, G. W. | en_US |
dc.contributor.author | Kac, M. | en_US |
dc.contributor.author | Mazur, P. | en_US |
dc.date.accessioned | 2010-05-06T22:39:25Z | |
dc.date.available | 2010-05-06T22:39:25Z | |
dc.date.issued | 1965-04 | en_US |
dc.identifier.citation | Ford, G. W.; Kac, M.; Mazur, P. (1965). "Statistical Mechanics of Assemblies of Coupled Oscillators." Journal of Mathematical Physics 6(4): 504-515. <http://hdl.handle.net/2027.42/70752> | en_US |
dc.identifier.uri | https://hdl.handle.net/2027.42/70752 | |
dc.description.abstract | It is shown that a system of coupled harmonic oscillators can be made a model of a heat bath. Thus a particle coupled harmonically to the bath and by an arbitrary force to a fixed center will (in an appropriate limit) exhibit Brownian motion. Both classical and quantum mechanical treatments are given. | en_US |
dc.format.extent | 3102 bytes | |
dc.format.extent | 854921 bytes | |
dc.format.mimetype | text/plain | |
dc.format.mimetype | application/pdf | |
dc.publisher | The American Institute of Physics | en_US |
dc.rights | © The American Institute of Physics | en_US |
dc.title | Statistical Mechanics of Assemblies of Coupled Oscillators | en_US |
dc.type | Article | en_US |
dc.subject.hlbsecondlevel | Physics | en_US |
dc.subject.hlbtoplevel | Science | en_US |
dc.description.peerreviewed | Peer Reviewed | en_US |
dc.contributor.affiliationum | Department of Physics, University of Michigan, Ann Arbor, Michigan | en_US |
dc.contributor.affiliationother | The Rockefeller Institute, New York, New York | en_US |
dc.contributor.affiliationother | Lorentz Institute for Theoretical Physics, Leiden, The Netherlands | en_US |
dc.description.bitstreamurl | http://deepblue.lib.umich.edu/bitstream/2027.42/70752/2/JMAPAQ-6-4-504-1.pdf | |
dc.identifier.doi | 10.1063/1.1704304 | en_US |
dc.identifier.source | Journal of Mathematical Physics | en_US |
dc.identifier.citedreference | There is an extensive literature on the motion of coupled oscillators, mostly concerned with motion in a lattice with nearest‐neighbor interactions. Some of the more recent articles which have a bearing on our work are: P. Mazur and E. Montroll, J. Math. Phys. 1, 70 (1960); P. C. Hemmer, “Dynamic and Stochastic Types of Motion in the Linear Chain,” thesis, Norges Tekniske Høgskole, Trondheim, Norway (1959); R. J. Rubin, J. Math. Phys. 1, 309 (1960); 2, 373 (1961); M. Toda and Y. Koguri, Suppl. Progr. Theoret. Phys. (Kyoto) 23, 157 (1962); R. E. Turner, Physica 26, 274 (1960). | en_US |
dc.identifier.citedreference | The classic papers on the phenomological theory of Brownian motion are: G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823 (1930); M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945). | en_US |
dc.identifier.citedreference | See the article by H. Wergeland in Fundamental Problems in Statistical Mechanics, edited by E. G. D. Cohen (North‐Holland Publishing Company, Amsterdam, 1962). | en_US |
dc.identifier.citedreference | That is, there are no memory effects. | en_US |
dc.identifier.citedreference | We use the notation ‖M‖jk‖M‖jk for the element in the jth row and kth column of a matrix M. | en_US |
dc.identifier.citedreference | See, e.g., G. Kowalewski, Determinantentheorie (Chelsea Publishing Company, New York, 1948), 3rd ed., p. 105. | en_US |
dc.identifier.citedreference | What is unique is the spectrum of eigenfrequencies: g(ω) = 2ω/(πf′(θ)),g(ω)=2ω∕(πf′(θ)), where θ is the function of ω obtained by inverting the equation ω2 = f(θ).ω2=f(θ). For (25), g(ω) = (2f/π)⋅(ω2+f2)−1.g(ω)=(2f∕π)⋅(ω2+f2)−1. | en_US |
dc.identifier.citedreference | See, e.g., Wang and Uhlenbeck, Ref. 2. | en_US |
dc.identifier.citedreference | See, e.g., G.‐C. Wick, Phys. Rev. 80, 268 (1950). | en_US |
dc.identifier.citedreference | Essentially the same point is made in connection with the quantum description of statistical light beams by E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963). See also R. J. Glauber, Phys. Rev. Letters 10, 84 (1963). | en_US |
dc.identifier.citedreference | The result of the average over the states of the heat bath is still in general an operator function of q0(0)q0(0) and p0(0).p0(0). We trust that our definition of this average, which involves an average over the initial coordinate and momentum of a fictitious Brownian particle, is not confusing. | en_US |
dc.identifier.citedreference | For elementary discussion of the Nyquist formula and its quantum generalization see C. Kittel, Elementary Statistical Physics (John Wiley & Sons, Inc., New York, 1958), pp. 141–153. See also J. Lawson and G. E. Uhlenbeck, Threshold Noise Signals (McGraw‐Hill Book Company, Inc., New York, 1950), especially pp. 64–79. | en_US |
dc.identifier.citedreference | The Brownian motion of a quantum oscillator is considered in a paper by J. Schwinger, J. Math. Phys. 2, 407 (1961). | en_US |
dc.identifier.citedreference | See, e.g., H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1945), p. 118. | en_US |
dc.identifier.citedreference | These properties of the operators are discussed in many textbooks on quantum mechanics. See, e.g., A. Messiah, Quantum Mechanics (North‐Holland Publishing Company, Amsterdam, 1961), Chap. 12. | en_US |
dc.owningcollname | Physics, Department of |
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