DOI: 10.1063/1.1703637. Poincaré Cycles, Ergodicity, and Irreversibility in Assemblies of Coupled Harmonic Oscillators

Poincaré Cycles, Ergodicity, and Irreversibility in Assemblies of Coupled Harmonic Oscillators

10.1063/1.1703637

Crossref journal-article

AIP Publishing

Journal of Mathematical Physics (317)

Abstract

The transport coefficients (diffusion constant, electrical conductivity, etc.) associated with irreversible processes in an assembly of particles can be expressed as integrals over certain time relaxed correlation functions between small numbers of variables of the assembly. The scattering of slow neutrons is also a measure of time relaxed correlation functions. Irreversibility is a consequence of the vanishing of the correlation coefficients as the relaxation time becomes infinite. On the other hand these coefficients have Poincaré cycles so that any value which they take on is repeated an infinite number of times. It is shown that, in the case of fluctuations of 0(N−½) from zero (N being the number of degrees of freedom), the period of Poincaré cycles is of the order of the mean period of normal mode vibrations while for fluctuations of a magnitude independent of N the period is of the order of CN where C is a constant which is greater than 1. The time relaxed correlation coefficients of a pair of particles separated by r lattice spacings decays as t−m/2, m being the number of dimensions of the assembly. The statistics of the decay of the momentum of a particle from a preassigned initial value to its equipartition value are discussed.

Bibliography

Mazur, P., & Montroll, E. (1960). PoincarÃ© Cycles, Ergodicity, and Irreversibility in Assemblies of Coupled Harmonic Oscillators. Journal of Mathematical Physics, 1(1), 70â84.

Authors 2

Peter Mazur (first)
Elliott Montroll (additional)

References 22 Referenced 135

{'key': '2024021103243768000_r1'}
{'key': '2024021103243768000_r2', 'first-page': '239', 'volume': '105', 'year': '1859', 'journal-title': 'Pogg. Ann.'} / Pogg. Ann. (1859)
10.1103/PhysRev.95.249 / Phys. Rev. (1954)
10.1016/S0031-8914(53)80008-X / Physica (1953)
{'key': '2024021103243768000_r4a', 'first-page': '299', 'volume': '21', 'year': '1955', 'journal-title': 'Physica'} / Physica (1955)
{'key': '2024021103243768000_r5'}
10.1103/RevModPhys.15.1 / Revs. Modern Phys. (1943)
{'key': '2024021103243768000_r7', 'first-page': '948', 'volume': '21', 'year': '1958', 'journal-title': 'Progr. Theoret. Phys.'} / Progr. Theoret. Phys. (1958)
{'key': '2024021103243768000_r8'}
10.1103/PhysRev.112.1829 / Phys. Rev. (1958)
{'key': '2024021103243768000_r10', 'first-page': '570', 'volume': '12', 'year': '1957', 'journal-title': 'J. Phys. Soc. (Japan)'} / J. Phys. Soc. (Japan) (1957)
10.1063/1.1740082 / J. Chem. Phys. (1954)
{'key': '2024021103243768000_r12', 'first-page': '604', 'volume': '190', 'year': '1934', 'journal-title': 'Math. Ann.'} / Math. Ann. (1934)
{'issue': '12', 'key': '2024021103243768000_r13', 'volume': '28B', 'year': '1942', 'journal-title': 'Ack. Mat. Astr. Fys.'} / Ack. Mat. Astr. Fys. (1942)
{'key': '2024021103243768000_r14'}
{'key': '2024021103243768000_r15', 'first-page': '1607', 'volume': '21', 'year': '1953', 'journal-title': 'J. Chem. Phys.'} / J. Chem. Phys. (1953)
{'key': '2024021103243768000_r16', 'first-page': '209', 'volume': '3', 'year': '1957', 'journal-title': 'Proc. Third Berkeley Symposium on Math. Stat. and Prob.'} / Proc. Third Berkeley Symposium on Math. Stat. and Prob. (1957)
10.2307/2371869 / Am. J. Math. (1943)
10.1017/S0305004100020739 / Proc. Cambridge Phil. Soc. (1939)
10.1103/PhysRev.89.1189 / Phys. Rev. (1953)
{'key': '2024021103243768000_r20'}
10.2307/2371267 / Am. J. Math. (1938)

Dates

Type	When
Created	20 years, 7 months ago (Jan. 6, 2005, 12:26 p.m.)
Deposited	1 year, 6 months ago (Feb. 10, 2024, 10:24 p.m.)
Indexed	2 weeks ago (Aug. 6, 2025, 8:07 a.m.)
Issued	65 years, 7 months ago (Jan. 1, 1960)
Published	65 years, 7 months ago (Jan. 1, 1960)
Published Print	65 years, 7 months ago (Jan. 1, 1960)

Funders 0

None

BibTeX

@article{Mazur_1960, title={Poincaré Cycles, Ergodicity, and Irreversibility in Assemblies of Coupled Harmonic Oscillators}, volume={1}, ISSN={1089-7658}, url={http://dx.doi.org/10.1063/1.1703637}, DOI={10.1063/1.1703637}, number={1}, journal={Journal of Mathematical Physics}, publisher={AIP Publishing}, author={Mazur, Peter and Montroll, Elliott}, year={1960}, month=jan, pages={70–84} }

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