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Long-term behaviour of large mechanical systems with random initial data. (English) Zbl 1032.70009

Summary: We study the long-time behaviour of large systems of ordinary differential equations with random data. Our main focus is a Hamiltonian system which describes a distinguished particle attached by springs to a large collection of heat particles. In the limit where the size of the heat bath tends to infinity, the trajectory of the distinguished particle can be weakly approximated, on finite time intervals, by a Langevin stochastic differential equation. We examine the long-term behaviour of these trajectories, both analytically and numerically. We find ergodic behaviour manifest in both the long-time empirical measures and in the resulting auto-correlation functions.

MSC:

70F45 The dynamics of infinite particle systems
70H05 Hamilton’s equations
70L05 Random vibrations in mechanics of particles and systems
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
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[1] DOI: 10.1063/1.467427 · doi:10.1063/1.467427
[2] DOI: 10.1002/(SICI)1096-987X(199912)20:16<1760::AID-JCC8>3.0.CO;2-2 · Zbl 05428581 · doi:10.1002/(SICI)1096-987X(199912)20:16<1760::AID-JCC8>3.0.CO;2-2
[3] DOI: 10.1023/A:1004667325896 · Zbl 0958.82001 · doi:10.1023/A:1004667325896
[4] DOI: 10.1007/BF01011142 · Zbl 0682.60069 · doi:10.1007/BF01011142
[5] DOI: 10.1002/cpa.3160260405 · Zbl 0253.60065 · doi:10.1002/cpa.3160260405
[6] Papanicolaou G. C., Comm. Pure Appl. Math pp 641– (1974)
[7] DOI: 10.1063/1.1704304 · Zbl 0127.21605 · doi:10.1063/1.1704304
[8] DOI: 10.1023/A:1007307617547 · Zbl 0894.60051 · doi:10.1023/A:1007307617547
[9] DOI: 10.1143/PTP.33.423 · Zbl 0127.45002 · doi:10.1143/PTP.33.423
[10] DOI: 10.1007/BF01008729 · doi:10.1007/BF01008729
[11] DOI: 10.1073/pnas.97.7.2968 · Zbl 0968.60036 · doi:10.1073/pnas.97.7.2968
[12] DOI: 10.1006/jcph.2001.6722 · Zbl 0990.70003 · doi:10.1006/jcph.2001.6722
[13] DOI: 10.1007/s10208001002 · doi:10.1007/s10208001002
[14] DOI: 10.1002/cpa.1014 · Zbl 1017.86001 · doi:10.1002/cpa.1014
[15] DOI: 10.2307/2371267 · Zbl 0019.33404 · doi:10.2307/2371267
[16] Dvoretzky A., Proc. Fourth Int. Symp. Datyon Ohio 1975 pp 23– (1977)
[17] DOI: 10.1137/0134036 · Zbl 0392.93040 · doi:10.1137/0134036
[18] Beck C., Physica 72 pp 211– (1994)
[19] DOI: 10.1007/BF01014351 · Zbl 1255.60174 · doi:10.1007/BF01014351
[20] DOI: 10.1073/pnas.230433997 · Zbl 0969.35117 · doi:10.1073/pnas.230433997
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