October  2015, 35(10): 4831-4838. doi: 10.3934/dcds.2015.35.4831

Ergodicity of two particles with attractive interaction

1. 

Institut für Statistik und Wahrscheinlichkeitstheorie, TU Wien, 1040 Wien, Austria

2. 

Institut für Theoretische Physik, TU Wien, 1040 Wien, Austria

Received  January 2014 Revised  January 2015 Published  April 2015

We study the ergodic properties of a classical two-particle system with square-well pair potential in an interval.
Citation: Karl Grill, Christian Tutschka. Ergodicity of two particles with attractive interaction. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4831-4838. doi: 10.3934/dcds.2015.35.4831
References:
[1]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1989. doi: 10.1007/978-1-4757-2063-1.

[2]

R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press, 1982.

[3]

L. Boltzmann, Vorlesungen Über Gastheorie, Barth, 1896.

[4]

M. Boshernitzan, A condition for minimal interval exchange maps to be uniquely ergodic, Duke Mathematical Journal, 52 (1985), 723-752. doi: 10.1215/S0012-7094-85-05238-X.

[5]

G. R. Brannock and J. K. Percus, Wertheim cluster development of free energy functionals for general nearest-neighbor interactions in $D=1$, The Journal of Chemical Physics, 105 (1996), 614-627. doi: 10.1063/1.471920.

[6]

J. A. Cuesta and C. Tutschka, Overcomplete free energy functional for $D=1$ particle systems with next neighbor interactions, Journal of Statistical Physics, 111 (2003), 1125-1148. doi: 10.1023/A:1023096031180.

[7]

K. F. Herzfeld and M. Goeppert-Mayer, On the states of aggregation, The Journal of Chemical Physics, 2 (1934), 38-45. doi: 10.1063/1.1749355.

[8]

M. Keane, Interval exchange transformations, Mathematische Zeitschrift, 141 (1975), 25-31. doi: 10.1007/BF01236981.

[9]

S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Annals of Mathematics, 124 (1986), 293-311. doi: 10.2307/1971280.

[10]

A. I. Khinchin, Mathematical Foundations of Statistical Mechanics, Dover, 1949.

[11]

H. Masur, Interval exchange transformations and measured foliations, Annals of Mathematics, 115 (1982), 169-200. doi: 10.2307/1971341.

[12]

A. van der Poorten, Fermat's four squares theorem,, 2007. Available from: , (). 

[13]

D. Ruelle, Statistical Mechanics: Rigorous Results, Benjamin, 1969.

[14]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Annals of Mathematics, 115 (1982), 201-242. doi: 10.2307/1971391.

show all references

References:
[1]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1989. doi: 10.1007/978-1-4757-2063-1.

[2]

R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press, 1982.

[3]

L. Boltzmann, Vorlesungen Über Gastheorie, Barth, 1896.

[4]

M. Boshernitzan, A condition for minimal interval exchange maps to be uniquely ergodic, Duke Mathematical Journal, 52 (1985), 723-752. doi: 10.1215/S0012-7094-85-05238-X.

[5]

G. R. Brannock and J. K. Percus, Wertheim cluster development of free energy functionals for general nearest-neighbor interactions in $D=1$, The Journal of Chemical Physics, 105 (1996), 614-627. doi: 10.1063/1.471920.

[6]

J. A. Cuesta and C. Tutschka, Overcomplete free energy functional for $D=1$ particle systems with next neighbor interactions, Journal of Statistical Physics, 111 (2003), 1125-1148. doi: 10.1023/A:1023096031180.

[7]

K. F. Herzfeld and M. Goeppert-Mayer, On the states of aggregation, The Journal of Chemical Physics, 2 (1934), 38-45. doi: 10.1063/1.1749355.

[8]

M. Keane, Interval exchange transformations, Mathematische Zeitschrift, 141 (1975), 25-31. doi: 10.1007/BF01236981.

[9]

S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Annals of Mathematics, 124 (1986), 293-311. doi: 10.2307/1971280.

[10]

A. I. Khinchin, Mathematical Foundations of Statistical Mechanics, Dover, 1949.

[11]

H. Masur, Interval exchange transformations and measured foliations, Annals of Mathematics, 115 (1982), 169-200. doi: 10.2307/1971341.

[12]

A. van der Poorten, Fermat's four squares theorem,, 2007. Available from: , (). 

[13]

D. Ruelle, Statistical Mechanics: Rigorous Results, Benjamin, 1969.

[14]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Annals of Mathematics, 115 (1982), 201-242. doi: 10.2307/1971391.

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