Abstract
A system of noninteracting classical spins coupled to the modes of a chain of one-dimensional classical harmonic oscillators via −S z Σc k x k was investigated to see whether the spin-bath coupling could relax the spins toward equilibrium. We considered two different cases for the initial conditions on the classical harmonic oscillator variables when we took the harmonic oscillator bath variable averages for the total spin components. In the first case, the harmonic oscillators were initially in equilibrium while they did not recognize the presence of the spin, whereas in the second case, the spins were in equilibrium and did recognize the presence of the spin. For the first case, the bath variable averages of the x- and the y-components of the total spin showed that the effective angular velocity transiently slowed down after an initial increase; then, it recovered its initial angular velocity continually. In the second case, the effective angular velocity was fixed. For both cases, the z-component of the total spin vector remained constant. If the x- and the y-components of the single spins were randomly distributed, we would get equilibrium values. For the z-components of single spins, unless they are at equilibrium from the beginning, they do not attain equilibrium.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. J. Rubin, J. Math. Phys. 1, 309 (1960); ibid 2, 373 (1961).
P. Mazur and E. Montroll, J. Math. Phys. 1, 70 (1960).
E. Teramoto, Prog. Theor. Phys. 28, 1059 (1962).
M. A. Huerta and H. S. Robertson, J. Stat. Phys. 1, 393 (1969); ibid 3, 171 (1971); J. Math. Phys. 12, 2305 (1971).
G. W. Ford, M. Kac and P. Mazur, J. Math. Phys. 6, 504 (1965).
M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945).
A. O. Caldeira and A. J. Legget, Phys. Rev. Lett. 46, 211 (1981).
Vincent Hakim and V. Ambegaokar, Phys. Rev. A 32, 423 (1985).
R. D. Carlitz and R. Chakrabarti, Phys. Rev. A 35, 3156 (1987).
G. W. Ford, J. T. Lewis and R. F. O’Connell, Phys. Rev. A 37, 4419 (1988).
P. Nielaba, J. L. Lebowitz, H. Spohn and J. L. Vallés, J. Stat. Phys. 55, 745 (1989).
Hänggi, G.-L. Ingold and P. Talkner, New J. Phys. 10, 115008 (2008).
J. R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge University Presss, Cambridge, 1999).
R. Klages, Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics (World Scientific, Singapore, 2007).
M. E. Tuckerman, Statistical Mechanics: Theory and Molecular Simulation (Oxford University Presss, Oxford, 2010).
S. K. Oh, New Phys: Sae Mulli 57, 16 (2008).
K. -H. Yang and J. O. Hirschfelder, Phy. Rev. A 22, 1814 (1980).
N. -N. Huang, H. Cao, T. Yan and F.-R. Xu, J. Nonlinear Math. Phys. 13, 302 (2006).
J. -P. Amiet and M. B. Cibils, J. Phys. A: Math. Gen. 24, 1515 (1991).
K. F. Riley, M. P. Hobson and S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University Press, Cambridge, 2006).
N. W. Ashcroft and N. D. Mermin, Solid State Physics (Sauders College, Philadelphia, 1976).
D. J. Amit and Y. Verbin, Statistical Physics: An Introductory Course (World Scientific, Singapore, 1999).
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (Eds.), NIST Handbook of Mathematical Funcions (NIST and Cambridge University Press, Cambridge, 2010).
A. Lasota and M. C. Mackey, Probabilistic Properties of Deterministic Systems (Cambridge University Press, Cambridge, 1985); M. C. Mackey, Time’s Arrow: The Origins of Thermodynamic Behavior (Dover, Mineola, 2003).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Oh, S.K. Noninteracting classical spins coupled to a heat bath of one-dimensional classical harmonic oscillators: Exact bath variable average. Journal of the Korean Physical Society 63, 1892–1900 (2013). https://doi.org/10.3938/jkps.63.1892
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3938/jkps.63.1892