Abstract
In this paper we analyze the stability of a gyroscopic oscillator interacting with a finite- and infinite-dimensional heat bath in both the classical and quantum cases. We consider a finite gyroscopic oscillator model of a particle on a rotating disc and a particle in a magnetic field and we examine stability before and after coupling to a heat bath. The heat bath is modelled in the finite-dimensional setting by a system of independent oscillators with mass. It is shown that if the oscillator is gyroscopically stable, coupling to a sufficiently massive heat bath induces instability even in the finite-dimensional setting. The key mechanism for instability in this paper is thus not induced by damping. The meaning of these ideas in the quantum context is discussed. The model extends the exact diagonalization analysis of an oscillator and field of Ford, Lewis, and O'Connell to the gyroscopic setting. We also discuss the interesting role that damping of Landau type plays in the infinite limit.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
V. Bach, J. Fröhlich, and I. Sigal, Return to equilibrium, J. Math. Phys. 41:3985–4060 (2000).
J. Baillieul and M. Levi, Constrained relative motions in rotational mechanics, Arch. Rational Mech. Anal. 115:101–135 (1991).
A. M. Bloch, Dissipation dynamics in classical and quantum conservative systems, in Mathematical Systems Theory in Biology, Communications, Computation, and Finance, J. Rosenthal and D. Gilliam, eds., (IMA, Springer Verlag, 2003), 121–156.
A. M. Bloch, P. J. Hagerty, A. G. Rojo, and M. Weinstein, Classical and quantum gyroscopic instability and heat baths, in Proc. 40th IEEE Conference on Decision and Control (2001).
A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, and T. S. Ratiu, Dissipation induced instabilities, Ann. Inst. H. Poincarè Anal. Non Linèaire 11:37–90 (1994).
L. S. Brown and G. Gabrielse, Geonium theory: Physics of a single electron or ion in a Penning trap, Rev. Modern Phys. 58:233–310 (1986).
A. Caldeira and A. Leggett, Path integral approach to quantum Brownian motion, Physica A 121:587–616 (1983).
N. G. Chetaev, The Stability of Motion (Pergamon Press, New York, 1961).
J. Derezinski and V. Jaksic, Return to equilibrium for Pauli-Fierz systems, to appear in Ann. H. Poincarè (2002).
J. Derezinski, V. Jaksic, and C.-A. Pillet, Perturbation theory of W*-dynamics, Liouvilleans, and KMS-states, Rev. Math. Phys. 15:447–489 (2003).
G. W. Ford, J. T. Lewis, and R. F. O'Connell, Independent oscillator model of a heat bath: Exact diagonalization of the Hamiltonian, J. Stat. Phys. 53:439–455 (1988).
P. J. Hagerty, A. M. Bloch, and M. I. Weinstein, Radiation induced instability in interconnected systems, in Proc. 38th IEEE Conference on Decision and Control (1999).
P. J. Hagerty, A. M. Bloch, and M. I. Weinstein, Radiation induced instability, to appear in SIAM J. Appl. Math. (2003).
W. Hahn, Stability of Motion (Springer-Verlag, New York, 1967).
V. Jaksic and C.-A. Pillet, On a model for quantum friction I: Fermi's golden rule and dynamics at zero temperature, Ann. Inst. H. Poincarè Phys. Thèor. 62:47–68 (1995).
V. Jaksic and C.-A. Pillet, Ergodic properties of the non-Markovian Langevin equation, Lett. Math. Phys. 41:1757–1780 (1997).
N. Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics (North-Holland, Amsterdam, 1967).
E. Kirr and M. I. Weinstein, Parametrically excited Hamiltonian partial differential equations, Siam J. Math. Anal. 33:16–52 (2001).
A. Komech, M. Kunze, and H. Spohn, Effective dynamics for a mechanical particle coupled to a wave field, Comm. Math. Phys. 203:1–19 (1999).
H. Lamb, On the peculiarity of the wave-system due to the free vibrations of a nucleus in an extended medium, Proc. London Math. Soc. 32:208–211 (1900).
L. Landau, On the vibrations of the electronic plasma, J. Phys. (USSR) 10:25(1946).
J. Lebowitz, Microscopic reversibility and macrosopic behavior: Physical explanations and mathematical derivations, in Lecture Notes in Physics, J. J. Brey et al., eds. (Springer-Verlag, 1994).
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry (Springer-Verlag, New York, 1999).
M. Merkli and I. Sigal, On time-dependent theory of quantum resonances, Comm. Math. Phys. 201:549–576 (1999).
A. Messiah, Quantenmechanik, Band 2 (de Gruyter, Berlin, 1990).
A. Messiah, Quantenmechanik, Band 1 (de Gruyter, Berlin, 1991).
J. C. Simo, D. Lewis, and J. E. Marsden, Stability of relative equilibria. I. The reduced energy-momentum method, Arch. Rational Mech. Anal. 115:15–59 (1991).
P. Smereka, Synchronization and relaxation for a class of globally coupled oscillator systems, Physica D 124:104–125 (1998).
A. Soffer and M. I. Weinstein, Nonautonomous Hamiltonians, J. Stat. Phys. 93:359–391 (1998).
A. Soffer and M. I. Weinstein, Time dependent resonance theory, Geom. Funct. Anal. 8:1086–1128 (1998).
A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136:9–74 (1999).
L. Thomson and P. G. Tait, Principles of Mechanics and Dynamics (Cambridge University Press, 1897).
W. Unruh and W. Zurek, Reduction of a wave packet in quantum Brownian motion, Phys. Rev. 400:1071–1094 (1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bloch, A.M., Hagerty, P., Rojo, A.G. et al. Gyroscopically Stabilized Oscillators and Heat Baths. Journal of Statistical Physics 115, 1073–1100 (2004). https://doi.org/10.1023/B:JOSS.0000022367.36305.d3
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000022367.36305.d3