Abstract
A quantum spin system is discussed where a heat flow between infinite reservoirs takes place in a finite region. A time-dependent force may also be acting. Our analysis is based on a simple technical assumption concerning the time evolution of infinite quantum spin systems. This assumption, physically natural but currently proved for few specific systems only, says that quantum information diffuses in space-time in such a way that the time integral of the commutator of local observables converges: ∫0 −∞ dt ‖[B, α t A]‖<∞. In this setup one can define a natural nonequilibrium state. In the time-independent case, this nonequilibrium state retains some of the analyticity which characterizes KMS equilibrium states. A linear response formula is also obtained which remains true far from equilibrium. The formalism presented here does not cover situations where (for time-independent forces) the time-translation invariance and uniqueness of the natural nonequilibrium state are broken.
Similar content being viewed by others
REFERENCES
O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, II (Springer, New York, 1979–1981) [There is a 2nd ed. (1997) of Vol. II].
C. P. Dettmann and G. P. Morriss, Proof of Lyapunov exponent pairing for systems at constant kinetic energy, Phys. Rev. E 53:R5541–5544 (1996).
J. R. Dorfman, An introduction to chaos in non-equilibrium statistical mechanics (Springer, Berlin), to appear.
J.-P. Eckmann, C.-A. Pillet, and L. Rey-Bellet, Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Commun. Math. Phys. 201:657–697 (1999).
J.-P. Eckmann, C.-A. Pillet, and L. Rey-Bellet, Entropy production in non-linear, thermally driven Hamiltonian systems, J. Statist. Phys. 95:305–331 (1999).
D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Fluids (Academic Press, New York, 1990).
F. Fidaleo and C. Liverani, Ergodic properties for a quantum non linear dynamics, preprint.
G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics, Phys. Rev. Letters 74:2694–2697 (1995).
G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in stationary states, J. Statist. Phys. 80:931–970 (1995).
K. Hepp and E. H. Lieb, Phase transitions in reservoir-driven open systems with applications to lasers and superconductors, Helvetica Physica Acta 46:573–603 (1973).
W. G. Hoover, Molecular Dynamics, Lecture Notes in Physics, Vol. 258 (Springer, Heidelberg, 1986).
V. Jakš ić and C.-A. Pillet, Spectral theory of thermal relaxation, J. Math. Phys. 38:1757–1780 (1997).
D. Ruelle, General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium, Phys. Lett. A 245:220–224 (1998).
D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, J. Statist. Phys. 95:393–468 (1999).
H. Spohn and J. L. Lebowitz, Stationary non-equilibrium states of infinite harmonic systems, Commun. Math. Phys. 54:97–120 (1977).
M. P. Wojtkowski and C. Liverani, Conformally symplectic dynamics and symmetry of the Lyapunov spectrum, Commun. Math. Phys. 194:47–60 (1998).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ruelle, D. Natural Nonequilibrium States in Quantum Statistical Mechanics. Journal of Statistical Physics 98, 57–75 (2000). https://doi.org/10.1023/A:1018618704438
Issue Date:
DOI: https://doi.org/10.1023/A:1018618704438