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1.
We study a small quantum system (e.g., a simplified model for an atom or molecule) interacting with two bosonic or fermionic
reservoirs (say, photon or phonon fields). We show that the combined system has a family of stationary states parametrized
by two numbers, T
1 and T
2 (‘reservoir temperatures’). If T
1 ≠ T
2, then these states are non-equilibrium stationary states (NESS). In the latter case we show that they have nonvanishing heat
fluxes and positive entropy production and are dynamically asymptotically stable. The latter means that the evolution with
an initial condition, normal with respect to any state where the reservoirs are in equilibria at temperatures T
1 and T
2, converges to the corresponding NESS. Our results are valid for the temperatures satisfying the bound min (T
1,T
2) > g
2 + α, where g is the coupling constant and 0 < α < 1 is a power related to the infra-red behaviour of the coupling functions.
Submitted: March 20, 2006. Revised: March 19, 2007. Accepted: May 11, 2007.
Marco Merkli: Partly supported by an NSERC PDF, the Institute of Theoretical Physics of ETH Zürich, Switzerland, the Departments
of Mathematics of McGill University and the University of Toronto, Canada.
Matthias Mück: Supported by DAAD under grant HSP III.
Israel Michael Sigal: Supported by NSERC under grant NA7901. 相似文献
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Marco Merkli 《Journal of Mathematical Analysis and Applications》2007,327(1):376-399
Level shift operators describe the second-order displacement of eigenvalues under perturbation. They play a central role in resonance theory and ergodic theory of open quantum systems at positive temperatures. We exhibit intrinsic properties of level shift operators, properties which stem from the structure of open quantum systems at positive temperatures and which are common to all such systems. They determine the geometry of resonances bifurcating from eigenvalues of positive temperature Hamiltonians and they relate the Gibbs state, the kernel of level shift operators, and zero energy resonances. We show that degeneracy of energy levels of the small part of the open quantum system causes the Fermi Golden Rule Condition to be violated and we analyze ergodic properties of such systems. 相似文献
5.
The general theory of simple transport processes between quantum mechanical
reservoirs is reviewed and extended. We focus on thermoelectric phenomena,
involving exchange of energy and particles. The theory is illustrated on the
example of two reservoirs of free fermions coupled through a local interaction. We
construct a stationary state and determine energy and particle currents with the
help of a convergent perturbation series.
We explicitly calculate several interesting quantities to lowest order, such as
the entropy production rate, the resistance, and the heat conductivity. Convergence
of the perturbation series allows us to prove that they are strictly positive
under suitable smallness and regularity assumptions on the interaction between the
reservoirs.
Communicated by Gian Michele Graf
submitted 15/01/03, accepted: 25/02/03 相似文献
6.
We consider random analytic functions defined on the unit disk of the complex plane f(z) = ?n=0¥ an Xn znf(z) = \sum_{n=0}^{\infty} a_{n} X_{n} z^{n}, where the X
n
’s are i.i.d., complex-valued random variables with mean zero and unit variance. The coefficients a
n
are chosen so that f(z) is defined on a domain of ℂ carrying a planar or hyperbolic geometry, and Ef(z)[`(f(w))]\mathbf{E}f(z)\overline{f(w)} is covariant with respect to the isometry group. The corresponding Gaussian analytic functions have been much studied, and
their zero sets have been considered in detail in a monograph by Hough, Krishnapur, Peres, and Virág. We show that for non-Gaussian
coefficients, the zero set converges in distribution to that of the Gaussian analytic functions as one transports isometrically
to the boundary of the domain. The proof is elementary and general. 相似文献
7.
In the context of an idealized model describing an atom coupled to black-body radiation at a sufficiently high positive temperature,
we show that the atom will end up being ionized in the limit of large times. Mathematically, this is translated into the statement
that the coupled system does not have any time-translation invariant state of positive (asymptotic) temperature, and that
the expectation value of an arbitrary finite-dimensional projection in an arbitrary initial state of positive (asymptotic)
temperature tends to zero, as time tends to infinity. These results are formulated within the general framework of W
*-dynamical systems, and the proofs are based on Mourre's theory of positive commutators and a new virial theorem. Results
on the so-called standard form of a von Neumann algebra play an important role in our analysis. 相似文献
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We consider quantum systems consisting of a “small” system coupled to two reservoirs (called open systems). We show that such systems have no equilibrium states normal with respect to any state of the decoupled system in which the reservoirs are at different temperatures, provided that either the temperatures or the temperature difference divided by the product of the temperatures are not too small. Our proof involves an elaborate spectral analysis of a general class of generators of the dynamics of open quantum systems, including quantum Liouville operators (“positive temperature Hamiltonians”) which generate the dynamics of the systems under consideration. 相似文献
10.
Marco Merkli 《Communications in Mathematical Physics》2001,223(2):327-362
The method of positive commutators, developed for zero temperature problems over the last twenty years, has been an essential
tool in the spectral analysis of Hamiltonians in quantum mechanics. We extend this method to positive temperatures, i.e. to
non-equilibrium quantum statistical mechanics. We use the positive commutator technique to give an alternative proof of a
fundamental property of a certain class of large quantum systems, called Return to Equilibrium. This property says that equilibrium states are (asymptotically) stable: if a system is slightly perturbed from its equilibrium
state, then it converges back to that equilibrium state as time goes to infinity.
Received: 27 December 2000 / Accepted: 21 June 2001 相似文献