97 research outputs found
Origin of the Canonical Ensemble: Thermalization with Decoherence
We solve the time-dependent Schrodinger equation for the combination of a
spin system interacting with a spin bath environment. In particular, we focus
on the time development of the reduced density matrix of the spin system. Under
normal circumstances we show that the environment drives the reduced density
matrix to a fully decoherent state, and furthermore the diagonal elements of
the reduced density matrix approach those expected for the system in the
canonical ensemble. We show one exception to the normal case is if the spin
system cannot exchange energy with the spin bath. Our demonstration does not
rely on time-averaging of observables nor does it assume that the coupling
between system and bath is weak. Our findings show that the canonical ensemble
is a state that may result from pure quantum dynamics, suggesting that quantum
mechanics may be regarded as the foundation of quantum statistical mechanics.Comment: 12 pages, 4 figures, accepted for publication by J. Phys. Soc. Jp
The foundations of statistical mechanics from entanglement: Individual states vs. averages
We consider an alternative approach to the foundations of statistical
mechanics, in which subjective randomness, ensemble-averaging or time-averaging
are not required. Instead, the universe (i.e. the system together with a
sufficiently large environment) is in a quantum pure state subject to a global
constraint, and thermalisation results from entanglement between system and
environment. We formulate and prove a "General Canonical Principle", which
states that the system will be thermalised for almost all pure states of the
universe, and provide rigorous quantitative bounds using Levy's Lemma.Comment: 12 pages, v3 title changed, v2 minor change
Long-Time Behavior of Macroscopic Quantum Systems: Commentary Accompanying the English Translation of John von Neumann's 1929 Article on the Quantum Ergodic Theorem
The renewed interest in the foundations of quantum statistical mechanics in
recent years has led us to study John von Neumann's 1929 article on the quantum
ergodic theorem. We have found this almost forgotten article, which until now
has been available only in German, to be a treasure chest, and to be much
misunderstood. In it, von Neumann studied the long-time behavior of macroscopic
quantum systems. While one of the two theorems announced in his title, the one
he calls the "quantum H-theorem", is actually a much weaker statement than
Boltzmann's classical H-theorem, the other theorem, which he calls the "quantum
ergodic theorem", is a beautiful and very non-trivial result. It expresses a
fact we call "normal typicality" and can be summarized as follows: For a
"typical" finite family of commuting macroscopic observables, every initial
wave function from a micro-canonical energy shell so evolves that for
most times in the long run, the joint probability distribution of these
observables obtained from is close to their micro-canonical
distribution.Comment: 34 pages LaTeX, no figures; v2: minor improvements and additions. The
English translation of von Neumann's article is available as arXiv:1003.213
Geometric dynamical observables in rare gas crystals
We present a detailed description of how a differential geometric approach to
Hamiltonian dynamics can be used for determining the existence of a crossover
between different dynamical regimes in a realistic system, a model of a rare
gas solid. Such a geometric approach allows to locate the energy threshold
between weakly and strongly chaotic regimes, and to estimate the largest
Lyapunov exponent. We show how standard mehods of classical statistical
mechanics, i.e. Monte Carlo simulations, can be used for our computational
purposes. Finally we consider a Lennard Jones crystal modeling solid Xenon. The
value of the energy threshold turns out to be in excellent agreement with the
numerical estimate based on the crossover between slow and fast relaxation to
equilibrium obtained in a previous work by molecular dynamics simulations.Comment: RevTeX, 19 pages, 6 PostScript figures, submitted to Phys. Rev.
Remarks on Shannon's Statistical Inference and the Second Law in Quantum Statistical Mechanics
We comment on a formulation of quantum statistical mechanics, which
incorporates the statistical inference of Shannon.
Our basic idea is to distinguish the dynamical entropy of von Neumann, , in terms of the density matrix ,
and the statistical amount of uncertainty of Shannon, , with in the representation where the total
energy and particle numbers are diagonal. These quantities satisfy the
inequality . We propose to interprete Shannon's statistical inference
as specifying the {\em initial conditions} of the system in terms of . A
definition of macroscopic observables which are characterized by intrinsic time
scales is given, and a quantum mechanical condition on the system, which
ensures equilibrium, is discussed on the basis of time averaging.
An interesting analogy of the change of entroy with the running coupling in
renormalization group is noted. A salient feature of our approach is that the
distinction between statistical aspects and dynamical aspects of quantum
statistical mechanics is very transparent.Comment: 16 pages. Minor refinement in the statements in the previous version.
This version has been published in Journal of Phys. Soc. Jpn. 71 (2002) 6
Wishing for deburdening through a sustainable control after bariatric surgery
The aim of this study was an in-depth investigation of the change process experienced by patients undergoing bariatric surgery. A prospective interview study was performed prior to as well as 1 and 2 years after surgery. Data analyses of the transcribed interviews were performed by means of the Grounded Theory method. A core category was identified: Wishing for deburdening through a sustainable control over eating and weight, comprising three related categories: hoping for deburdening and control through surgery, feeling deburdened and practising control through physical restriction, and feeling deburdened and trying to maintain control by own willpower. Before surgery, the participants experienced little or no control in relation to food and eating and hoped that the bariatric procedure would be the first brick in the building of a foundation that would lead to control in this area. The control thus achieved in turn affected the participants' relationship to themselves, their roles in society, and the family as well as to health care. One year after surgery they reported established routines regarding eating as well as higher self-esteem due to weight loss. In family and society they set limits and in relation to health care staff they felt their concern and reported satisfaction with the surgery. After 2 years, fear of weight gain resurfaced and their self-image was modified to be more realistic. They were no longer totally self-confident about their condition, but realised that maintaining control was a matter of struggle to obtaining a foundation of sustainable control. Between 1 and 2 years after surgery, the physical control mechanism over eating habits started to more or less fade for all participants. An implication is that when this occurs, health care professionals need to provide interventions that help to maintain the weight loss in order to achieve a good long-term outcome
Long-Time Tails and Anomalous Slowing Down in the Relaxation of Spatially Inhomogeneous Excitations in Quantum Spin Chains
Exact analytic calculations in spin-1/2 XY chains, show the presence of
long-time tails in the asymptotic dynamics of spatially inhomogeneous
excitations. The decay of inhomogeneities, for , is given in the
form of a power law where the relaxation time
and the exponent depend on the wave vector ,
characterizing the spatial modulation of the initial excitation. We consider
several variants of the XY model (dimerized, with staggered magnetic field,
with bond alternation, and with isotropic and uniform interactions), that are
grouped into two families, whether the energy spectrum has a gap or not. Once
the initial condition is given, the non-equilibrium problem for the
magnetization is solved in closed form, without any other assumption. The
long-time behavior for can be obtained systematically in a form
of an asymptotic series through the stationary phase method. We found that
gapped models show critical behavior with respect to , in the sense that
there exist critical values , where the relaxation time
diverges and the exponent changes discontinuously. At those points, a
slowing down of the relaxation process is induced, similarly to phenomena
occurring near phase transitions. Long-lived excitations are identified as
incommensurate spin density waves that emerge in systems undergoing the Peierls
transition. In contrast, gapless models do not present the above anomalies as a
function of the wave vector .Comment: 25 pages, 2 postscript figures. Manuscript submitted to Physical
Review
Towards the fast scrambling conjecture
Many proposed quantum mechanical models of black holes include highly
nonlocal interactions. The time required for thermalization to occur in such
models should reflect the relaxation times associated with classical black
holes in general relativity. Moreover, the time required for a particularly
strong form of thermalization to occur, sometimes known as scrambling,
determines the time scale on which black holes should start to release
information. It has been conjectured that black holes scramble in a time
logarithmic in their entropy, and that no system in nature can scramble faster.
In this article, we address the conjecture from two directions. First, we
exhibit two examples of systems that do indeed scramble in logarithmic time:
Brownian quantum circuits and the antiferromagnetic Ising model on a sparse
random graph. Unfortunately, both fail to be truly ideal fast scramblers for
reasons we discuss. Second, we use Lieb-Robinson techniques to prove a
logarithmic lower bound on the scrambling time of systems with finite norm
terms in their Hamiltonian. The bound holds in spite of any nonlocal structure
in the Hamiltonian, which might permit every degree of freedom to interact
directly with every other one.Comment: 34 pages. v2: typo correcte
Role of chaos for the validity of statistical mechanics laws: diffusion and conduction
Several years after the pioneering work by Fermi Pasta and Ulam, fundamental
questions about the link between dynamical and statistical properties remain
still open in modern statistical mechanics. Particularly controversial is the
role of deterministic chaos for the validity and consistency of statistical
approaches. This contribution reexamines such a debated issue taking
inspiration from the problem of diffusion and heat conduction in deterministic
systems. Is microscopic chaos a necessary ingredient to observe such
macroscopic phenomena?Comment: Latex, 27 pages, 10 eps-figures. Proceedings of the Conference "FPU
50 years since" Rome 7-8 May 200
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