1,419 research outputs found
H-Theorems from Autonomous Equations
The H-theorem is an extension of the Second Law to a time-sequence of states
that need not be equilibrium ones. In this paper we review and we rigorously
establish the connection with macroscopic autonomy.
If for a Hamiltonian dynamics for many particles, at all times the present
macrostate determines the future macrostate, then its entropy is non-decreasing
as a consequence of Liouville's theorem. That observation, made since long, is
here rigorously analyzed with special care to reconcile the application of
Liouville's theorem (for a finite number of particles) with the condition of
autonomous macroscopic evolution (sharp only in the limit of infinite scale
separation); and to evaluate the presumed necessity of a Markov property for
the macroscopic evolution.Comment: 13 pages; v1 -> v2: Sec. 1-2 considerably rewritten, minor
corrections in Sec. 3-
An extension of the Kac ring model
We introduce a unitary dynamics for quantum spins which is an extension of a
model introduced by Mark Kac to clarify the phenomenon of relaxation to
equilibrium. When the number of spins gets very large, the magnetization
satisfies an autonomous equation as function of time with exponentially fast
relaxation to the equilibrium magnetization as determined by the microcanonical
ensemble. This is proven as a law of large numbers with respect to a class of
initial data. The corresponding Gibbs-von Neumann entropy is also computed and
its monotonicity in time discussed.Comment: 15 pages, v2 -> v3: minor typographic correctio
Quantum Macrostates, Equivalence of Ensembles and an H-Theorem
Before the thermodynamic limit, macroscopic averages need not commute for a
quantum system. As a consequence, aspects of macroscopic fluctuations or of
constrained equilibrium require a careful analysis, when dealing with several
observables. We propose an implementation of ideas that go back to John von
Neumann's writing about the macroscopic measurement. We apply our scheme to the
relation between macroscopic autonomy and an H-theorem, and to the problem of
equivalence of ensembles. In particular, we show how the latter is related to
the asymptotic equipartition theorem. The main point of departure is an
expression of a law of large numbers for a sequence of states that start to
concentrate, as the size of the system gets larger, on the macroscopic values
for the different macroscopic observables. Deviations from that law are
governed by the entropy.Comment: 16 pages; v1 -> v2: Sec. 3 slightly rewritten, 2 references adde
Approach to ground state and time-independent photon bound for massless spin-boson models
It is widely believed that an atom interacting with the electromagnetic field
(with total initial energy well-below the ionization threshold) relaxes to its
ground state while its excess energy is emitted as radiation. Hence, for large
times, the state of the atom+field system should consist of the atom in its
ground state, and a few free photons that travel off to spatial infinity.
Mathematically, this picture is captured by the notion of asymptotic
completeness. Despite some recent progress on the spectral theory of such
systems, a proof of relaxation to the ground state and asymptotic completeness
was/is still missing, except in some special cases (massive photons, small
perturbations of harmonic potentials). In this paper, we partially fill this
gap by proving relaxation to an invariant state in the case where the atom is
modelled by a finite-level system. If the coupling to the field is sufficiently
infrared-regular so that the coupled system admits a ground state, then this
invariant state necessarily corresponds to the ground state. Assuming slightly
more infrared regularity, we show that the number of emitted photons remains
bounded in time. We hope that these results bring a proof of asymptotic
completeness within reach.Comment: 45 pages, published in Annales Henri Poincare. This archived version
differs from the journal version because we corrected an inconsequential
mistake in Section 3.5.1: to do this, a new paragraph was added after Lemma
3.
Non-equilibrium work relations
This is a brief review of recently derived relations describing the behaviour
of systems far from equilibrium. They include the Fluctuation Theorem,
Jarzynski's and Crooks' equalities, and an extended form of the Second
Principle for general steady states. They are very general and their proofs
are, in most cases, disconcertingly simple.Comment: Brief Summer School Lecture Note
'Return to equilibrium' for weakly coupled quantum systems: a simple polymer expansion
Recently, several authors studied small quantum systems weakly coupled to
free boson or fermion fields at positive temperature. All the approaches we are
aware of employ complex deformations of Liouvillians or Mourre theory (the
infinitesimal version of the former). We present an approach based on polymer
expansions of statistical mechanics. Despite the fact that our approach is
elementary, our results are slightly sharper than those contained in the
literature up to now. We show that, whenever the small quantum system is known
to admit a Markov approximation (Pauli master equation \emph{aka} Lindblad
equation) in the weak coupling limit, and the Markov approximation is
exponentially mixing, then the weakly coupled system approaches a unique
invariant state that is perturbatively close to its Markov approximation.Comment: 23 pages, v2-->v3: Revised version: The explanatory section 1.7 has
changed and Section 3.2 has been made more explici
Fluctuation theorems for stochastic dynamics
Fluctuation theorems make use of time reversal to make predictions about
entropy production in many-body systems far from thermal equilibrium. Here we
review the wide variety of distinct, but interconnected, relations that have
been derived and investigated theoretically and experimentally. Significantly,
we demonstrate, in the context of Markovian stochastic dynamics, how these
different fluctuation theorems arise from a simple fundamental time-reversal
symmetry of a certain class of observables. Appealing to the notion of Gibbs
entropy allows for a microscopic definition of entropy production in terms of
these observables. We work with the master equation approach, which leads to a
mathematically straightforward proof and provides direct insight into the
probabilistic meaning of the quantities involved. Finally, we point to some
experiments that elucidate the practical significance of fluctuation relations.Comment: 48 pages, 2 figures. v2: minor changes for consistency with published
versio
Fluctuations in Nonequilibrium Statistical Mechanics: Models, Mathematical Theory, Physical Mechanisms
The fluctuations in nonequilibrium systems are under intense theoretical and
experimental investigation. Topical ``fluctuation relations'' describe
symmetries of the statistical properties of certain observables, in a variety
of models and phenomena. They have been derived in deterministic and, later, in
stochastic frameworks. Other results first obtained for stochastic processes,
and later considered in deterministic dynamics, describe the temporal evolution
of fluctuations. The field has grown beyond expectation: research works and
different perspectives are proposed at an ever faster pace. Indeed,
understanding fluctuations is important for the emerging theory of
nonequilibrium phenomena, as well as for applications, such as those of
nanotechnological and biophysical interest. However, the links among the
different approaches and the limitations of these approaches are not fully
understood. We focus on these issues, providing: a) analysis of the theoretical
models; b) discussion of the rigorous mathematical results; c) identification
of the physical mechanisms underlying the validity of the theoretical
predictions, for a wide range of phenomena.Comment: 44 pages, 2 figures. To appear in Nonlinearity (2007
Relating the thermodynamic arrow of time to the causal arrow
Consider a Hamiltonian system that consists of a slow subsystem S and a fast
subsystem F. The autonomous dynamics of S is driven by an effective
Hamiltonian, but its thermodynamics is unexpected. We show that a well-defined
thermodynamic arrow of time (second law) emerges for S whenever there is a
well-defined causal arrow from S to F and the back-action is negligible. This
is because the back-action of F on S is described by a non-globally Hamiltonian
Born-Oppenheimer term that violates the Liouville theorem, and makes the second
law inapplicable to S. If S and F are mixing, under the causal arrow condition
they are described by microcanonic distributions P(S) and P(S|F). Their
structure supports a causal inference principle proposed recently in machine
learning.Comment: 10 page
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