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Photon number variance in isolated cavities
We consider a strictly isolated single-mode optical cavity resonating at
angular frequency omega containing atoms whose one-electron level energies are
supposed to be: hbar*omega, 2*hbar*omega,...B*hbar\omega, and m photons. If
initially the atoms are in their highest energy state and m=0, we find that at
equilibrium: variance(m)/mean(m)=(B+1)/6, indicating that the internal field
statistics is sub-Poissonian if the number of atomic levels B does not exceed
4. Remarkably, this result does not depend on the number of atoms, nor on the
number of electrons that each atom incorporates. Our result has application to
the statistics of the light emitted by pulsed lasers and nuclear magnetic
resonance. On the mathematical side, the result is based on the restricted
partitions of integers.Comment: 4 pages, to be submitted to Journal of Physics
Statistics of non-interacting bosons and fermions in micro-canonical, canonical and grand-canonical ensembles: A survey
The statistical properties of non-interacting bosons and fermions confined in
trapping potentials are most easily obtained when the system may exchange
energy and particles with a large reservoir (grand-canonical ensemble). There
are circumstances, however, where the system under consideration may be
considered as being isolated (micro-canonical ensemble). This paper first
reviews results relating to micro-canonical ensembles. Some of them were
obtained a long time ago, particularly by Khinchin in 1950. Others were
obtained only recently, often motivated by experimental results relating to
atomic confinement. A number of formulas are reported for the first time in the
present paper. Formulas applicable to the case where the system may exchange
energy but not particles with a reservoir (canonical ensemble) are derived from
the micro-canonical ensemble expressions. The differences between the three
ensembles tend to vanish in the so-called Thermodynamics limit, that is, when
the number of particles and the volume go to infinity while the particle number
density remains constant. But we are mostly interested in systems of moderate
size, often referred to as being mesoscopic, where the grand-canonical
formalism is not applicable. The mathematical results rest primarily on the
enumeration of partitions of numbers.Comment: 18 pages, submitted to J. Phys.
A simple quantum heat engine
Quantum heat engines employ as working agents multi-level systems instead of
gas-filled cylinders. We consider particularly two-level agents such as
electrons immersed in a magnetic field. Work is produced in that case when the
electrons are being carried from a high-magnetic-field region into a
low-magnetic-field region. In watermills, work is produced instead when some
amount of fluid drops from a high-altitude reservoir to a low-altitude
reservoir. We show that this purely mechanical engine may in fact be considered
as a two-level quantum heat engine, provided the fluid is viewed as consisting
of n molecules of weight one and N-n molecules of weight zero. Weight-one
molecules are analogous to electrons in their higher energy state, while
weight-zero molecules are analogous to electrons in their lower energy state.
More generally, fluids consist of non-interacting molecules of various weights.
It is shown that, not only the average value of the work produced per cycle,
but also its fluctuations, are the same for mechanical engines and quantum
(Otto) heat engines. The reversible Carnot cycles are approached through the
consideration of multiple sub-reservoirs.Comment: RevTeX 9 pages, 4 figures, paper shortened, improved presentatio
On Classical Ideal Gases
The ideal gas laws are derived from the democritian concept of corpuscles
moving in vacuum plus a principle of simplicity, namely that these laws are
independent of the laws of motion aside from the law of energy conservation. A
single corpuscle in contact with a heat bath and submitted to a and
-invariant force is considered, in which case corpuscle
distinguishability is irrelevant. The non-relativistic approximation is made
only in examples. Some of the end results are known but the method appears to
be novel. The mathematics being elementary the present paper should facilitate
the understanding of the ideal-gas law and more generally of classical
thermodynamics. It supplements importantly a previously published paper: The
stability of ideal gases is proven from the expressions obtained for the force
exerted by the corpuscle on the two end pistons of a cylinder, and the internal
energy. We evaluate the entropy increase that occurs when the wall separating
two cylinders is removed and show that the entropy remains the same when the
separation is restored. The entropy increment may be defined at the ratio of
heat entering into the system and temperature when the number of corpuscles (0
or 1) is fixed. In general the entropy is defined as the average value of
where denotes the probability of a given state. Generalization to
-dependent weights, or equivalently to arbitrary static potentials, is made.Comment: Generalization of previous versions to questions of stabilit
Comment on: "Sadi Carnot on Carnot's theorem"
Carnot established in 1824 that the efficiency of reversible
engines operating between a hot bath at absolute temperature and a
cold bath at temperature is equal to . Carnot
particularly considered air as a working fluid and small bath-temperature
differences. Plugging into Carnot's expression modern experimental values,
exact agreement with modern Thermodynamics is found. However, in a recently
published paper ["Sadi Carnot on Carnot's theorem", \textit{Am. J. Phys.}
\textbf{70}(1), 42-47, 2002], Guemez and others consider a "modified cycle"
involving two isobars that they mistakenly attribute to Carnot. They calculate
an efficiency considerably lower than and suggest that Carnot made
compensating errors. Our contention is that the Carnot theory is, to the
contrary, perfectly accurate.Comment: Submitted to American Journal of Physic
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