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A  Bientôt

Jacques

# COMMENT ON: “SADI CARNOT ON CARNOT’S THEOREM”

26/01/2014

ABSTRACT

Carnot established in 1824 that the efficiency η C of reversible engines operating between a hot bath at absolute temperature T hot and a cold bath at temperature T cold is equal to 1−T cold/T hot. 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”, Am. J. Phys. 70 (1), 42–47, 2002], G ̈u ́emez and others consider a “modified cycle” involving two isobars that they mistakenly attribute to Carnot. They calculate an efficiency considerably lower than η C and suggest that Carnot made compensating errors. Our contention is that the Carnot theory is, to the contrary, perfectly accurate.

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# ON CLASSICAL IDEAL GASES

Jacques Arnaud, Laurent Chusseau, Fabrice Philippe

Entropy 2013, 15, 960-971; doi:10.3390/e15030960

23/04/2013

ABSTRACT

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 z and t-invariant force −w 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 ln(p) where p denotes the probability of a given state. Generalization to z-dependent weights, or equivalently to arbitrary static potentials, is made.

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# A NEW PERSPECTIVE ON CLASSICAL IDEAL GASES

Jacques Arnaud, Laurent Chusseau, Fabrice Philippe

Entropy 2013, 15, 3379-3395

ABSTRACT

The ideal-gas barometric and pressure laws are derived from the Democritian concept of independent corpuscles moving in vacuum, plus a principle of simplicity, namely that these laws are independent of the kinetic part of the Hamiltonian. A single corpuscle in contact with a heat bath in a cylinder and submitted to a constant force (weight) is considered.

The paper importantly supplements a previously published paper: First, the stability of ideal gases is established. Second, we show that when walls separate the cylinder into parts and are later removed, the entropy is unaffected. We obtain full agreement with Landsberg’s and others’ (1994) classical  thermodynamic result for the entropy of a column of gas submitted to gravity.

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# ON THE IDEAL GAS LAW

Jacques Arnaud, Laurent Chusseau, Fabrice Philippe

Arxiv 19/07/2011 version 2

ABSTRACT

The air density on earth decays as a function of altitude z approximately according to an exp(−wz/θ)-law, where w denotes the weight of a nitrogen molecule and $\theta=\kB T$ where kB is a constant and T the hermodynamic temperature. To derive this law one usually invokes the Boltzmann factor, itself derived from statistical considerations. We show that this (barometric) law may be derived solely from the democritian concept of corpuscles moving in vacuum. We employ a principle of simplicity, namely that this law is \emph{independent} of the law of corpuscle motion. This view-point puts aside restrictive assumptions that are source of confusion. Similar observations apply to the ideal-gas law. In the absence of gravity, when a cylinder terminated by a piston, containing a single corpuscle and with height h has temperature θ, the average force that the corpuscle exerts on the piston is: $\ave{F}=\theta/h$. This law is valid at any temperature, except at very low temperatures when quantum effects are significant and at very high temperatures because the corpuscle may then split into smaller parts. It is usually derived under the assumption that the temperature is proportional to the corpuscle kinetic energy, or else, from a form of the quantum theory. In contradistinction, we show that it follows solely from the postulate this it is independent of the law of corpuscle motion. On the physical side we employ only the concept of potential energy. A consistent picture is offered leading to the barometric law when wh≫θ, and to the usual ideal-gas law when wh≪θ. The mathematics is elementary. The present paper should accordingly facilitate the understanding of the physical meaning of the barometric and ideal-gas laws.

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# DEMOCRITE ET LA PUISSANCE MOTRICE DU FEU

Jacques Arnaud, Laurent Chusseau, Fabrice Philippe

arXiv:1104.0836v1 [physics.hist-ph] 5 Apr 2011

ABSTRACT

Cet article, dont le but est principalement historique et pédagogique, suggère que les concepts introduits dans la Grèce antique par Anaximandre (terre plate isolée) et Démocrite (corpuscules et vide) permettent d’obtenir, à partir d’observations qualitatives et de généralisations plausibles, le rendement maximal et le travail d’une machine thermique obtenus à l’époque moderne par Carnot (1824). Un prologue présente un modèle simple de machine thermique. Après avoir introduit la notion d’équilibre thermique, un modèle de machine thermique constitué de deux réservoirs de corpuscules est étudié de manière classique.

Nous établissons ensuite l’équation des gaz parfaits suivant le concept corpusculaire de Daniel Bernoulli (1738) en utilisant la notion d’action d’un corpuscule. L’énergie interne d’un gaz à toute température est déterminée.

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# A SIMPLE MODEL FOR CARNOT HEAT ENGINES

Jacques Arnaud, Laurent Chusseau, Fabrice Philippe

American Journal of Physics, American Association of Physics Teachers, 2010, 78, pp.106-110

ABSTRACT

We discuss a model consisting of two reservoirs, each with N possible ball locations, at heights Eh and ElEh in a gravitational ﬁeld. The two reservoirs contain nh and nl weight 1 balls. Empty locations are treated as weight 0 balls. The reservoirs are shaken so that all possible ball conﬁgurations are equally likely to occur. A cycle consists of exchanging a ball randomly chosen from the higher reservoir and a ball randomly chosen from the lower reservoir. We relate this system to a heat engine and show that the efﬁciency, which is deﬁned as the ratio of the average work produced to the average energy lost by the higher reservoir, is 1−El/Eh. When nl is comparable to nh, the efﬁciency is found to coincide with the maximum efﬁciency 1−Tl/Th, where the temperatures Tl and Th are deﬁned from a simple expression for the entropy. We also discuss the evaluation of ﬂuctuations and the history of the Carnot discovery.

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# QUIET LASERS

Jacques Arnaud, Laurent Chusseau, Fabrice Philippe

Arxiv, 8 juin 2009

ABSTRACT

We call « quiet laser » a stationary laser that generates in detectors regular photo-electrons (sub-Poisson statistics). It follows from the law of conservation of energy that this is so when the laser power supply does not fluctuate. Various configurations are analyzed on the basis of the Planck (1907) semi-classical concept: « I am not seeking the meaning of light quanta in the vacuum but rather in places where emission and absorption occur, and I assume that what happens in the vacuum is rigorously described by Maxwell’s equations ». Exact agreement with Quantum Optics results is noted. Comments welcome !

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# GRAND-MOTHER CLOCKS AND QUIET LASERS

Jacques Arnaud, Laurent Chusseau, Fabrice Philippe

09/01/2009

ABSTRACT

Galileo noted in the 16th century that the period of oscillation of a pendulum is almost independent of the amplitude. However, such a pendulum is damped by air friction. The latter may be viewed as resulting from air molecules getting in contact with the pendulum. It follows that air friction, not only damps the oscillation, but also introduces randomness. In the so-called “grand-mother” clock, discovered by Huygens in the 18th century, damping is compensated for, on the average, by an escapement mechanism driven by a falling weight. The purpose of this paper is to show that such a clock is, in its idealized form, a quiet oscillator. By “quiet” we mean that in spite of the randomness introduced by damping, the dissipated power (viewed as the oscillator output) does not fluctuate slowly. Comparison is made with quiet laser oscillators discovered theoretically in 1984. Because the input power does not fluctuate in both the mechanical oscillator and the quiet laser oscillator, the output power does not fluctuate at small Fourier frequencies, irrespectively of the detailed mechanisms involved.

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# MECHANICAL EQUIVALENT OF QUANTUM HEAT ENGINES

Jacques Arnaud, Laurent Chusseau, Fabrice Philippe

Physical Review E 77, 061102, 03/06/2008

ABSTRACT

Quantum heat engines employ as working agents multilevel systems instead of classical gases. We show that under some conditions quantum heat engines are equivalent to a series of reservoirs at different altitudes containing balls of various weights. A cycle consists of picking up at random a ball from one reservoir and carrying it to the next, thereby performing or absorbing some work. In particular, quantum heat engines, employing two-level atoms as working agents, are modeled by reservoirs containing balls of weight 0 or 1. The mechanical model helps us prove that the maximum efficiency of quantum heat engines is the Carnot efficiency. Heat pumps and negative temperatures are considered.

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