arXiv subjects

Jacques Arnaud

Publications and source records attributed to Jacques Arnaud.

16 recordsLinked to original sources

arXiv2013

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 $z$ 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.

physics.class-ph↗

arXiv2012

Democritus and the motive power of fire

The present work is a translation from french to english of our previous \g{Démocrite et la puissance motrice du feu}, amended on a number of respects. It is mainly of historical and pedagogical interest. We suggest that the concepts introduced in the ancien Greece by Anaximander (flat earth) and Democritus (corpuscles moving in vacuum) allow us to obtain through qualitative observations and plausible generalizations the maximum efficiency and work of heat engines, results that were firmly established around 1824 by Carnot. A prologue introduces the subject. We next present the concept of thermal equilibrium and consider a model consisting of two reservoirs located at different altitudes, each with $g$ sites. Each site may contain a specified number of corpuscles. One particular site plays the role of \g{working agent}. We subsequently consider an alternative model consisting of independent corpuscles submitted to gravity and in contact with heat baths. Only average quantities are considered, leaving out fluctuations and questions of stability.

physics.hist-ph↗

arXiv2011

On the ideal gas law

The air density on earth decays as a function of altitude $z$ approximately according to an $\exp(-w\,z/θ)$-law, where $w$ denotes the weight of a nitrogen molecule and $θ=\kB T$ where $k_B$ is a constant and $T$ the thermodynamic 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: $\ave{F}=θ/h$. 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 $w\,h\ggθ$, and to the usual ideal-gas law when $w\,h\llθ$. The mathematics is elementary. The present paper should accordingly facilitate the understanding of the physical meaning of the barometric and ideal-gas laws.

physics.hist-ph↗

arXiv2009

A simple model for Carnot heat engines

We present a (random) mechanical model consisting of two lottery-like reservoirs at altitude $E_h$ and $E_l<E_h$, respectively, in the earth's gravitational field. Both reservoirs consist of $N$ possible ball locations. The upper reservoir contains initially $n_h\le N$ weight-1 balls and the lower reservoir contains initially $n_l\le N$ weight-1 balls. Empty locations are treated as weight-0 balls. These reservoirs are being shaken up so that all possible ball configurations are equally likely to occur. A cycle consists of exchanging a ball randomly picked from the higher reservoir and a ball randomly picked from the lower reservoir. It is straightforward to show that the efficiency, defined as the ratio of the average work produced to the average energy lost by the higher reservoir is $η=1-E_l/E_h$. We then relate this system to a heat engine. This thermal interpretation is applicable only when the number of balls is large. We define the entropy as the logarithm of the number of ball configurations in a reservoir, namely $S(n)=\ln[N!/n!(N-n)!]$, with subscripts $h,l$ appended to $S$ and to $n$. When $n_l$ does not differ much from $n_h$, the system efficiency quoted above is found to coincide with the maximum efficiency $η=1-T_l/T_h$, where the $T$ are absolute temperatures defined from the above expression of $S$. Fluctuations are evaluated in Appendix A, and the history of the Carnot discovery (1824) is recalled in Appendix B. Only elementary physical and mathematical concepts are employed.

cond-mat.stat-mech↗

arXiv2009

Quiet Lasers

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!

quant-ph↗

arXiv2009

Grand-mother clocks and quiet lasers

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.

quant-ph↗

arXiv2007

Sub-Poissonian laser emission from a single-electron permanently interacting with a single-mode cavity

Quiet (or sub-Poissonian) oscillators generate a number of dissipation events whose variance is less than the mean. It was shown in 1984 by Golubev and Sokolov that lasers driven by regular pumps are quiet in that sense. The purpose of this paper is to show that, as long as the laser-detector system is strictly stationary, quantization of the optical field is not required to explain such phenomena. The theory presented here is semi-classical, yet exact. Previous theories considering excited-state atoms regularly-injected in resonators, on the other hand, do require in principle light quantization. Specifically, we consider a laser involving a single electron permanently interacting with the field and driven by a constant-potential battery, and point out a similarity with reflex klystrons. The detected noise is found to be only 7/8 of the shot-noise level. It is therefore sub-Poissonian. Our calculations are related to resonance-fluorescence treatments but with different physical interpretations.

quant-ph↗

arXiv2007

Semi-classical theory of quiet lasers. Short version

This article is the shorten version of quant-phys/0610106 with a supplemented theory and new results concerning a single-electron laser driven by a constant-potentiel battery. "Quiet (or sub-Poissonian) oscillators generate a number of dissipation events whose variance is less than the mean. It was shown in 1984 by Golubev and Sokolov that lasers driven by regular pumps are quiet in that sense. We consider in the present paper two oscillators that should exhibit in principle the same property. First, a reflex klystron, a vacuum tube operating in the microwave range of frequency. Second a laser involving a single electron permanently interacting with the field. It is unnecessary to quantize the optical field, that is, the theory is semi-classical, yet exact. As an example, the battery-driven one-electron laser delivers a detected noise of 7/8 of the shot-noise level, and is therefore sub-Poissonian. Our calculations are related to resonance-fluorescence treatments but with a different physical interpretation. Previous theories considering excited-state atoms regularly-injected in low-loss resonators, on the other hand, do require light quantization. The theory presented here is restricted to above-threshold stationary single-mode oscillators. The paper is written in such a way that readers should be able to follow it without having to refer to quantum-optics texts."

quant-ph↗

arXiv2007

Semi-classical theory of quiet lasers. I: Principles

When light originating from a laser diode driven by non-fluctuating electrical currents is incident on a photo-detector, the photo-current does not fluctuate much. Precisely, this means that the variance of the number of photo-electrons counted over a large time interval is much smaller that the average number of photo-electrons. At non-zero Fourier frequency $Ω$ the photo-current power spectrum is of the form $Ω^2/(1+Ω^2)$ and thus vanishes as $Ω\to 0$, a conclusion equivalent to the one given above. The purpose of this paper is to show that results such as the one just cited may be derived from a (semi-classical) theory in which neither the optical field nor the electron wave-function are quantized. We first observe that almost any medium may be described by a circuit and distinguish (possibly non-linear) conservative elements such as pure capacitances, and conductances that represent the atom-field coupling. The theory rests on the non-relativistic approximation. Nyquist noise sources (in which the Planck term $\hbarω/2$ is being restored) are associated with positive or negative conductances, and the law of average-energy conservation is enforced. We consider mainly second-order correlations in stationary linearized regimes.

quant-ph↗

arXiv2003

Comment on: "Sadi Carnot on Carnot's theorem"

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", \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 $η_{C}$ and suggest that Carnot made compensating errors. Our contention is that the Carnot theory is, to the contrary, perfectly accurate.

physics.hist-ph↗

arXiv2003

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.

quant-ph↗

arXiv2002

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.

math-ph↗

arXiv2002

Two-level laser light statistics

The statistics of the light emitted by two-level lasers is evaluated on the basis of generalized rate equations. According to that approach, all fluctuations are interpreted as being caused by the jumps that occur in active and detecting atoms. The intra-cavity Fano factor and the photo-current spectral density are obtained analytically for Poissonian and quiet pumps. The algebra is simple and the formulas hold for small as well as large pumping rates. Lasers exhibit excess noise at low pumping levels.

quant-ph↗

arXiv2002

Rate-equation approach to atomic-laser light statistics

We consider three- and four-level atomic lasers that are either incoherently (unidirectionally) or coherently (bidirectionally) pumped, the single-mode cavity being resonant with the laser transition. The intra-cavity Fano factor and the photo-current spectral density are evaluated on the basis of rate equations. According to that approach, fluctuations are caused by jumps in active and detecting atoms. The algebra is considerably simpler than the one required by Quantum-Optics treatments. Whenever a comparison can be made, the expressions obtained coincide. The conditions under which the output light exhibits sub-Poissonian statistics are considered in detail. Analytical results, based on linearization, are verified by comparison with Monte Carlo simulations. An essentially exhaustive investigation of sub-Poissonian light generation by three- and four-level atoms lasers has been performed. Only special forms were reported earlier.

quant-ph↗

arXiv2002

Rate equation theory of sub-Poissonian laser light

Lasers essentially consist of single-mode optical cavities containing two-level atoms with a supply of energy called the pump and a sink of energy, perhaps an optical detector. The latter converts the light energy into a sequence of electrical pulses corresponding to photo-detection events. It was predicted in 1984 on the basis of Quantum Optics and verified experimentally shortly thereafter that when the pump is non-fluctuating the emitted light does not fluctuate much. Precisely, this means that the variance of the number of photo-detection events observed over a sufficiently long period of time is much smaller than the average number of events. Light having that property is said to be ``sub-Poissonian''. The theory presented rests on the concept introduced by Einstein around 1905, asserting that matter may exchange energy with a wave at angular frequency $ω$ only by multiples of $\hbarω$. The optical field energy may only vary by integral multiples of $\hbarω$ as a result of matter quantization and conservation of energy. A number of important results relating to isolated optical cavities containing two-level atoms are first established on the basis of the laws of Statistical Mechanics. Next, the laser system with a pump and an absorber of radiation is treated. The expression of the photo-current spectral density found in that manner coincides with the Quantum Optics result. The concepts employed in this paper are intuitive and the algebra is elementary. The paper supplements a previous OQE tutorial paper in establishing a connection between the theory of laser noise and Statistical Mechanics.

quant-ph↗

arXiv2001

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*hbarω, 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.

physics.optics↗