Weak order for the discretization of the stochastic heat equation

In this paper we study the approximation of the distribution of $X_t$ Hilbert--valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as $$dX_t+AX_t dt = Q^{1/2} d W_t, \quad X_0=x \in H, \quad t\in[0,T],$$ driven by a Gaussian space time noise whose covariance operator $Q$ is given. We assume that $A^{-\alpha}$ is a finite trace operator for some $\alpha>0$ and that $Q$ is bounded from $H$ into $D(A^\beta)$ for some $\beta\geq 0$. It is not required to be nuclear or to commute with $A$. The discretization is achieved thanks to finite element methods in space (parameter $h>0$) and implicit Euler schemes in time (parameter $\Delta t=T/N$). We define a discrete solution $X^n_h$ and for suitable functions $\phi$ defined on $H$, we show that |\E \phi(X^N_h) - \E \phi(X_T) | = O(h^{2\gamma} + \Delta t^\gamma) \noindent where $\gamma<1- \alpha + \beta$. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations

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

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.Comment: 7 pages, submitted to Phys Rev

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!Comment: 186 page

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 $\Omega$ the photo-current power spectrum is of the form $\Omega^2/(1+\Omega^2)$ and thus vanishes as $\Omega\to 0$, 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 $\hbar\omega/2$ 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.Comment: 116 pages Second draft of a book project. To be completed by a part II incuding extended details on application of the theor

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 $z$ 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.Comment: Generalization of previous versions to questions of stabilit

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 $\eta=1-E_l/E_h$. 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 $\eta=1-T_l/T_h$, 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.Comment: To appear in American Journal of Physic

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

Carnot established in 1824 that the efficiency $\eta_{C}$ 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 $\eta_{C}$ 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