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Publication# Propriétés hors-équilibre d'une paroi adiabatique

Résumé

We study the evolution of a system composed of N non-interacting particles of mass m distributed in a cylinder of length L. The cylinder is separated into two parts by an adiabatic piston of a mass M ≫ m. The length of the cylinder is a fix parameter and can be finite or infinite (in this case N is infinite). For the infinite case we carry out a perturbative analysis using Boltzmann's equation based on a development of the velocity distribution of the piston in function of a small dimensionless parameter ε = √(m/M). The non-stationary case is solved up to the order ε ;; our analysis shows that the system tends exponentially fast towards a stationary state where the piston has an average velocity V. The characteristic time scale for this relaxation is proportional to the mass of the piston (τ0 = M/A where A is the cross-section of the piston). We show that for equal pressures the collisions of the particles induce asymmetric fluctuations of the velocity of the piston which leads to a macroscopic movement of the piston in the direction of the higher temperature. In the case of the finite model a perturbative approach based on Liouville's equation (using the parameter α = 2m/(M + m)) shows that the evolution towards thermal equilibrium happens on two well separated time scales. The first relaxation step is a fast, deterministic and adiabatic evolution towards a state of mechanical equilibrium with approximately equal pressures but different temperatures. The movement of the piston is more or less damped. This damping qualitatively depends on whether the ratio R = Mgas/M between the total mass of the gas and the mass of the piston is small (R < 2) or large (R > 4). The second part of the evolution is much slower ; the typical time scales are proportional to the mass of the piston. There is a stochastic evolution including heat transfer leading to thermal equilibrium. A microscopic analysis yields the relation XM(t) = L(1/2 - ξ(at)) where the function ξ is independent of M. Using the hypothesis of homogeneity (i.e. the values of the densities, pressures and temperatures at the surface of the piston can be replaced by their respective average values) introduced in the previous analysis the observed damping does not show up. This can be explained by shock waves propagating between the piston and the walls at the extremities of the cylinder. In order to study the behaviour of the system there is hence a need to adequately describe the non-equilibrium fluids around the piston. We carry out an analysis of the infinite case, based on the perturbative approach introduced earlier. In this case the initial conditions are chosen in such a manner that the piston on average stays at the origin. It is shown that it is possible to describe the evolution of the fluids in such a way that it is coherent with the two laws of thermodynamics and the phenomenological relationships. Finally we study the case of a constant velocity of the piston in a finite cylinder. Such a condition and elastic collisions allow us to derive an explicit expression for the distribution of the fluids and hence for the hydrodynamics fields. This expression reveals the presence of shock waves between the piston and the extremities of the cylinder.

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Low Pressure Plasma Spraying (LPPS) processes use a DC plasma jet expanding at low pressure for fast deposition of dense coatings in a controlled atmosphere. The LPPS technology is widely used industrially in particular in the aeronautics and medical industries among others. Unlike atmospheric pressure plasma jets, which have been extensively studied experimentally and theoretically, the interest in low pressure DC plasma jets only occurred recently. However, the process development has been mainly based on empirical methods and the basics of the physical mechanisms that govern them still remain to be investigated. Further improvement of the processes requires, in particular, the knowledge of physical properties of the plasma jet such as the temperature, flow velocity and plasma density. Low pressure plasma jets present unconventional properties such as low collisionality, large dimensions and supersonic flow. Therefore specific diagnostics have to be adapted to these conditions. In this study, argon plasma jets at pressures between 2 and 100 mbar are investigated. Imaging has been used to allow a qualitative description of the plasma jet topology for different pressures and torch parameters. Low pressure plasma jets are most of the time supersonic, compressible and in an aerodynamic non-equilibrium, which results in visible successive compression and expansion zones corresponding to a variation of the local pressure, temperature and density. Imaging, combined with pressure measurements inside the plasma torch, has evidenced three different types of flow regimes with respect to the chamber pressure. For chamber pressures below 45 mbar, the flow is under-expanded and is characterized by an exit pressure higher than the chamber pressure. For pressures above 45 mbar, the plasma jet is over-expanded, in this case the exit pressure is lower than the chamber pressure. When the exit pressure is equal to the chamber pressure, the plasma jet is in the so-called design pressure regime. A diagnostic tool, extensively applied on atmospheric plasma jets, the enthalpy probe system, has been modified in order to allow gas sampling from the plasma jet at low pressures. A shock wave appears in front of the probe when it is immersed in a supersonic plasma jet, making the interpretation of enthalpy measurements more difficult.The free-stream properties, like the Mach number, temperature and free-stream enthalpy have to be inferred from stagnation measurements. Two interpretations of enthalpy probe measurements are described in this study. The first method uses the energy conservation equation and LTE assumptions with calorically perfect gas and neglecting the aerodynamic non-equilibrium, whereas the second method, uses a complementary measurement of the static pressure just after the shock using a specially developed tool: the Post Shock Static Pressure Probe (PSSPP). This allows the use of the conservation equations to determine the free-stream properties of the plasma jet without the assumption of calorically perfect gas and aerodynamic non-equilibrium. Determination of the free-stream enthalpy, Mach number and temperature were possible on over-expanded jets for pressures higher than 40 mbar. At 100 mbar with torch parameters of 400 A and 40 SLPM argon flow, the temperature of the plasma jet reaches 10000 K and the velocity is about 3000 m/s on axis. Measurement of plasma jet properties such as the Mach number, electron density and temperature, were performed using double Langmuir probes and Mach probes. In particular, under-expanded jets are studied in detail by performing complete mappings of plasma jet properties at 10 and 2 mbar chamber pressure. These results show that the measured physical properties are consistent with the jet flow phenomenology such as the presence of periodic expansion and compression zones, the effect of the pressure and the location of the shocks. It is shown in particular that for highly under-expanded jets at 2 mbar, the Mach number reaches 2.8 in the first expansion zone followed by a strong drop to subsonic flow revealing the presence of a Mach reflection. The flow is accelerated further and a periodic structure of compression/expansion cells is observed until the local static pressure is in equilibrium with the surrounding pressure. Another diagnostic often used in plasma spraying is optical emission spectroscopy (OES) which is non-intrusive and gives information about the plasma excited species.However, the determination of the excitation temperature, obtained by the Boltzmann plot method, relies on the assumption of local thermodynamic equilibrium (LTE), which is no longer satisfied at low working pressures. The result of the deviation from LTE is that the heavy particle, electron and excitation temperatures are different. In this study, Boltzmann plots have been used to evaluate the deviation from LTE as a function of the working pressure and the location in the plasma jet. It has been shown that the plasma jet is closer to LTE in the compression zones and close to the axis. Measurements of spectral line broadening due to the Stark effect allowed to determine the electron density for under-expanded jets and give results similar than with electrostatic probe measurements.On the other hand, excitation temperatures are systematically lower than the electron temperature for the same plasma conditions. For a 10 mbar plasma jet, the excitation temperature of argon is between 0.73 and 0.78 eV whereas the electron temperature is between 0.7 and 1.2 eV. This shows that at low pressure the plasma jets are not in LTE.These results contribute to the understanding of the supersonic plasma jet behavior at low pressure and can be used to quantify the deviation from local thermodynamic equilibrium (LTE). The extensive mapping of the measured physical properties of the jet will also serve as input for modeling.

This thesis presents a theoretical description of the phase transition, with formation of long-range spatial coherence, occurring in a gas of exciton-polaritons in a semiconductor microcavity structure. The results and predictions of the theories developed in this thesis suggest that this phase transition, recently observed in experiments, can be interpreted as the Bose-Einstein Condensation (BEC) of microcavity polaritons. Our theoretical framework is conceived as a generalization to the microcavity polariton system of the standard theories describing the BEC of a weakly interacting Bose gas. These latter are reviewed in Chapter 2, where an introduction to the physics of polaritons is also given. The polariton system is peculiar, basically due to three main features, i.e. the composite nature of polaritons, which are a linear superposition of photon and exciton states, their intrinsic 2-D nature, and the presence of two-body interactions, arising both from the mutual interaction between excitons and from the saturation of the exciton oscillator strength. Therefore it is not clear whether the observed phase transition can be properly described in terms of BEC of a trapped gas. To clarify this point, one has to describe self-consistently the linear exciton-photon coupling giving rise to polariton quasiparticles, and the exciton-nonlinearities. This is made in Chapter 3, where a bosonic theory is developed by generalizing the Hartree-Fock-Popov description of BEC to the case of two coupled Bose fields at thermal equilibrium. Hence, we derive the classical equations describing the condensate wave function and the Dyson-Beliaev equations for the field of collective excitations. In this way, for each value of the temperature and of the total polariton density, a self-consistent solution can be obtained, fixing the populations of the condensate and of the excited states. In particular, the theory allows to describe simultaneously the properties of the polariton, the exciton and the photon fields, this latter being directly investigated in the typical optical measurements. The predicted phase diagram, the energy shifts, the population energy distribution and the behavior of the resulting first order spatial correlation function agree with the recent experimental findings [Kasprzak 06, Balili 07]. These results thus support the idea that the observed experimental signatures are a clear evidence of polariton BEC. However, from a quantitative pint of view, the measured coherence amount in the condensed regime is significantly lower than the predicted one. This discrepancy could be due to deviations from the weakly interacting Bose gas picture and/or to deviations from the thermal equilibrium regime. In particular, these latter are expected to be strong in current experiments, because polaritons have a short radiative lifetime, while the rate of the energy-relaxation mechanisms is very slow. To investigate how the deviations from equilibrium could affect the condensate fraction and the formation of off-diagonal long-range correlations, in Chapter 4, we develop a kinetic theory of the polariton condensation, accounting for both the relaxation mechanisms and for the field dynamics of fluctuations. Within the Hartree-Fock-Bogoliubov limit, we derive a set of coupled equations of motion for the one-particle populations and for the two particle correlations describing quantum fluctuations. We account for the relaxation processes due both to the polariton-phonon coupling and to the exciton-exciton scattering. The actual spectrum of the system is evaluated within the Popov limit, during the relaxation kinetics. Within this model, we solve self-consistently the populations kinetics and the dynamics of the excitation field, for typical experimental conditions. In particular, we show that the role of quantum fluctuations is amplified by non-equilibrium, resulting in a significant condensate depletion. This behavior could explain the partial suppression of off-diagonal long-range coherence reported in experiments [Kasprzak 06, Balili 07]. We complete the analysis, by studying how the deviations from equilibrium depend on the system parameters. Our results show that the polariton lifetime plays a crucial role. In particular, we expect that the increase of the polariton lifetime above 10 ps would lead to thermal-equilibrium polariton BEC in realistic samples. In Chapter 5, devoted to the conclusions, we discuss which issues of BEC could be clarified, by achieving polariton BEC at thermal-equilibrium, and which extensions of the present work would be most promising in this respect.

Jean-Philippe Ansermet, François Reuse, Jacques Van Der Klink

We present a detailed discussion of the evolution of a statistical ensemble of quantum mechanical systems coupled weakly to a bath. The Hilbert space of the full system is given by the tensor product between the Hilbert spaces associated with the bath and the bathed system. The statistical states of the ensemble are described in terms of density matrices. Supposing the bath to be held at some - not necessarily thermal - statistical equilibrium and tracing over the bath degrees of freedom, we obtain reduced density matrices defining the statistical states of the bathed system. The master equations describing the evolution of these reduced density matrices are derived under the most general conditions. On time scales that are large with respect to the bath correlation time τB corr and with respect to the reciprocal transition frequencies of the bathed system, the resulting evolution of the reduced density matrix of the bathed system is of Markovian type. The detailed balance relations valid for a thermal equilibrium of the bath are derived and the conditions for the validity of the fluctuation-dissipation theorem are given. Based on the general approach, we investigate the non-linear response of the bathed subsystem to a time-periodic perturbation. Summing the perturbation series we obtain the coherences and the populations for arbitrary strengths of the perturbation. © 2003 Springer-Verlag Berlin/Heidelberg.

2003