Equilibrium statistical mechanics for single waves and wave spectra in Langmuir wave-particle interaction

Firpo, Marie-Christine; Leyvraz, F.; Attuel, Guillaume

doi:10.1063/1.2397039

Cited by 9 publications

(27 citation statements)

References 35 publications

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“…In the limiting case where just one wave is allowed for, the canonical solution displays a second order phase transition, as reported in the literature [11,14,17], with the associated critical temperature…”

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confidence: 66%

Ensemble inequivalence in systems with wave-particle interaction

2014

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The classical wave-particle Hamiltonian is considered in its generalized version, where two modes are assumed to interact with the co-evolving charged particles. The equilibrium statistical mechanics solution of the model is worked out analytically, both in the canonical and the microcanonical ensembles. The competition between the two modes is shown to yield ensemble inequivalence, at variance with the standard scenario where just one wave is allowed to develop. As a consequence, both temperature jumps and negative specific heat can show up in the microcanonical ensemble. The relevance of these findings for both plasma physics and Free Electron Laser applications is discussed. [6]. For these systems energy is non additive implying that entropy could be non concave in some range of values of macroscopic extensive parameters [7]. This is at the origin of ensemble inequivalence, which in turn instigates peculiar thermodynamic properties, like negative specific heat and temperature jumps in the microcanonical ensemble. Ensemble inequivalence has been reported to occur in the past for gravitational systems [8], spin models [9] and twodimensional flows [10]. These are extremely interesting models per se, as well as for their theoretical implications, but in general they do not allow for a straightforward experimental verification of the predictions drawn.A paradigmatic example of long range interacting system is represented by the so called wave-particle Hamiltonian, which describes the self-consistent interaction of N charged particles with a co-evolving wave [11]. This is a rather general descriptive scenario, often invoked in different fields of investigations where the mutual coupling between particles and waves proves central, as e.g. in plasma physics [12] and the Free Electron Laser (FEL) [13]. The equilibrium statistical mechanics solution of the celebrated wave-particle Hamiltonian has been so far solely carried out for the simple setting where just one isolated wave is allowed to exist [14,15]. Working under this limiting, and in many respects, unrealistic assumption, it can be shown that the canonical and microcanonical solutions coincide [16]. In real experiments, however, several modes are simultaneously present and interact with the bunch of co-evolving particles. Motivated by this observation and to eventually bridge the gap between theory and experiments, we here consider the generalized equilibrium solution of the reference wave-particle Hamiltonian when a second harmonic is allowed to self-consistently develop. As we shall analytically demonstrate, the insertion of an additional wave causes the ensemble inequivalence to rise, within a specific range of the coupling constants which yield a first order phase transition in the canonical ensemble. This conclusion is reached for a model of marked experimental value, paving the way to direct verifications and, possibly, exploitations of the general concept of ensemble inequivalence.The starting point of our discussion is the universal wave particle Hamil...

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confidence: 66%

Ensemble inequivalence in systems with wave-particle interaction

2014

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“…In particular, the evolution of a beam (composed by N particles) interacting with R electrostatic modes (labeled by the index n) is found to be [11,23] self-consistently governed by the set of the following equations (for the sake of simplicity, we drop the rescaled-variable notation)…”

Section: Multi-wave Modelmentioning

confidence: 99%

Beam–plasma instability and fast particles: the Lynden-Bell approach

Carlevaro

Fanelli

Garbet

et al. 2014

Plasma Phys. Control. Fusion

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Abstract. The beam-plasma instability, i.e., the response of the plasma bulk to the injection of supra thermal charged-particle beams, results to be appropriately characterized by a long-range interaction system. This physical system hosts a number of very interesting phenomena and, in particular, the emergence of long-lived quasi-stationary states. We characterize the self-consistent distribution functions of such out-of-equilibrium states by means of the Lynden-Bell's theory. The prediction of this theory, based on the statistical mechanics of the Vlasov equation, are checked against the outcomes of numerical simulations of the discrete system. Moreover, a phenomenological study of the effective resonance band for the system response is also addressed. A threshold value has been found in the initial spread of beam-particle momenta. This threshold allows to discriminate between the resonant and non-resonant regimes. The analysis of the thermalization of a few percents of the beam population characterized by large initial momenta (with respect to the main part of the beam itself) is also performed and it confirms and deepens the understanding of the physical meaning of the mentioned threshold. General backgroundLong-range interactions are characterized by two-body interaction potentials which are inversely proportional to a power of the inter-particle distance which is smaller than the number of spatial dimension (otherwise, they are said to be short-range interactions) [1]. In this case, it can be shown that the energy per particle diverges for large sizes and the contribution due to the boundary surfaces of the system cannot be neglected with respect to the bulk. This implies that the evolution of an element of the treated model is affected by all the other ones, instead of by its neighbors only, as is the case of short-range interactions. Several recent physical experiments involving long range interactions call for the search of novel theoretical tools to describe the physics these systems.

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“…30 For instance the existence of negative specific heat in a magnetically self-confined plasma torus [84]. The saturation of the cold and water-bag beam-plasma instability can be computed analytically by using Hamiltonian (1) introduced in section III with a single wave (cold: [70], water-bag: [8,9]). The mean-field derivation of Vlasov equation was already mentioned in section II A 2.…”

Section: A Limited Capabilities Of Modelsmentioning

confidence: 99%

“…37 This academic issue has a broader relevance since the QL approximation is used everywhere in plasma physics. 38 For a plateau with a finite width, the small remaining source brings a further evolution of the wave-particle system toward a Gibbsian state where the wave spectrum collapses toward small wavelengths together with the escape of initially resonant particles towards low bulk plasma thermal speeds [70]. This corresponds to a further step toward a new thermal equilibrium of the N -body system corresponding to the initial beam-plasma system.…”

Section: Dynamics When the Distribution Is A Plateaumentioning

confidence: 99%

How to Face the Complexity of Plasmas?

Escande

2013

Nonlinear Systems and Complexity

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This paper has two main parts. The first part is subjective and aims at favoring a brainstorming in the plasma community. It discusses the present theoretical description of plasmas, with a focus on hot weakly collisional plasmas. It comprises two sub-parts. The first one deals with the present status of this description. In particular, most models used in plasma physics are shown to have feet of clay, there is no strict hierarchy between them, and a principle of simplicity dominates the modeling activity. At any moment the description of plasma complexity is provisional and results from a collective and somewhat unconscious process. The second sub-part considers possible methodological improvements, some of them specific to plasma physics, some others of possible interest for other fields of science. The proposals for improving the present situation go along the following lines: improving the way papers are structured and the way scientific quality is assessed in the referral process, developing new data bases, stimulating the scientific discussion of published results, diversifying the way results are made available, assessing more quality than quantity, making available an incompressible time for creative thinking and non purpose-oriented research. Some possible improvements for teaching are also indicated. The suggested improvement of the structure of papers would be for each paper to have a "claim section" summarizing the main results and their most relevant connection to previous literature. One of the ideas put forward is that modern nonlinear dynamics and chaos might help revisiting and unifying the overall presentation of plasma physics.The second part of this chapter is devoted to one instance where this idea has been developed for three decades: the description of Langmuir wave-electron interaction in one-dimensional plasmas by a finite dimensional Hamiltonian. This part is more specialized, and is written like a classical scientific paper. This Hamiltonian approach enables recovering Vlasovian linear theory with a mechanical understanding. The quasilinear description of the weak warm beam is discussed, and it is shown that self-consistency vanishes when the plateau forms in the tail distribution function. This leads to consider the various diffusive regimes of the dynamics of particles in a frozen spectrum of waves with random phases. A recent numerical simulation showed that diffusion is quasilinear when the plateau sets in, and that the variation of the phase of a given wave with time is almost non fluctuating for random realizations of the initial wave phases. This led to new analytical calculations of the average behavior of the self-consistent dynamics when the initial wave phases are random. Using Picard iteration technique, they confirm numerical results, and exhibit a spontaneous emission of spatial inhomogeneities.Non quia difficilia sunt, non audemus, sed quia non audemus, difficilia sunt. It is not because things are difficult that we do not dare, it is because we do not dare that things a...

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Equilibrium statistical mechanics for single waves and wave spectra in Langmuir wave-particle interaction

Cited by 9 publications

References 35 publications

Ensemble inequivalence in systems with wave-particle interaction

Ensemble inequivalence in systems with wave-particle interaction

Beam–plasma instability and fast particles: the Lynden-Bell approach

How to Face the Complexity of Plasmas?

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