Ensemble inequivalence in systems with wave-particle interaction

Abstract

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...

Ensemble Inequivalence and Maxwell Construction in the Self-Gravitating Ring Model

“…Besides self-gravitating systems, examples of real physical situations with the occurrence of a convex intruder in the entropy are two-dimensional quasi-geostrophic flows [14], waveparticle interaction in a plasma in the presence of two harmonics [15,16] and magnetically self-confined plasma torus [17]. They are also are examples of long-range interacting systems, with interacting potentials decaying at long-distances as 1/r α with α < D, D being the spatial dimension [18][19][20][21][22][23].…”

Generalized maximum entropy approach to quasistationary states in long-range systems

“…QSS have been reported to occur for a wide plethora of physical systems for which longrange couplings are at play. These include plasma wave instabilities [3] relevant for fusion devices, self-gravitating systems [4] invoked in the study of non baryonic large scale structures formation in the Universe, and Free Electron Laser [5,6], light sources of paramount importance for their intrinsic flexibility. To elucidate the mechanisms which drive the emergence of QSS proved a challenging task, that stimulated a vigorous, still open, debate.…”

“…which differs from the Boltzmann-Gibbs expression because of the "Fermionic" denominator that originates from the specific nature of the entropy (6). By inserting expression (8) into energy, momentum and normalization constraints and by making use of the definition of magnetization, one can obtain a set of implicit equations in the unknowns β, λ, µ, M x and M y .…”

Ensemble inequivalence in a mean-fieldXYmodel with ferromagnetic and nematic couplings

“…A striking feature of long-range systems distinct from shortrange ones is that of non-additivity, whereby thermodynamic quantities scale superlinearly with the system size. Non-additivity manifests in static properties like negative microcanonical specific heat [10,11], inequivalence of statistical ensembles [12][13][14][15][16][17][18][19], and other rich possibilities [20]. As for the dynamics, long-range systems often exhibit broken ergodicity [16,21], and slow relaxation towards equilibrium [8,16,[22][23][24][25].Here, we demonstrate ensemble inequivalence in a model of long-range systems that has mean-field interaction (i.e., α = 0) and two coupling modes.…”