Collisionless relaxation of a Lynden-Bell plasma

Ewart, Robert James; Brown, Andrew R.; Adkins, T.; Schekochihin, A. A.

doi:10.1017/s0022377822000782

Cited by 13 publications

(24 citation statements)

References 33 publications

(68 reference statements)

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“…There is a wealth of interesting physics that can arise from the ballistic response, see, e.g. Ewart et al (2022) and references therein, in the context of the Vlasov-Poisson system. However, this is not the focus of the present work and so will not be discussed further.…”

Section: Analytic Continuation For the Inverse Laplace Transformmentioning

confidence: 99%

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An analytical form of the dispersion function for local linear gyrokinetics in a curved magnetic field

Ivanov

Adkins

2023

J. Plasma Phys.

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Starting from the equations of collisionless linear gyrokinetics for magnetised plasmas with an imposed inhomogeneous magnetic field, we present the first known analytical, closed-form solution for the resulting velocity-space integrals in the presence of resonances due to both parallel streaming and constant magnetic drifts. These integrals are written in terms of the well-known plasma dispersion function (Faddeeva & Terent'ev, Tables of Values of the Function $w(z)=\exp (-z^2)(1+2i/\sqrt {\pi }\int _0^z \exp (t^2) \,\mathrm {d} t)$ for Complex Argument, 1954. Gostekhizdat. English translation: Pergamon Press, 1961; Fried & Conte, The Plasma Dispersion Function, 1961. Academic Press), rendering the subsequent expressions simpler to treat analytically and more efficient to compute numerically. We demonstrate that our results converge to the well-known ones in the straight-magnetic-field and two-dimensional limits, and show good agreement with the numerical solver by Gürcan (J. Comput. Phys., vol. 269, 2014, p. 156). By way of example, we calculate the exact dispersion relation for a simple electrostatic, ion-temperature-gradient-driven instability, and compare it with approximate kinetic and fluid models.

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Section: Analytic Continuation For the Inverse Laplace Transformmentioning

confidence: 99%

“…Ewart et al. (2022) and references therein, in the context of the Vlasov–Poisson system. However, this is not the focus of the present work and so will not be discussed further.…”

Section: Analytic Continuation For the Inverse Laplace Transformmentioning

confidence: 99%

An analytical form of the dispersion function for local linear gyrokinetics in a curved magnetic field

Ivanov

Adkins

2023

J. Plasma Phys.

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“…If the evolution of the system is ergodic, the QSS can be determined by the Lynden-Bell statistical theory [9]. The kinetic theory of violent collisionless relaxation is discussed in [10,11]. On a longer timescale, the mean DF evolves through a sequence of QSSs sourced by the noise.…”

Section: Stochastic Damped Vlasov Equationmentioning

confidence: 99%

“…This process takes place on a few dynamical times t D . If the system mixes efficiently, the QSS can be predicted by the statistical theory of Lynden-Bell [9] which assumes that the system has reached the most mixed state consistent with the collisionless evolution, i.e., the one that maximizes a mixing entropy while conserving all the constraints of the Vlasov equation: the energy and all the Casimirs (see [10,11] for recent studies on the Lynden-Bell theory of violent relaxation). This maximum entropy principle relies on an assumption of ergodicity which is not always fulfilled in practice.…”

Section: Introductionmentioning

confidence: 99%

Kinetic theory of two-dimensional point vortices and fluctuation–dissipation theorem

Chavanis

2023

Eur. Phys. J. Plus

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We complete the kinetic theory of inhomogeneous systems with long-range interactions initiated in previous works. We use a simpler and more physical formalism. We consider a system of particles submitted to a small external stochastic perturbation and determine the response of the system to the perturbation. We derive the diffusion tensor and the friction by polarization of a test particle. We introduce a general Fokker-Planck equation involving a diffusion term and a friction term. When the friction by polarization can be neglected, we obtain a secular dressed diffusion (SDD) equation sourced by the external noise. When the external perturbation is created by a discrete collection of N field particles, we obtain the inhomogeneous Lenard-Balescu kinetic equation reducing to the inhomogeneous Landau kinetic equation when collective effects are neglected. We consider a multispecies system of particles. When the field particles are at statistical equilibrium (thermal bath), we establish the proper expression of the fluctuation-dissipation theorem for systems with long-range interactions relating the power spectrum of the fluctuations to the response function of the system. In that case, the friction and diffusion coefficients satisfy the Einstein relation and the Fokker-Planck equation reduces to the inhomogeneous Kramers equation. We also consider a gas of Brownian particles with long-range interactions described by N coupled stochastic Langevin equations and determine its mean and mesoscopic evolution. We discuss the notion of stochastic kinetic equations and the role of fluctuations possibly triggering random transitions from one equilibrium state to the other. Our presentation parallels the one given for two-dimensional point vortices in a previous paper [P.H. Chavanis, Eur. Phys. J. Plus 138, 136 (2023)].

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“…1), yielding a quantity that has the same physical dimensions as the phase-space volume. The dimensional entropies are well suited as a diagnostic for irreversibility, and may be applied to elucidate the role of generalized entropy in a range of physical systems, such as collisionless plasmas [e.g., 6,26,27], strongly coupled plasmas [28], gravitationally interacting systems [e.g., [29][30][31][32][33], collisionless systems with long-range interactions [34,35], classical turbulence [36][37][38], quantum processes [39][40][41][42], chemical reactions [43], and biophysics [44].…”

Section: Introductionmentioning

confidence: 99%

Dimensional measures of generalized entropy

Zhdankin

2023

J. Phys. A: Math. Theor.

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Entropy is useful in statistical problems as a measure of irreversibility, randomness, mixing, dispersion, and number of microstates. However, there remains ambiguity over the precise mathematical formulation of entropy, generalized beyond the additive definition pioneered by Boltzmann, Gibbs, and Shannon (applicable to thermodynamic equilibria). For generalized entropies to be applied rigorously to nonequilibrium statistical mechanics, we suggest that there is a need for a physically interpretable (dimensional) framework that can be connected to dynamical processes operating in phase space. In this work, we introduce dimensional measures of entropy that admit arbitrary invertible weight functions (subject to curvature and convergence requirements). These ``dimensional entropies'' have physical dimensions of phase-space volume and represent the extent of level sets of the distribution function. Dimensional entropies with power-law weight functions (related to Renyi and Tsallis entropies) are particularly robust, as they do not require any internal dimensional parameters due to their scale invariance. We also point out the existence of composite entropy measures that can be constructed from functionals of dimensional entropies. We calculate the response of the dimensional entropies to perturbations, showing that for a structured distribution, perturbations have the largest impact on entropies weighted at a similar phase-space scale. This elucidates the link between dynamics (perturbations) and statistics (entropies). Finally, we derive corresponding generalized maximum-entropy distributions. Dimensional entropies may be useful as a diagnostic (for irreversibility) and for theoretical modeling (if the underlying irreversible processes in phase space are understood) in chaotic and complex systems, such as collisionless systems of particles with long-range interactions.

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Collisionless relaxation of a Lynden-Bell plasma

Cited by 13 publications

References 33 publications

An analytical form of the dispersion function for local linear gyrokinetics in a curved magnetic field

An analytical form of the dispersion function for local linear gyrokinetics in a curved magnetic field

Kinetic theory of two-dimensional point vortices and fluctuation–dissipation theorem

Dimensional measures of generalized entropy

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