| Summary: | <p>A fundamental tenet of thermodynamics is that chaotic systems will relax to maximum-entropy states. In plasmas, the chaos is conventionally provided by interparticle collisions and the universal maximum-entropy equilibrium is a Maxwellian distribution. However, in collisionless plasmas, the chaotic state is due to collective turbulent dynamics that, while certainly disordered, does not maximise the conventional entropy which would give rise to the Maxwellian. The key result of this thesis is the identification and numerical verification of a class of universal equilibria towards which collisionless plasmas relax. These are still maximum-entropy equilibria, but of a different thermodynamics---that proposed by Lynden-Bell (1967) to describe chaotic relaxation in a collisionless system which conserves an infinite family of invariants, known as Casimir invariants (Ye & Morrison, 1992). We show theoretically that, in sufficiently turbulent plasmas, the Lynden-Bell equilibria become highly universal, forming distributions possessing a power-law tail with exponent -2. The conservation of the Casimir invariants however, is not precise due to the presence of a small amount of collisionality. As a result, the conservation laws on short times endow the system with a partial ‘memory’ of its prior conditions, but the development of a turbulent cascade to small scales, which breaks the precise conservation of phase volume, make this short-term memory imprecise on long times scales. The equilibria are still determined by the short-time collisionless invariants, but the invariants themselves are driven to a universal form by the nature of the turbulence. We demonstrate this effect numerically using the largest, longest run, and best resolved simulations of the two-stream instability to date, showing strong agreement with our theory.</p>
<p>Finally we derive quasilinear collision integrals which, by virtue of possessing an H-theorem, naturally push the system towards a Lynden-Bell equilibrium. This acts as the dynamical counterpart to the thermodynamic formalism we have developed. The collision integrals we derive are similar in form to those derived in collisional and collisionless contexts (Landau, 1936; Balescu, 1960; Lenard, 1960; Kadomtsev & Pogutse, 1970; Severne & Luwel, 1980; Chavanis, 2004, 2022), relying on the scattering of particles off decorrelated fluctuations which, however, are statistically aware of the conservation of the collisionless invariants. We show that relaxation mediated by such a collision integral gives rise to a number of novel effects such as anomalous interspecies drag, and spontaneous current generation.</p>
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