This issue of the Journal of Plasma Physics is dedicated to Radu Balescu on the occasion of his 65th birthday.

Of the more than 170 articles published worldwide by Professor Balescu to date, more than 25 have been published in this journal. Readers of the Journal of Plasma Physics will thus be well acquainted with his contributions to the study of the interplay between plasma turbulence and the theory of transport phenomena in magnetically confined plasmas. A vast community of students, teachers and researchers will also have read or consulted the many monographs, five to date, written by Balescu and world-famous for their clarity. In his first monograph, Statistical Mechanics of Charged Particles (Interscience, New York), published in 1963, the whole of plasma physics, including the transport equation to which his name will remain attached, was put into the broader context of statistical physics. The monograph Equilibrium and Non-Equilibrium Statistical Mechanics (Wiley-Interscience, New York) followed in 1975, and still inspires many teachers today.

From 1990, the boronized TEXTOR tokamak was characterized by an improved confinement (coined the ‘I mode’) at high power that was substantially better than the L mode, but densities had to be limited to n[bar]e0/nGR[lsim ]0.5–0.6, where nGR is the Greenwald density limit. With the injection of Ne, Si or Ar in order to increase the edge radiation and provided that γ=Prad/Ptot[greater, similar]0.5, PNBI-co/Ptot[greater, similar]0.25 and n[bar]e0/nGR[greater, similar]0.75, a further improved confinement called the radiative improved mode (RI mode) was discovered in 1993 on TEXTOR, a tokamak of intermediate size, and confirmed on TEXTOR-94. The radiated power fraction γ can reach 0.9, and the radiation is nearly isotropically distributed over the torus wall. The RI mode is characterized by its ability to obtain simultaneously and stationarily high densities and high confinement. It is linked to a substantial lowering of edge ne, Te and Ti, a reduction in particle transport and a peaking of the density profile. The RI-mode confinement scales on TEXTOR as τE= (n[bar]e0/nGR)τITERH93-P and values up to n[bar]e0/nGR≈1.2 are obtained. There is no detrimental concentration of the seeded impurity at the centre of the plasma. Results of three different interpretative and modelling approaches are in agreement with the improved confinement features; the preliminary indications are that ITG turbulence is strongly reduced. The Z mode observed on ISX-B has a clear resemblance to the RI mode. The very favourable features of the RI mode justify efforts in trying to establish it on larger machines to verify if the present scaling then holds.

The well-known van Kampen–Case treatment of the Vlasov equation leads to a spectrum on the real axis. In this paper we show that, by going to a ‘rigged’ Hilbert space, we can derive a spectral representation that is complex and breaks time symmetry. This leads to a semigroup description in which the decay rates due to the Landau damping appear explicitly in the spectrum. Moreover, we can then define an entropy. In this way, the relation between Landau damping and irreversibility is made explicit. The analogy with the well-known Friedrichs model is stressed.

An understanding of the mechanisms leading to the symmetry breaking of the dynamical description of a large system with respect to the direction of time is necessary, but not sufficient to ensure the finding of a functional of the state of the system that would satisfy the requirements placed by the Second Law of Thermodynamics upon the non-equilibrium entropy S.

The Van Kampen mode discussion of plasma oscillations is briefly reviewed alongside theories of non-equilbrium statistical mechanics and attention is drawn to illuminating connections between them.

The infinite set of moments of the two-particle distribution function is found exactly for the uniform cooling state of a hard-sphere gas with inelastic collisions. Their form shows that velocity correlations cannot be neglected, and consequently the ‘molecular chaos’ hypothesis leading to the inelastic Boltzmann and Enskog kinetic equations must be questioned.

A brief review of two alternative descriptions of non-equilibrium processes in a fully ionized plasma is presented. The example of ‘Landau damping’ is used to demonstrate that some fundamental problems remain unsolved in the kinetic theory of plasmas. These arise from inconsistent descriptions of the transition from the reversible equations of the mechanics of charged particles and fields to the irreversible equations for a continuous medium in the theory of plasmas. These difficulties can be overcome through consistently defining the structure of the continuous medium and the characteristics of dynamic instability of motion of particles. This leads to generalized irreversible kinetic equations that provide the basis for a unified description of non-equilibrium processes in a plasma on kinetic and hydrodynamic scales.

Taking into account the first non-Markovian correction to the Balescu–Lenard equation, we derive an expression for the pair correlation function and a nonlinear kinetic equation valid for a weakly non-ideal polarized classical plasma. This equation allows for the description of the correlational energy evolution and exhibits global conservation of energy with dynamical polarization.

For low-dimensional Hamiltonian systems with chaotic dynamics, we discuss the differences in the kinetic description, and the necessity of windowing of the time and parameter space, which is imposed by the differences in the phase-space topology and by the stickiness of trajectories at island boundaries.

Stochastic modelling of turbulence is considered. Exact and approximate solution procedures based on projection techniques are briefly reviewed. Procedures leading to equations that are local in time are discussed. Two different approximations of this type are presented, whose performances are tested and compared on a simple model case.

A synthesized formulation of classical, neoclassical and anomalous transport in toroidal confinement systems with electromagnetic fluctuations and large mean flows is presented. The positive-definite entropy production rate and the conjugate flux–force pairs are rigorously defined for each transport process. The Onsager symmetries of the classical and neoclassical transport matrices are derived from the self-adjointness of the linearized collision operator. The linear gyrokinetic equation with given electromagnetic fluctuations determines the anomalous fluxes with the quasilinear anomalous transport matrix, which satisfies the Onsager symmetry.

It is shown that particle trapping plays an important role in the appearance of anomalous or strange diffusion in turbulent fields, mainly by inducing a non-Gaussian statistics of the stochastic displacements. Trapping processes are responsible for the important deviation observed between the traditional prediction of Bohm scaling for diffusion in low-frequency turbulence and the numerically checked prediction by Isichenko. Also, several theoretical models are proposed or reviewed.

The Hamiltonian dynamics of a two-degree-of-freedom system is shown to mimic the structure of an incompressible fluid if the fluid streamlines are considered instead of the pathlines used by many authors. This approach is formally equivalent to the determination of magnetic field lines, which satisfy the same mathematical equations. Turbulence in this case is seen as Hamiltonian chaotic dynamics perturbed by time, which plays the role of the codimension parameter of such chaos. Any spatial Lie symmetry leads to a laminar flow, whereas stationary flow could be considered chaotic but not turbulent. The transition from laminar to turbulent flow is associated with the destruction of KAM trajectories produced by time evolution.

Hamiltonian vortices and reconnection in magnetized plasmas are investigated analytically and numerically using a two-fluid model. The equations are written in the Lagrangian form of three fields that are advected with different velocities. This system can be considered as a generalization and extension of the two-dimensional Euler equation for an ordinary fluid. It is pointed out that these equations allow solutions in the form of singular current-vortex filaments, drift-Alfvén vortices and magnetic islands, and admit collisionless magnetic reconnection where magnetic flux is converted into electron momentum and ion vorticity.

Two-field models for Rayleigh–Taylor modes are investigated. The changes due to external velocity shear (without flow curvature) are reviewed, and the influences of the various terms in the models are discussed. It is shown that, in principle, velocity shear in combination with dissipation leads to the suppression of linear Rayleigh–Taylor modes in the long-time limit. The long-wavelength modes first seem to be damped; however, later they show an algebraic growth in time, before ultimately the exponential viscous damping wins. In general, the amplitudes become very large, and therefore the often-quoted stability of Rayleigh–Taylor modes in the presence of velocity shear is more a mathematical artefact than a real physical process. Vortices, on the other hand, can lead (together with velocity shear) to a quite different dynamical behaviour. Because of a locking of the wave vectors, pronounced oscillations appear. This effect is demonstrated by a simple model calculation. When vortices and velocity shear are generated from linear instability, the resulting oscillatory state finally becomes unstable with respect to Rayleigh–Taylor modes on a long time scale (‘secondary instability’).

Transmission, reflection and absorption of a transverse electromagnetic wave by a thermal plasma slab of finite width are investigated theoretically. By solving the Maxwell–Boltzmann equations using a mode-analysis method under appropriate boundary conditions, the coefficients of transmission and reflection of the electromagnetic wave and the absorption rate by plasma electrons are obtained as Fourier series. We discuss the relevant physics of the results.

E×B guiding-centre (GC) motion in a special configuration of three low-frequency electrostatic waves can be considered as a paradigmatic Hamiltonian system for studying adiabatic motion and separatrix crossings. A peculiarity of this system is that a single initial condition gives rise to two stroboscopic phase-space trajectories. According to the classical Hamiltonian theory, the proportion of points on the stroboscopic trajectories is a function of the time evolution of the surfaces enclosed by the separatrices in the phase space. This behaviour is qualitatively observed in test-particle numerical experiments. The ability of numerical integration methods like the ‘classical’ fourth-order Runge–Kutta integration scheme or a third-order symplectic integrator to reproduce the statistics is analysed.