Violent Relaxation and Gentle Dissipation: Turbulent Lives and Equilibrium States of "Collisionless" Systems

Since, in the late 1960s, Lynden-Bell proposed an entropy-maximisation principle for phase-volume conserving
gravitating systems [1] and Kadomtsev & Pogutse quickly realised that it was portable to
collisionless plasmas [2], Lyndon-Bell statistics has come to be viewed as a beautiful but disappointing idea:
real-life particle distributions in either galaxies or plasmas appeared to have little interest in relaxing to
equilibrium states predicted for them. Was that because the phase space was an obstacle course, hard for
the system to sample fully when subjected to the continuum of constraints (“Casimirs”) encoding the
conservation of the phase-space volumes occupied by the level sets of the particles’ probability distribution?
Or because the multitude of these constraints rendered the hopes for a universal, initial-condition-independent
outcome futile? Having embarked on a fool’s errand of revisiting the 60-year-old idea in the
company of several students and postdocs who should have got better advice but instead made a success
of one they received, I can report some considerable progress. In both plasmas and gravitating systems
where a highly unstable initial state is set up (e.g., counterpropagating electron beams [3], cold dark-matter
streams [4,6], or merging galactic halos [5]) and binary collisions are too infrequent to pin the particle
distribution to a local Maxwellian, the phase-space density of particles does evolve dynamically to maximise
the entropy associated with its entire functional shape, subject to the continuum of Casimirs [3,5,6]. The
fixed points to which this dynamical (“violent”) relaxation pushes the system are indeed the Lynden-Bell
equilibria, which are convolutions of the Fermi distributions of mutually excluding elements of phase-space
density weighted by the values of the density. However, the system cannot stay truly collisionless for long:
the conservation of the Casimirs is gently broken on timescales that are longer than the dynamical times by
roughly the log of the particle number, so the constraints themselves evolve gradually in a certain universal
direction [3,5,6]. How to predict this evolution precisely is not as yet a solved problem, but its ultimate
destination, in a closed system, appears to be such a set of constraints that the ground state of the system
under them has the energy that matches the actual (conserved) energy of the system – and it is the Lynden-
Bell phase-space density corresponding to them, not to the initial condition, that is the long-term
equilibrium. Thus, a (closed) “collisionless” system contrives simultaneously to achieve a state of minimum energy and maximum entropy [5]. “Under the hood” of this process is a turbulent cascade of a particular kind [5,7]: in
phase space, but reminiscent of the Batchelor scalar turbulence in fluid dynamics [8] – it is this cascade that
activates collisional dissipation in a “collisionless” system on a timescale only logarithmically longer than
the dynamical mixing time. Furthermore, numerical simulations [3,4,5,6] tell us that before the final
equilibrium state is achieved, the system can go through a long transient period, also of a universal nature,
and perhaps long enough to be the only relevant thing in some physical conditions. During this period,
“unmixed”, quasi-stable structures (“phase-space holes” in plasmas, persistent halos or streams in
gravitating kinetics) endure for a long time, have their own dynamics of phase-space peregrination and
mergers, and in the process mix the ambient distribution into “hot” quasi-equilibrium nonthermal states –
in the case of plasmas, these exhibit a universal high-energy E² power-law tail, predicted analytically from
the Lynden-Bell statistics far above the ground state [9]. This raises the possibility that observed power-law
distributions (e.g., in the solar wind, for the cosmic rays, or even such phenomena as the universal NFW
profiles in dark-matter halos) might be understood as the outcome of the interplay between generic
collisionless-relaxation and energy-injection processes. Similar statistical ideas now also appear to be taking
hold in the treatment of certain fluid-dynamical nonlinear relaxation processes, e.g., magnetised stratified
atmospheres [10].

References

[1] Lynden-Bell, D. 1967 Statistical mechanics of violent relaxation in stellar systems. MNRAS 136, 101

[2] Kadomtsev, B. B. & Pogutse, O. P. 1970 Collisionless relaxation in systems with Coulomb interactions. PRL 25,
1155

[3] Ewart, R. J., Nastac, M. L., Bilbao, P. J., Silva, T., Silva, L. O. & Schekochihin, A. A. 2025 Relaxation to universal
non-Maxwellian equilibria in a collisionless plasma, PNAS 122, e2417813122

[4] Ginat, Y. B., Nastac, M. L., Ewart, R. J., Konrad, S., Bartelmann, M. & Schekochihin, A. A. 2025, Gravitational
turbulence: the small-scale limit of the cold-dark-matter power spectrum, PRD 112, 063501

[5] Nastac, M. L., Ginat, Y. B., Ewart, R. J., Barnes, M. & Schekochihin, A. A. 2026 Violent relaxation redux, in
preparation

[6] Ginat, Y. B., Nastac, M. L., Ewart, R. J. & Schekochihin, A. A. 2026 Some like it hot: maximum-entropy and
minimum-energy statistics in collisionless kinetic systems, in preparation

[7] Nastac, M. L., Ewart, R. J., Juno, J., Barnes, M. & Schekochihin, A. A. 2025 Universal fluctuation spectrum of
Vlasov-Poisson turbulence, e-print arXiv:2503.17278

[8] Batchelor, G. K. 1959 Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1.
General discussion and the case of small conductivity. JFM 5, 113

[9] Ewart, R. J., Nastac, M. L. & Schekochihin, A. A. 2023 Non-thermal particle acceleration and power-law tails via
relaxation to universal Lynden-Bell equilibria. JPP 89, 905890516

[10] Hosking, D. N., Wasserman, D. & Cowley, S. C. 2025 Metastability of stratified magnetohydrostatic equilibria
and their relaxation. JPP 91, E35