This paper explores how orbits in a galactic potential can be impacted by large amplitude time-dependences of the form that one might associate with galaxy or halo formation or strong encounters between pairs of galaxies. A period of time-dependence with a strong, possibly damped, oscillatory component can give rise to large amounts of transient chaos, and it is argued that chaotic phase mixing associated with this transient chaos could play a major role in accounting for the speed and efficiency of violent relaxation. Analysis of simple toy models involving time-dependent perturbations of an integrable Plummer potential indicates that this chaos results from a broad, possibly generic, resonance between the frequencies of the orbits and harmonics thereof and the frequencies of the time-dependent perturbation. Numerical computations of orbits in potentials exhibiting damped oscillations suggest that, within a period of 10 dynamical times t_D or so, one could achieve simultaneously both `near-complete' chaotic phase mixing and a nearly time-independent, integrable end state.Comment: 11 pages and 12 figures: an extended version of the original manuscript, containing a modified title, one new figure, and approximately one page of additional text, to appear in Monthly Notices of the Royal Astronomical Societ

We analyze N-body simulations of halo mergers to investigate the mechanisms responsible for driving mixing in phase space and the evolution to dynamical equilibrium. We focus on mixing in energy and angular momentum and show that mixing occurs in a steplike fashion following pericenter passages of the halos. This makes mixing during a merger unlike other well-known mixing processes such as phase mixing and chaotic mixing, whose rates scale with local dynamical time. We conclude that the mixing process that drives the system to equilibrium is primarily a response to energy and angular momentum redistribution that occurs due to impulsive tidal shocking and dynamical friction rather than a result of chaotic mixing in a changing potential. We also analyze the merger remnants to determine the degree of mixing at various radii by monitoring changes in radius, energy, and angular momentum of particles. We confirm previous findings that show that the majority of particles retain strong memory of their original kinetic energies and angular momenta, but do experience changes in their potential energies owing to the tidal shocks they experience during pericenter passages. Finally, we show that a significant fraction of mass (%40%) in the merger remnant lies outside its formal virial radius, and that this matter is ejected roughly uniformly from all radii outside the inner regions. This highlights the fact that mass, in its standard virial definition, is not additive in mergers. We discuss the implications of these results for our understanding of relaxation in collisionless dynamical systems.

We study the evolution of phase‐space density during the hierarchical structure formation of Λ cold dark matter (CDM) haloes. We compute both a spherically averaged surrogate for phase‐space density (Q=ρ/σ3) and the coarse‐grained distribution function f(x, v) for dark matter (DM) particles that lie within ∼2 virial radii of four Milky Way sized dark matter haloes. The estimated f(x, v) spans over four decades at any radius. DM particles that end up within 2 virial radii of a Milky Way sized DM halo at z= 0 have an approximately Gaussian distribution in log (f) at early redshifts, but the distribution becomes increasingly skewed at lower redshifts. The value fpeak corresponding to the peak of the Gaussian decreases as the evolution progresses and is well described by fpeak(z) ∝ (1 +z)4.5 for z > 1. The highest values of f (responsible for the skewness of the profile) are found at the centres of dark matter haloes and subhaloes, where f can be an order of magnitude higher than in the centre of the main halo. We confirm that Q(r) can be described by a power law with a slope of −1.8 ± 0.1 over 2.5 orders of magnitude in radius and over a wide range of redshifts. This Q(r) profile likely reflects the distribution of entropy (K≡σ2/ρ2/3DM∝r1.2), which dark matter acquires as it is accreted on to a growing halo. The estimated f(x, v), on the other hand, exhibits a more complicated behaviour. Although the median coarse‐grained phase‐space density profile F(r) can be approximated by a power law, ∝r−1.6±0.15, in the inner regions of haloes (<0.6 rvir), at larger radii the profile flattens significantly. This is because phase‐space density averaged on small scales is sensitive to the high‐f material associated with surviving subhaloes, as well as relatively unmixed material (probably in streams) resulting from disrupted subhaloes, which contribute a sizable fraction of matter at large radii.

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