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An N-body simulation approximates a dynamical system by evolving $N$ interacting particles. In a collisionless galaxy calculation these particles sample a smooth mass distribution rather than represent individual physical stars. Their larger individual masses enhance artificial two-body relaxation; softening, force accuracy and convergence with particle number must be checked over the intended duration.

For the Zeldovich approximation, form the Jacobian matrix of the Lagrangian map:Let ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3$ be the eigenvalues of this potential Hessian matrix, with the convention that positive $\lambda$ means compression. Before shell crossing, conservation of the mass in a Lagrangian volume givesIf ${\lambda }_1>0$ and is distinct from the other eigenvalues, the first singularity occurs at $b=1/{\lambda }_1$. One principal length collapses while the other two remain nonzero. The resulting cosmological caustic is locally a sheet, conventionally a pancake, rather than a simultaneous spherical point collapse. Degenerate equal eigenvalues can instead give line- or pointlike special cases.

The predicted density divergence signals failure of the single-stream description. If the straight Lagrangian displacements are simply continued, particles cross the sheet and keep moving along their original displacement directions. They are not slowed and trapped by the evolved self-gravitating multistream structure. The streams therefore separate again, producing excessively broad pancakes at late times. In an N-body simulation, the changing gravitational field pulls crossing streams back, produces repeated crossings and binds matter into more compact structures. This is the limitation described by Zeldovich pancake thickness after shell crossing. The initial-displacement extrapolation is useful through the onset of collapse, but does not supply the self-consistent post-crossing gravitational dynamics or virialization. After crossing, the physical density would require summing inverse absolute Jacobians over all streams, not using one negative determinant as a density.

The stellar relaxation time is the time on which accumulated discrete gravitational encounters change a typical star's velocity by an amount comparable to its original velocity. It concerns two-body relaxation, rather than the much faster orbital evolution in a smooth gravitational potential. A collisionless stellar system requires this time to greatly exceed the duration being studied.

In the straight-line impulse approximation, an encounter at relative speed $u$ has transverse acceleration $Gmb/(b^2+u^2{t}^2{)}^{3/2}$. ThusA small-angle gravitational encounter needs $\Delta v_{\perp }\ll u$. The transition to order-one deflections is therefore $b\sim Gm/u^2$, ignoring factors such as the equal-mass relative deflection. For the spherical estimate $u\sim v$,This is the lower cutoff of the weak-scattering estimate; closer encounters actually occur and require strong-scattering treatment.

The number of encounters in time $T$ with impact parameters in $[b,b+db]$ is $2\pi bdbnuT$. Independent random transverse directions make mean kicks cancel while their variances add. ConsequentlyEach logarithmic interval contributes equally, giving the Coulomb logarithm in stellar dynamics. Defining a deflection time by variance $v^2$ and setting $u=v$ gives $v^3/(8\pi G^2{m}^{2}n\log \Lambda )$. The paper instead uses an order-one normalization three quarters of this estimate:Both have the same physical scaling. The specified characteristic speed, the word “comparable”, the velocity-distribution average and strong-encounter cutoff do not fix that numerical coefficient uniquely. The crude equal-speed impulse calculation must not be claimed to determine $3/(32\pi )$ exactly.

For a self-gravitating, approximately virialized system, the virial theorem gives $v^2\sim GNm/R$. With $n=3N/(4\pi R^3)$, $\Lambda \sim N$ and the stellar crossing time $t_{\mathrm{dyn}}=R/v$, substitution into the paper's convention givesAn externally dominated gravitational potential or a different structural constant changes this substitution. For an N-body simulation lasting $T=K{t}_{\mathrm{dyn}}$, a tolerable fractional velocity-squared diffusion $\epsilon \ll 1$ requires $T/t_{\mathrm{rel}}\lesssim \epsilon$, henceThere is no unique smallest $N$ without $K$, the error tolerance and the force prescription. For scale, equality at $t_{\mathrm{rel}}/t_{\mathrm{dyn}}=100$ occurs near $N=7.1\times 10^3$; $N=10^5$ gives about $1.1\times 10^3$ crossing times. A calculation lasting 100 crossing times therefore needs substantially more than the first threshold to have negligible relaxation. Gravitational softening can raise the effective cutoff and reduce artificial scattering, but it also sets the spatial force resolution.