  Gravitational collapse in cosmological context produces an intricate cosmic
web of voids, walls, filaments and nodes. The anisotropic nature of
collisionless collapse leads to the emergence of an anisotropic velocity
dispersion, or stress, that absorbs most of the kinetic energy after
shell-crossing. In this paper, we measure this large-scale velocity dispersion
tensor $\sigma^2_{ij}$ in $N$-body simulations using the phase-space
interpolation technique. We study the environmental dependence of the amplitude
and anisotropy of the velocity dispersion tensor field, and measure its spatial
correlation and alignment. The anisotropy of $\sigma^2_{ij}$ naturally encodes
the collapse history and thus leads to a parameter-free identification of the
four dynamically distinct cosmic web components. We find this purely dynamical
classification to be in good agreement with some of the existing classification
methods. In particular, we demonstrate that $\sigma^2_{ij}$ is well aligned
with the large-scale tidal field. We further investigate the influence of small
scale density fluctuations on the large scale velocity dispersion, and find
that the measured amplitude and alignments are dominated by the largest
perturbations and thus remain largely unaffected. We anticipate that these
results will give important new insight into the anisotropic nature of
gravitational collapse on large scales, and the emergence of anisotropic stress
in the cosmic web.
