We study the evolution of phase-space density during the hierarchical structure formation of \Lambda CDM halos .
We compute both a spherically-averaged surrogate for phase-space density ( Q = \rho / \sigma ^ { 3 } ) and the coarse-grained distribution function f ( \mathbf { x } , \mathbf { v } ) for dark matter particles that lie within \sim 2 virial radii of four Milky-Way-sized dark matter halos .
The estimated f ( \mathbf { x } , \mathbf { v } ) spans over four decades at any radius .
Dark matter particles that end up within two 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 f _ { peak } corresponding to the peak of the Gaussian decreases as the evolution progresses and is well described by f _ { peak } ( z ) \propto ( 1 + z ) ^ { 4.3 \pm 1.1 } .
The decrease is due both to the dynamical mixing as the matter accreted by halos is virialized , and due to the overall decrease of space-density of the unprocessed material due to expansion of the Universe .
The highest values of f , ( responsible for the skewness of the profile ) are found at the centers of dark matter halos and subhalos , where f can be an order of magnitude higher than in the center of the main halo .
We confirm that Q ( r ) can be described by a power-law with the slope of \beta = -1.8 \pm 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 \equiv \sigma ^ { 2 } / \rho _ { dm } ^ { 2 / 3 } \propto r ^ { 1.2 } ) , which dark matter acquires as it is accreted onto a growing halo .
The estimated f ( \mathbf { x } , \mathbf { v } ) , on the other hand , exhibits a more complicated behavior .
Although the median coarse-grained phase-space density profile F ( r ) can be approximated by a power-law , \propto r ^ { -1.6 \pm 0.15 } , in the inner regions of halos ( < 0.6 r _ { vir } ) , 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 subhalos , as well as relatively unmixed streams resulting from disrupted subhalos , which constitute a sizeable fraction of matter at large radii .