A version of the virial theorem, which takes into account the effects of the noncompact extra dimensions, is derived in the framework of the brane world models. In the brane world scenario, the four-dimensional effective Einstein equation has some extra terms, called dark radiation and dark pressure, respectively, which arise from the embedding of the 3-brane in the bulk. To derive the generalized virial theorem, we use a method based on the collisionless Boltzmann equation. The dark radiation term generates an equivalent mass term (the dark mass), which gives an effective contribution to the gravitational energy. This term may account for the well-known virial theorem mass discrepancy in actual clusters of galaxies. An approximate solution of the vacuum field equations on the brane, corresponding to weak gravitational fields, is also obtained, and the expressions for the dark radiation and dark mass are derived. The qualitative behavior of the dark mass is similar to that of the observed virial mass in clusters of galaxies. We compare our model with the observational data for galaxy clusters, and we express all the physical parameters of the model in terms of observable quantities. In particular, we predict that the dark mass must extend far beyond the presently considered virial radius. The behavior of the galaxy cluster velocity dispersion in brane world models is also considered. Therefore the study of the matter distribution and velocity dispersion at the extragalactic scales could provide an efficient method for testing the multidimensional physical models.

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