One-dimensional Calogero-Moser-Sutherland particle models with N particles can be regarded as diffusions on suitable subsets of $\mathbb R^N$ like Weyl chambers and alcoves with second order differential operators as generators which are singular on the boundaries of the state spaces. The most relevant examples are multivariate Bessel processes and Heckman-Opdam processes which are related to special functions associated with root systems. These models include Dyson's Brownian motions and multivariate Jacobi processes and, for fixed times, $\beta$-Hermite, Laguerre, and Jacobi ensembles. The processes depend on parameters which have the interpretation of an inverse temperature. We review several freezing limits for fixed N when one or several parameters tend to $\infty$. Usually, the limits are normal distributions and, in the process case, Gaussian processes where the parameters of the limit distributions are described in terms of solutions of ordinary differential equations which appear as frozen versions of the particle diffusions. We also discuss connections of these ODEs with the zeros of the classical orthogonal polynomials and polynomial solutions of some associated one-dimensional inverse heat equations.