  Temperature-dependent Smoluchowski equations describe the ballistic
agglomeration. In contrast to the standard Smoluchowski equations for the
evolution of cluster densities with constant rate coefficients, the
temperature-dependent equations describe both -- the evolution of the densities
as well as cluster temperatures, which determine the aggregation rates. To
solve these equations, we develop a novel Monte Carlo technique based on the
low-rank approximation for the aggregation kernel. Using this highly effective
approach, we perform a comprehensive study of the phase diagram of the system
and reveal a few surprising regimes, including permanent temperature growth and
"density separation", with a large gap in the size distribution for middle-size
clusters. We perform classification of the aggregation kernels for the
temperature-dependent equations and conjecture the lack of gelation. The
results of our scaling analysis agree well with the simulation data.
