The phase space distribution function (or phase space density) is supposed to be the probability density of finding a particle around a given phase space point. But, classically, through Hamilton's equations, the system's time evolution is completely determined once the initial conditions are specified. So for a 2D phase space, why isn't the distribution function always the same:
$$f(x,p,t)=\delta(x-x(t)) \ \delta(p-p(t))$$
I know that this thinking has to be wrong, and I am definitely confusing some things. I would like to ask for clarification.