Determinisitic system with a probablisitic initial condition Consider a deterministic system like a spring mass damper. Lets say we do not know the exact initial condition but we are given a probability distribution function (PDF), $p(x,v,t = 0)$ of the mass's initial state. How do I calculate the evolution of this PDF.
I tried imagining this is phase space. For a spring mass system the trajectories will the an ellipse, so if we were to plot the PDF with x and v on the horizontal plane and the probability density on the vertical axis I think the time evolution would look like the PDF surface rotating about the vertical axis (sort of). Is this correct and how can I formulate this mathematically?
 A: If I understand correctly, you want to find $p(x,\dot{x}, t)$ given $p(x,\dot{x}, t=0)$?
When the system is deterministic there is nothing interesting happening, if you impose some kind of a probability distribution on the initial conditions. Simply, you can think of the each possible initial condition evolving for itself. 
To mathematically describe this, consider the following approach using complex numbers (your physical variables are the real parts of all the complex variables):


*

*Represent $x$ as $x(t=0) = A e^{i\phi}$, this notation captures both $x$ and $\dot{x}$

*The time evolution is very simple to describe: $x(t) = A e^{i(\phi + \omega\cdot t)}$

*You can introduce the time evolution operator: $\hat{U}(t): x(0) \rightarrow x(t)$, for which you can easily deduce that $\hat{U}(t) = e^{i\omega\cdot t}$

*You can, then, find the inverse operator $\hat{U}^{-1}(t) = e^{-i\omega\cdot t}: x(t) \rightarrow x(0)$
Now, since this operator maps states in one-to-one correspondence you can write the following:
$$p(x(t),\dot{x}(t),t) = p(\hat{U}^{-1}x(0), \dot{\hat{U}^{-1}}x(0), t=0)$$
As you can see, this is very easy to determine. A mathematically more interesting question could be, what is $p(x, t)$ given $p(x, \dot{x}, t=0)$? In this case there is not only one possibility, but a large number of possible trajectories that end up at specific $x$ at some $t$. However, I am not aware of any theories involving such situations.
Lastly, I believe that your graphical interpretation is correct.
