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?


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.

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  • $\begingroup$ Thanks for you reply. I have one doubt though. Consider the spring mass damper system. Let the initial PDF be a normal distribution so that its integral over all of x and v equates to 1. The phase space trajectory of point is a spiral. This could imply that the PDF becomes thinner but not taller as p(0,0,t) is always the same. Wouldn't that imply that the integral of the PDF over all of x and v be less than 1? Is this example a wrong interpretation? $\endgroup$ – Rohit John Mar 24 at 15:28
  • $\begingroup$ I believe that does not change things much. Your time evolution operator would be: $e^{-\zeta \cdot t} e^{i\omega\cdot t}$. Thus, there is still one to one correspondence between the individual states at two different times. That literally means that if states A and B are in correspondence p(A) at t=0 is the same is p(B) at t=t'. Since all you can map all B states into some A states, and since you know that the total probability of A states is 1, the total probability of B states is also one. $\endgroup$ – Light Mar 24 at 19:12
  • $\begingroup$ You are technically right by saying that it does not get taller but that is only because the exponent itself is never zero. However, the density of probable points around zero gets larger and thus the the most of probability accumulates around (0,0) as the time passes. $\endgroup$ – Light Mar 24 at 19:14
  • $\begingroup$ I see. So then the PDF must be normalized each time we calculate it using the time evolution operator. I was wondering if there was a way to develop a partial differential equation for the PDF. In this case the time evolution operator easily solves the problem. However, how do you solve for a case when the dynamic system is governed by a non linear differential. Is there a way to incorporate this governing equation into a partial differential equation for the PDF? $\endgroup$ – Rohit John Mar 25 at 13:08

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