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Forgive me if this question is too elementary for this site, but I am wondering if there is any governing principle that guide us in assigning continuous probability distributions? I have encountered many situations where the authors assign a probability distribution whose motivation is not clear to me. For concreteness, consider a classical oscillator whose starting time is completely unknown. We are given that we know the total energy of the system. Then $p(x)dx$ is given by $2dt/T$, where $p(x)dx$ is the probability to find the oscillating mass in region dx and T is the time period of oscillation. This makes sense, but I am able to provide an argument for why this is the expression for $p(x)dx$, rather than say $dx/L$, where L is the total distance between the two extreme positions of the oscillator.

I mean both $2dt/T$ and $dx/L$ satisfy the normalization condition, but why is $p(x)dx=2dt/T$ rather than $p(x)dx=dx/L$ ?

Kid
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1 Answers1

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Probability of state being in some region of phase space is proportional to size of that region: probability is $p(x)dx$ where $p(x)$ is some function of $x$ and $dx$ is volume of the region.

That probability is also proportional to fraction of time the system will spend in that region of space. So it is $f(t)dt$ where $f(t)$ is some function of time and $dt$ is time it takes to move through that region. It is easy to show that for symmetric oscillation we have $f(t)dt = 2 dt/T$.

The reason why $f(t)$ is independent of $t$ is all time instants have the same probability, by assumption. But $p(x)$ is not independent of $x$ because all states $x$ do not have the same probability. The extreme points have greater probability because the system spends more time there. So it would be wrong to write $p(x)dx = cdx$ where $c$ is independent of $x$.

Ján Lalinský
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  • Thanks for the very clear answer. Would it be possible to point me to a resource where f(t)dt=2dt/T is proved for symmetric oscillations ? – Kid Jan 14 '21 at 16:16
  • It is obvious - probability for state to be in interval $x,x+dx$ is fraction of time $T$ spent there. When swinging to the left, the time fraction is $dt/T$, when swinging from the right, again $dt/T$. So total fraction of time is $2dt/T$. – Ján Lalinský Jan 14 '21 at 20:12