# Define statistical potential energy [closed]

I am looking for a statistical method that closely relates to the idea of potential energy.

Here is a quick google definition for potential energy

"...the energy possessed by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors."

The classic example is the gravitational potential energy, for example: the higher you lift a brick - the higher the ability to do work, or apply a force on another object.

Now here is a similar example in statistics. Let's say that we are studying lung cancer and we discriminate on how many cigarettes people used to smoke. If one smokes two packs a day - it is four times more likely to get lung cancer. If one smokes three packs a day - than it is nine times more likely; hence, we can model the "statistical potential" with the function y = x^2 where x - is the number of smokes.

Note that I am not looking for a probability... it must be a positive number from zero to infinity.

The closest thing I have found that resembles my idea is Relative Risk but typically is a binary choice - the group either smokes or it doesn't.

Edit

What is the continuous version or Relative Risk or Odds Ratio?

Edit 2 Here is my take if RR and OR was not a thing

Let A be the event one smoke x-number of cigarettes Let B the event one gets lung diseases Let x be the number of cigs one smokes in a fixed time period

By bayes rule we have

$$P(A|B) = \frac{P(B|A)P(B)}{ \intop\nolimits_{a}^{b} P(B|A)P(A)dA }$$

But we need some kind of partial expectation where we can see the difference of smoking 0 to x-many cigarettes. Then

$$V(x) = \intop\nolimits_{0}^{x} x*P(A|B) dx$$

Where $$V(x)$$ we can call the statistical potential of smoking x-cigarettes

Of course we can pick proper distributions to model the events.

Now - we can see the magnitude of the expectation for $$x_1 < x_2 < max_x$$

This really sums up my idea of the statistical potential.

Is this acceptable?

Perhaps this should be all the way around - the expectation should be on the consequence but it makes the point

Edit 3

I think I have convinced myself that RR is the way to go about. Here is the deal

Let $$RR(1)$$ = P(disease|smoke 1 cigs) / P(disease|smoke 0 cigs)

Let $$RR(2)$$ = P(disease|smoke 2 cigs) / P(disease|smoke 0 cigs) and so on

Then $$V(x) = RR(x)$$ = P(disease|smoke x cigs) / P(disease|smoke 0 cigs)

One can fit a function between the pairs such as in my example $$y=f(x)=V(x)$$

I want credit for coining the term "statistical potential energy" :)

## closed as unclear what you're asking by Michael Chernick, user158565, mkt, kjetil b halvorsen, Peter Flom♦Feb 7 at 11:14

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• What's wrong with odds ratio or relative risk? – Peter Flom Feb 7 at 11:13
• @PeterFlom Nothing wrong besides the fact that they are not continuous - I am asking for confirmation that RR fits my idea of potential energy. – Edv Beq Feb 7 at 12:06
• They are continuous. The OR and the RR can take any value from 0 to infinity. – Peter Flom Feb 7 at 12:09
• @PeterFlom okay thank you - which one do you think is more appropriate? – Edv Beq Feb 7 at 12:15
• I don't know. They are usually similar. – Peter Flom Feb 7 at 21:35

If you have a coordinate $$x$$ that separates the states of your system, and you can assign a probability $$P(x)$$ to finding your system at coordinate value $$x$$, then you can define a kind of energy as follows. Define $$E(x)$$ such that $$P(x) \sim \exp(-\beta E(x))$$ or $$E(x) = -\log(P(x)/\beta + E_0$$, where $$\beta$$ is a constant that can be ignored for now. That gets you an energy-like thing.
But how would you use it? Would you try to define a force $$F=-dE(x)/dx$$? Would you like to say that the average energy $$$$ is a constant of the system given the constancy of several intensive features, like $$\beta$$? Would you try to define dynamics in $$x$$, or kinetics? or diffusion? I'm genuinely curious where you are going with this . . .