# Are all discovered normal distribution in the physical world a result of central limit theorem?

Apart from the wave function in a harmonic well. Are all discovered normal distribution in the physical world a result of central limit theorem?

If it is the case, it may allow some reverse reasoning when we see another normal distribution. Though I guess it is already applied in some domain.

I didn't think it thoroughly, so it may be a nonsense question.

• According to Cunningham’s Law, it may be more effective to assert that all natural normal distributions arise from the CLT and see if anyone pushes back with a counterexample. – Chemomechanics Mar 31 '19 at 16:16

Normal distributions are found elsewhere too. jacob1729 noted one example of a Normal distribution resulting (in thermal equilibrium) from a quadratic energy, a very important scenario. Another interesting example is a quantum SHO's ground state, which is Normal in either $$x$$ or $$p$$-space; the reason is we have to solve $$\hat{a}|\psi\rangle=0$$.
The $$x$$ component of the velocity of an ideal gas is normally distributed for entirely different reasons, so no.
• @Chemomechanics No. If you had a gas of massless particles, the velocity is distributed like $e^{-|v|}$, but your logic would say it's also $e^{-v^2}$. – knzhou Apr 1 '19 at 16:43