# 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
• What does it mean to discover a normal distribution? I'd rather say that no normal distributions 'exist' in the physical world; they exist in people's heads. – innisfree Mar 5 at 5:04
• I suppose then the question means, are there cases where something is well approximated by a normal distribution, that cannot be explained as originating form a large number of independent variables, such that a CLT result could be anticipated? – innisfree Mar 5 at 5:07

## 2 Answers

There are several ways to motivate measurement errors being Gaussian (Jaynes has a whole chapter on them, summarised here).

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.

• That distribution can’t be explained by applying the CLT to a large number of random kicks from previous collisions? – Chemomechanics Mar 31 '19 at 20:42
• @Chemomechanics if you can find a sensible way to make that work I'd be interested. It's possible but the barriers I see are: (1) each collision essentially randomises the velocities of both atoms which makes the "many collisions" idea seem wrong and (2) why should that apply to components and not to the speed? – jacob1729 Mar 31 '19 at 20:56
• @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
• For a relativistic treatment, see here. – J.G. Apr 1 '19 at 18:20