I have come across central limit theorem in statistical mechanics texts. But I'm not being able to fathom its importance. Can someone explain with an example that appeal to a physics student?
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$\begingroup$ thermal noise in a resistor/amplifier/radio/tv ... $\endgroup$– hyportnexCommented May 15, 2017 at 14:57
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$\begingroup$ Renormalized actions on the lattice. $\endgroup$– Cosmas ZachosCommented May 15, 2017 at 15:03
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2$\begingroup$ This seems to be an open-ended list question with no identifiable correct answer, which are usually viewed as too broad. $\endgroup$– ACuriousMind ♦Commented May 15, 2017 at 16:17
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There are many examples you could give, I guess. I would like to contribute with a very simple one that you may find interesting: diffusion.
Consider a particle sitting at $x=0$ at time $t=0$. We work in discrete time and space. At every timestep $\delta t, 2\delta t,$ etc, the new position of the particle is given by:
$$ x_{t+\delta t} = x_t + S\delta x$$
where $S$ is a random variable which is +1 with probability 1/2 and -1 with probability 1/2. Clearly, after $n$ timesteps, starting at 0
$$x_{n\delta t} = (S_1 + S_2 + ... + S_n)\delta x$$
the $S_i$ are independent, identically distributed and have finite variance. Therefore the quantity in brackets tends to a Gaussian when $n$ is large. If we go to the continuum limit by having $\delta t \to 0$ and $\delta x \to 0$, the probability of our particle being at location $x$ at time $t$ is Gaussian, with variance increasing with time, whence the diffusion.
Note that this result can also be found using the diffusion partial differential equation, which is a completely different way and may be interesting for teaching purposes.
Apologies for not providing all mathematical details.
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