From statistical mechanics, for each degree of freedom \(q\) of a free system in thermal equilibrium at temperature \(T\), the corresponding thermal fluctuations of \(\dot{q}\) is given by

Here \(m\) is the mass associated in the kinetic energy expression written in terms of \(q\). Mapping this setup on the random brownian motion dynamics elaborated in the text, find a relation between \(T, \sigma\), and \(\alpha\); that is, a relation between temperature, random force strength, and friction. This is a form of the fluctuation-dissipation theorem.

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2. kT 2