In statistical thermodynamics we write $$f(v)\,dv = f(E)\,dE$$

where $v$ is velocity and $E= \frac12mv^2$ is energy

and $f$ refers to the distribution function

Can someone explain the logic behind?

  • $\begingroup$ Abuse of notation $f(v):=f(E(v))$; what happens here is a simple matter of substitution, that is $g:=f\circ\alpha$ with $\alpha(v):=\frac{1}{2}mv^2$ then $f(E)dE=f(\alpha(v))\cdot\alpha'(v)dv=g(v)\cdot\alpha'(v)dv$. $\endgroup$ Jul 6, 2014 at 6:49

1 Answer 1


The distribution function has the property: $f(v) = \frac{dn}{ndv} $ & $f(E) = \frac{dn}{ndE} $ Hence $f(v)dv=f(E)dE=\frac{dn}{n}$


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