$$Z=\sum_i e^{-\beta E_i}$$ I am relatively new to statistical mechanics, and I am wondering if the individual energies ($E_i$) in the equation above are free energies associated with each microstate? If it is, would it be Gibb's Free Energy or Helmholtz's Free Energy?

  • $\begingroup$ It's not free energy, it's energy. $\endgroup$
    – valerio
    Feb 27, 2018 at 15:22

1 Answer 1


No, the $E_i$s are energies. In statistical mechanics it can be show (see for example here) that the Helmotz's free energy $F$ is related to the partition function of the system in the following way $$ F = -kT\log Z $$ and all other thermodynamic relations, e.g. that with the Gibbs's free energy, can be found starting from it.

  • $\begingroup$ Is it possible to obtain the free energy associated with a particular microstate, and not the entire ensemble? $\endgroup$
    – Astronomer
    Feb 27, 2018 at 17:05
  • $\begingroup$ @wanlei I'm not sure a possible definition on a single microstate would be meaningful, especially if I think at the relation between the free energy and the entropy of the system. $\endgroup$
    – ndrearu
    Feb 27, 2018 at 20:26

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