What is the physical meaning of fugacity? We know from the grand canonical ensemble formalism, the grand partition function is defined as
$$
\mathcal{Q} = \sum\limits_s e^{-\beta\varepsilon_s -\alpha N_s} = \sum\limits_N Z^N \sum\limits_s e^{-\beta\varepsilon_s} = \sum\limits_N Z^N Q_N
$$
where $Q_N$ is the partition function for a canonical ensemble consists of systems with $N$ particles, and $Z = e^{-\alpha}$ is defined as the fugacity. So my question is: what is the physical or statistical (if possible) meaning of fugacity. It seems to me that $Z$ is unitless, so I guess it only has a statistical meaning?
 A: This can be related to chemistry for a better understanding. If we have some species in an ideal solution, then we expect the chemical potential to depend on the concentration. However, in real life, we don't deal with ideal solutions, and instead the potential depends upon activity.
For example, due to ion-ion interactions and ion-$\mathrm{H_2O}$ interactions, the number of ions in a solution available for a reaction may actually be different than the number present. This is gauged by the activity $a$ rather than concentration.
For a real gas, the activity is the effective partial pressure and known as fugacity. It is precisely unity for an ideal gas, just as for an ideal solution the concentration and activity would be interchangeable because the activity coefficient $\gamma = 1$.

In summary, we have that the activity is,
$$a = \gamma x = \gamma \frac{c}{c_{\mathrm{std}}}$$
where $x$ is the molar fraction, $c$ is the concentration and $c_{\mathrm{std}}$ is the standard concentration. For a gas, we would define equivalently a dimensionless activity,
$$a = \frac{f}{p_{\mathrm{std}}} = \varphi y \frac{p}{p_{\mathrm{std}}}$$
where $f$ is the fugacity having dimensions of pressure as it is the effective partial pressure, $p_{\mathrm{std}}$ is standard pressure, $p$ the total pressure, $y$ the fraction of the gaseous mixture and $\varphi$ a dimensionless fugacity coefficient. Thus, when speaking of fugacity we may refer to it in the sense of the effective partial pressure $f$, the fugacity coefficient $\varphi$ or the activity $a$.
