In Riemanns monumental paper, he expresses a prime counting function as an inverse Mellin transform of the log of the function he analytically continued into the complex plane

$$\Pi(x) = \frac{1}{2\pi i} \int \log \zeta(s)\ x^s \frac{\mathrm{d}s}{s}$$

and the zeros of $\zeta$ are consequently of interest (Riemann hypothesis).

Associated quantities relate closely to concepts in statistical physics. Planck's law of for the spectral radiance ($B_\nu(\nu, T) = 2 h c^{-2}\frac{\nu^3}{e^{h\nu/k_\mathrm{B}T} - 1}$) Mellin transforms to the Riemann zeta function ($\zeta(s) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{x ^ {s-1}}{e ^ x - 1} \mathrm{d}x$) and physicists have their own polylog (mind the $z$) in the Fermi–Dirac integral.

Now the grand partition function $\mathcal{Z}$ relates microscopic statistical physics to thermodynamics via

$$-k_B T \ln \mathcal{Z} = \langle E \rangle - TS - \mu \langle N\rangle$$

and is often expressed as a power series in the fugacity $$z=\exp\left(\frac{\mu}{T}\right)$$ as $\mathcal{Z}(z) = \sum_{N_i} z^{N_i} Z(N_i).$

The zeros of $\mathcal{Z}$ in $z$ make the log $\ln \mathcal{Z}$ explode and this phenomenon is associated with phase transitions and people study and cook up result like the Lee-Yang-theorem. The fugacity as a parameter in $\mathbb C$, which controls the zeros, is actually on the cover of one of my favorite books. But I'd think the fugacity was defined without that context in mind.

What I wanted to know is when did "$z$" become a standard name for a //complex// variable and, more importantly, is it coincidence that the names fit?? Who names it and where there other names for the fugaciy?

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1 Answer 1


Yes, it is a coincidence.

The concept of fugacity (from the Latin for "fleetness, tendency to flee") was originally introduced by Gilbert Lewis in his 1901 paper "The Law of Physico-Chemical Change" for the pressure of an ideal gas which has the same chemical potential as a real gas. Lewis notated this with ψ, though these days the letter f is used.

The concept of complex fugacity was introduced by Lee and Lang in their 1952 paper "Statistical Theory of Equations of State and Phase Transitions". The complex fugacity plane was originally notated y, though as you note these days the letter z is used. I'm not sure precisely how old the use of z for a generic complex variable is but it dates to at least the 1890s (e.g. see here).

The concept of partition function was introduced by Planck in 1921. Its use of the letter Z comes from its original German name, Zustandssumme ("sum over states"). The English name is due to Darwin and Fowler (1922).


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