## Abstract

The notion of ideal liquid for the number of degrees of freedom less than 2, i.e., *γ* < 0, is introduced. The values of the pressure *P* and of the compressibility factor *Z* on the spinodal in the negative pressure region for the van der Waals equation determine the value of *γ*, *γ*(*T*) < 0, for *μ* = 0. For \(T \leqslant \frac{{{3^3}}}{{{2^5}}}{T_c}\), a relationship with the van der Waals equation is established.

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## References

- 1.
V. P. Maslov,

*Perturbation Theory and Asymptotic Methods*(Izd. Moskov. Univ., Moscow, 1965; Dunod, Paris, 1972) [in Russian and French]. - 2.
B. Ya. Frenkel,

*Yakov Il’ich Frenkel*(Nauka, Moscow–Leningrad, 1966) [in Russian]. - 3.
L. D. Landau and E. M. Lifshits,

*Statistical Physics*(Nauka, Moscow, 1964) [in Russian]. - 4.
P. Erdős and J. Lehner, “The distribution of the number of summands in the partitions of a positive integer,” Duke Math. J.

**8**(2), 335–345 (1941). - 5.
V. P. Maslov and V. E. Nazaikinskii, “On the distribution of integer random variables related by a certain linear inequality: I,” Mat. Zametki

**83**(1–2), 232–263 (2008) [Math. Notes 83 (1–2), 211–237 (2008)]. - 6.
V. P. Maslov and V. E. Nazaikinskii, “On the distribution of integer random variables related by a certain linear inequality: II,” Mat. Zametki

**83**(3), 381–401 (2008) [Math. Notes 83 (3–4), 345–363 (2008)]. - 7.
A. M. Vershik, “Statistical mechanics of combinatorial partitions, and their limit shapes,” Funktsional. Anal. i Prilozhen.

**30**(2), 19–39 (1996) [Functional Anal. Appl.**30**(2), 90–105 (1996)]. - 8.
V. P. Maslov, “Old mathematical errors in statistical physics,” Russ. J. Math. Phys.

**20**(2), 214–229 (2013). - 9.
V. P. Maslov, “Distribution corresponding to classical thermodynamics,” Phys.Wave Phenom.

**23**(2), 81–95 (2015). - 10.
V. P. Maslov, “Undistinguishing statistics of objectively distinguishable objects: Thermodynamics and superfluidity of classical gas,” Math. Notes

**94**(5–6), 722–813 (2013). - 11.
A. I. Burshtein,

*Molecular Physics*(Nauka, Novosibirsk, 1986) [in Russian]. - 12.
V.P Maslov, “Gas–amorphous solid and liquid–amorphous solid phase transitions. Introduction of negative mass and pressure from the mathematical viewpoint,” Math. Notes 97 (3–4), 423–430 (2015).

- 13.
I. A. Kvasnikov,

*Thermodynamics and Statistical Physics: Theory of Equilibrium Systems*(URSS, Moscow, 2002), Vol. 2 [in Russian]. - 14.
V. E. Nazaikinskii, “On the asymptotics of the number of states for the Bose–Maslov gas,” Math. Notes

**91**(5–6), 816–823 (2012). - 15.
V. P. Maslov, “Threshold levels in economics and time series,” Math. Notes

**85**(3–4), 305–321 (2009). - 16.
V. P. Maslov, Threshold Levels in Economics, arXiv:0903.4783v2 [q-fin. ST] 3 Apr 2009.

- 17.
T. Tate, “A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions,” Asymptot. Anal.

**67**(1–2), 101–123 (2010).

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Maslov, V.P. Case of less than two degrees of freedom, negative pressure, and the Fermi—Dirac distribution for a hard liquid.
*Math Notes* **98, **138–157 (2015). https://doi.org/10.1134/S0001434615070123

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### Keywords

- number of degrees of freedom
- negative pressure
- Fermi—Dirac distribution
- hard liquid