We propose a simple method for the approximation of the integrand in the integral representation of the free energy of a system of particles with two-particle interactions admitting a Fourier expansion, based on which the equation of state of a with a double exponential potential liquid is obtained without laborious calculations. The found temperature dependences of the equilibrium thermodynamic properties appropriately describe the corresponding experimental data. The effectiveness of the approach is demonstrated on the example of a model with double Yukawa potential.
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The author expresses his sincere gratitude to Professor A.Yu. Zakharov for helpful advice, valuable comments, and discussion of the results of the article.
Translated by L. Trubitsyna
To prove the inequality
in which 0 < D < 1, 0 < d < 1, D < d, in region G1, x > 0, let us consider the function
where 0 ≤ ν ≤ 1, p > 0.
Differentiating function (23) twice with respect to ν, we get
It follows from this equation that the curve φ(ν) is concave, and, since φ(0) = φ(1) = 0, then φ(ν) < 0 at 0 < ν < 1, or, as follows from (23):
By assuming in (24) ν = D/d and p = xd, we arrive at the inequality
Taking the logarithm of (25), we obtain inequality (22), from which inequality (9) follows. In conclusion, it should be noted that the values used in the proof, ν = 0 (D = 0) and ν = 1 (d = 1), correspond to the upper and lower boundaries of the region G1; however, the actual values of δ and ε are contained inside G1, where D ∈ (0; 1) and d ∈ (0; 1).
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Loktionov, I.K. Equation of State for a Liquid with Double Exponential Potential. High Temp 59, 184–191 (2021). https://doi.org/10.1134/S0018151X21020073