# Mayer and Virial Series at Low Temperature

- 188 Downloads
- 4 Citations

## Abstract

We analyze the Mayer and virial series (pressure as a function of the activity resp. the density) for a classical system of particles in continuous configuration space at low temperature. Particles interact via a finite range potential with an attractive tail. We propose physical interpretations of the Mayer and virial series’ radii of convergence, valid independently of the question of phase transition: the Mayer radius corresponds to a fast increase from very small to finite density, and the virial radius corresponds to a cross-over from monatomic to polyatomic gas. Our results are consistent with the Lee-Yang theorem for lattice gases and with the continuum Widom-Rowlinson model.

## Keywords

Classical statistical mechanics Mayer and virial series Phase transitions## Notes

### Acknowledgements

The author gratefully acknowledges very useful discussion with D. Ueltschi, D. Tsagkarogiannis, E. Presutti, E. Pulvirenti, B. Metzger and W. König, and also thanks E. Presutti for hospitality during a visit at the University of Rome “Tor Vergata”. This work was supported by the DFG Forschergruppe 718 “Analysis and stochastics in complex physical systems”.

## References

- 1.Au Yeung, Y., Friesecke, G., Schmidt, B.: Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape. Calc. Var. Partial Differ. Equ.
**44**, 81–100 (2012) zbMATHCrossRefGoogle Scholar - 2.Brydges, D., Federbush, P.: A new form of the Mayer expansion in classical statistical mechanics. J. Math. Phys.
**19**, 2064–2067 (1978) MathSciNetADSCrossRefGoogle Scholar - 3.Brydges, D., Martin, Ph.A.: Coulomb systems at low density: a review. J. Stat. Phys.
**96**, 1163–1330 (1999) MathSciNetADSzbMATHCrossRefGoogle Scholar - 4.Chayes, J.T., Chayes, L., Kotecký, R.: The analysis of the Widom-Rowlinson model by stochastic geometric methods. Commun. Math. Phys.
**172**, 551–569 (1995) ADSzbMATHCrossRefGoogle Scholar - 5.Collevecchio, A., König, W., Mörters, P., Sidorova, N.: Phase transitions for dilute particle systems with Lennard-Jones potential. Commun. Math. Phys.
**299**, 603–630 (2010) ADSzbMATHCrossRefGoogle Scholar - 6.Conlon, J.G., Lieb, E.H., Yau, H.-T.: The Coulomb gas at low temperature and low density. Commun. Math. Phys.
**125**, 153–180 (1989) MathSciNetADSzbMATHCrossRefGoogle Scholar - 7.Fefferman, C.L.: The atomic and molecular nature of matter. Rev. Mat. Iberoam.
**1**, 1–44 (1985) MathSciNetzbMATHCrossRefGoogle Scholar - 8.Hill, T.L.: Statistical Mechanics: Principles and Selected Applications. The McGraw-Hill Series in Advanced Chemistry. McGraw-Hill, New York (1956) zbMATHGoogle Scholar
- 9.Hill, T.L.: An Introduction to Statistical Thermodynamics. Addison-Wesley Series in Chemistry, Addison-Wesley, Reading–London (1960) Google Scholar
- 10.Jansen, S., König, W., Metzger, B.: Large deviations for cluster size distributions in a continuous classical many-body system. arXiv:1107.3670v1 [math.PR]
- 11.Lebowitz, J.L., Penrose, O.: Convergence of virial expansions. J. Math. Phys.
**5**, 841–847 (1964) MathSciNetADSCrossRefGoogle Scholar - 12.Lenard, A.: Exact statistical mechanics of a one-dimensional system with Coulomb forces. J. Math. Phys.
**2**, 682–693 (1961) MathSciNetADSzbMATHCrossRefGoogle Scholar - 13.Mayer, J.E., Mayer, M.G.: Statistical Mechanics. Wiley, New York (1940) zbMATHGoogle Scholar
- 14.Penrose, O.: Convergence of fugacity expansions for fluids and lattice gases. J. Math. Phys.
**4**, 1312–1320 (1963) MathSciNetADSzbMATHCrossRefGoogle Scholar - 15.Poghosyan, S., Ueltschi, D.: Abstract cluster expansion with applications to statistical mechanical systems. J. Math. Phys.
**50**, 053509 (2009, 17 pp.) MathSciNetADSCrossRefGoogle Scholar - 16.Pulvirenti, E., Tsagkarogiannis, D.: Cluster expansion in the canonical ensemble. arXiv:1105.1022v4 [math-ph]
- 17.Radin, C.: The ground state for soft disks. J. Stat. Phys.
**26**, 365–373 (1981) MathSciNetADSCrossRefGoogle Scholar - 18.Ruelle, D.: Statistical Mechanics: Rigorous Results. Benjamin, New York–Amsterdam (1969) zbMATHGoogle Scholar
- 19.Ruelle, D.: Existence of a phase transition in a continuous classical system. Phys. Rev. Lett.
**27**, 1040–1041 (1971) MathSciNetADSCrossRefGoogle Scholar - 20.Šamaj, L.: Widom-Rowlinson model (continuum and lattice). arXiv:0709.0617v1 [cond-mat.stat-mech]
- 21.Theil, F.: A proof of crystallization in two dimensions. Commun. Math. Phys.
**262**, 209–236 (2006) MathSciNetADSzbMATHCrossRefGoogle Scholar - 22.Widom, B., Rowlinson, J.S.: New model for the study of liquid-vapor phase transitions. J. Chem. Phys.
**52**, 1670–1684 (1970) ADSCrossRefGoogle Scholar - 23.Zeidler, E.: Applied Functional Analysis. Main Principles and Their Applications. Springer, New York (1995) zbMATHGoogle Scholar