Journal of Statistical Physics

, Volume 152, Issue 4, pp 599–618 | Cite as

Thermodynamics of Two-Component Log-Gases with Alternating Charges

  • Ladislav ŠamajEmail author


We consider a one-dimensional gas of positive and negative unit charges interacting via a logarithmic potential, which is in thermal equilibrium at the (dimensionless) inverse temperature β. In a previous paper [Šamaj, L. in J. Stat. Phys. 105:173–191, 2001], the exact thermodynamics of the unrestricted log-gas of pointlike charges was obtained using an equivalence with a (1+1)-dimensional boundary sine-Gordon model. The present aim is to extend the exact study of the thermodynamics to the log-gas on a line with alternating ± charges. The formula for the ordered grand partition function is obtained by using the exact results of the Thermodynamic Bethe ansatz. The complete thermodynamics of the ordered log-gas with pointlike charges is checked by a small-β expansion and at the collapse point β c =1. The inclusion of a small hard core around particles permits us to go beyond the collapse point. The differences between the unconstrained and ordered versions of the log-gas are pointed out.


Two-component log-gas Charge ordering Exact thermodynamics Thermodynamic Bethe ansatz 



I am grateful to Peter Forrester for directing my attention to works about equivalence between the grand partition functions of unconstrained and ordered log-gases. The support received from Grant VEGA No. 2/0049/12 is acknowledged.


  1. 1.
    Alastuey, A., Forrester, P.J.: Correlations in two-component log-gas systems. J. Stat. Phys. 81, 579–627 (1995) MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Anderson, P.W., Yuval, G.: Exact results in the Kondo problem: equivalence to a classical one-dimensional Coulomb gas. Phys. Rev. Lett. 23, 89–92 (1969) ADSCrossRefGoogle Scholar
  3. 3.
    Anderson, P.W., Yuval, G., Hamman, D.R.: Exact results in the Kondo problem. II. Scaling theory, qualitatively correct solution, and some new results on one-dimensional classical statistical models. Phys. Rev. B 1, 4464–4473 (1970) ADSCrossRefGoogle Scholar
  4. 4.
    Bazhanov, V.V., Lukyanov, S.L., Zamolodchikov, A.B.: Integrable structure of conformal field theory II. Q-operator and DDV equation. Commun. Math. Phys. 190, 247–278 (1997) MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Caldeira, A.O., Legget, A.J.: Path integral approach to quantum Brownian motion. Physica A 121, 587–616 (1983) MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Callan, C.G., Freed, D.: Phase diagram of the dissipative Hofstadter model. Nucl. Phys. B 374, 543–566 (1992) MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Cornu, F., Jancovici, B.: On the two-dimensional Coulomb gas. J. Stat. Phys. 49, 33–56 (1987) MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Cornu, F., Jancovici, B.: The electrical double layer: a solvabke model. J. Chem. Phys. 90, 2444–2452 (1989) ADSCrossRefGoogle Scholar
  9. 9.
    Fateev, V., Lukyanov, S.L., Zamolodchikov, A.B., Zamolodchikov, Al.B.: Expectation values of boundary fields in the boundary sine-Gordon model. Phys. Lett. B 406, 83–88 (1997) MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Fendley, P., Lesage, F., Saleur, H.: Solving 1D plasmas and 2D boundary problems using Jack polynomials and functional relations. J. Stat. Phys. 79, 799–819 (1995) MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Fendley, P., Ludwig, A.W.W., Saleur, H.: Exact nonequilibrium transport through point contacts in quantum wires and fractional quantum Hall devices. Phys. Rev. B 52, 8934–8950 (1995) ADSCrossRefGoogle Scholar
  12. 12.
    Fendley, P., Saleur, H.: Exact perturbative solution of the Kondo problem. Phys. Rev. Lett. 75, 4492–4495 (1995) ADSCrossRefGoogle Scholar
  13. 13.
    Fendley, P., Lesage, F., Saleur, H.: A unified framework for the Kondo problem and for an impurity in a Luttinger liquid. J. Stat. Phys. 85, 211–249 (1996) MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Forrester, P.J.: Charged rods in a periodic background: a solvable model. J. Stat. Phys. 42, 871–894 (1986) MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Forrester, P.J.: Positive and negative charged rods alternating along a line: exact results. J. Stat. Phys. 45, 153–169 (1986) MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Forrester, P.J.: Solvable isotherms for a two-component system of charged rods on a line. J. Stat. Phys. 51, 457–479 (1988) MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Forrester, P.J.: Exact results for correlations in a two-component log-gas. J. Stat. Phys. 59, 57–79 (1989) MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Forrester, P.J.: Yang-Lee theory and the conductor-insulator transition in asymmetric log-potential lattice gases. J. Stat. Phys. 60, 203–220 (1990) MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 5th edn. Academic Press, London (1994) zbMATHGoogle Scholar
  20. 20.
    Guinea, F., Hakim, V., Muramatsu, A.: Diffusion and localization of a particle in a periodic potential coupled to a dissipative environment. Phys. Rev. Lett. 54, 263–266 (1985) ADSCrossRefGoogle Scholar
  21. 21.
    Hansen, J.P., Viot, P.: Two-body correlations and pair formation in the two-dimensional Coulomb gas. J. Stat. Phys. 38, 823–850 (1985) MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Jancovici, B.: Pair correlation function in a dense plasma and pycnonuclear reactions in stars. J. Stat. Phys. 17, 357–370 (1977) ADSCrossRefGoogle Scholar
  23. 23.
    Kalinay, P., Šamaj, L.: Thermodynamic properties of the two-dimensional Coulomb gas in the low-density limit. J. Stat. Phys. 106, 857–974 (2002) zbMATHCrossRefGoogle Scholar
  24. 24.
    Kane, C.L., Fisher, M.P.A.: Transmission through barriers and resonant tunneling in an interacting one-dimensional electron gas. Phys. Rev. B 46, 15233–15262 (1992) ADSCrossRefGoogle Scholar
  25. 25.
    Martin, Ph.A.: Sum rules in charged fluids. Rev. Mod. Phys. 60, 1075–1127 (1988) ADSCrossRefGoogle Scholar
  26. 26.
    Šamaj, L., Travěnec, I.: Thermodynamic properties of the two-dimensional two-component plasma. J. Stat. Phys. 101, 713–730 (2000) zbMATHCrossRefGoogle Scholar
  27. 27.
    Šamaj, L.: Thermodynamic properties of the one-dimensional two-component log-gas. J. Stat. Phys. 105, 173–191 (2001) zbMATHCrossRefGoogle Scholar
  28. 28.
    Šamaj, L.: The statistical mechanics of the classical two-dimensional Coulomb gas is exactly solved. J. Phys. A, Math. Gen. 36, 5913–5920 (2003) ADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Schmid, A.: Diffusion and localization in a dissipative quantum system. Phys. Rev. Lett. 51, 1506–1509 (1983) ADSCrossRefGoogle Scholar
  30. 30.
    Schotte, K.D., Schotte, U.: Susceptibility pf the s-d model. Phys. Rev. B 4, 2228–2236 (1971) ADSCrossRefGoogle Scholar
  31. 31.
    Téllez, G.: Short-distance expansion of correlation functions for the charge-symmetric two-dimensional two-component plasma: exact results. J. Stat. Mech. 126, P10001 (2005) CrossRefGoogle Scholar
  32. 32.
    Téllez, G.: Equation of state in the fugacity format for the two-dimensional Coulomb gas. J. Stat. Phys. 126, 281–298 (2007) MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute of PhysicsSlovak Academy of SciencesBratislavaSlovakia

Personalised recommendations