Variational Methods in Statistical Thermodynamics—A Pedagogical Introduction

  • Zhen-Gang WangEmail author
Part of the Molecular Modeling and Simulation book series (MMAS)


This chapter presents a pedagogical introduction to the variational methods in statistical thermodynamics. We start with some general considerations of the variational nature of thermodynamics, which is rooted in the second law, and manifested in the maximum-term method in the evaluation of the partition function in statistical mechanics. We present two common mathematical variational techniques, one based on the Gibbs-Bogoliubov-Feynman (GBF) variational bound and one based on the saddle-point (or steepest-descent) method. We illustrate the use of these techniques in the derivation of the mean-field theory for Ising model and the Poisson-Boltzmann equation. We also show that the GBF method provides a self-consistent treatment of fluctuation effects in weakly correlated systems.


Free Energy Partition Function Saddle Point Helmholtz Free Energy Boltzmann Weight 
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  1. 1.
    The uncertainty principle also takes the form of an inequality, but it can be deduced from the fundamental equations of quantum mechanics, e.g., the Schrödinger equationGoogle Scholar
  2. 2.
    H.B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd edn. (Wiley, New York, 1985)zbMATHGoogle Scholar
  3. 3.
    A.H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981)zbMATHGoogle Scholar
  4. 4.
    T.L. Hill, Introduction to Statistical Thermodynamics (Dover, New York, 1986)Google Scholar
  5. 5.
    A. Ishihara, J. Phys. A Gen. Phys. 1, 539 (1968)CrossRefGoogle Scholar
  6. 6.
    R.P. Feynman, Statistical Mechanics: A Set of Lectures (Addison- Wesley, Redwood, 1972)zbMATHGoogle Scholar
  7. 7.
    H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, 2nd edn. (World Scientific, Singapore, 1995)CrossRefzbMATHGoogle Scholar
  8. 8.
    D. Amit, Field Theory, Critical Phenomena and Renormalization Group, 2nd edn. (World Scientific, Singapore, 1984)Google Scholar
  9. 9.
    P.-G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979)Google Scholar
  10. 10.
    G.H. Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers (Oxford University Press, New York, 2005)CrossRefzbMATHGoogle Scholar
  11. 11.
    A.L. Fetter, J.D. Walecka, Quantum Theory for Many-Particle Systems (Dover, Mineola, 2003)Google Scholar
  12. 12.
    J.-P. Hansen, I.R. McDonald, Theory of Simple Liquids, 2nd edn. (Academic Press, London, 1990)zbMATHGoogle Scholar
  13. 13.
    P.M. Chaikin, T.C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, 2000)Google Scholar
  14. 14.
    R.D. Coalson, A. Duncan, J. Chem. Phys. 97, 5653 (1992)CrossRefGoogle Scholar
  15. 15.
    R.R. Netz, H. Orland, Euro. Phys. J. E 1, 203 (2000)CrossRefGoogle Scholar
  16. 16.
    Because of the complex nature of the action in Eq. 8.23, the GBF bound does not hold, i.e., the true free energy is not always approached from above. D. Frydel has derived an set of exact relations based on the dual representation of the system—the original physical representation and the field-transformed representation. In the case of a two-parameter Gaussian reference, it is shown that the first two relations yield results identical to those by taking the stationary point of the variational action. See: D. Frydel, Eur. J. Phys.36, 065050 (2015)Google Scholar
  17. 17.
    R.R. Netz, H. Orland, Eur. Phys. J. E 11, 310 (2003)CrossRefGoogle Scholar
  18. 18.
    M.M. Hatlo, R.A. Curtis, L. Lue, J. Chem. Phys. 128, 164717 (2008)CrossRefGoogle Scholar
  19. 19.
    Z.-G. Wang, Phys. Rev. E 81, 021501 (2010)CrossRefGoogle Scholar
  20. 20.
    This can be understood by noting that the electrostatic energy of a charged body is \(\frac{1}{2} \int d {\bf r} \rho ({\bf r}) \psi ({\bf r})\) and the electrostatic potential is in turn \(\psi ({\bf r})=\int d {\bf r}^{\prime } G ({\bf r}, {\bf r}^{\prime }) \rho ({\bf r}^{\prime })\). For a point charge at \({\bf r}_1\), we then obtain the energy as \(\frac{1}{2}G({\bf r}_1,{\bf r}_1)\) Google Scholar
  21. 21.
    R. Wang, Z.-G. Wang, J. Chem. Phys. 142, 104705 (2015)CrossRefGoogle Scholar
  22. 22.
    R. Wang, Z.-G. Wang, J. Chem. Phys. 139, 124702 (2013)CrossRefGoogle Scholar
  23. 23.
    M. Doi, S.F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, New York, 1986)Google Scholar

Copyright information

© Springer Science+Business Media Singapore 2017

Authors and Affiliations

  1. 1.Division of Chemistry and Chemical EngineeringCalifornia Institute of TechnologyPasadenaUSA

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