Beyond the Boltzmann factor for corrections to scaling in ferromagnetic materials and critical fluids

Solid State and Materials


The Boltzmann factor comes from the linear change in entropy of an infinite heat bath during a local fluctuation; small systems have significant nonlinear terms. We present theoretical arguments, experimental data, and Monte-Carlo simulations indicating that nonlinear terms may also occur when a particle interacts directly with a finite number of neighboring particles, forming a local region that fluctuates independent of the infinite bath. A possible mechanism comes from the net force necessary to change the state of a particle while conserving local momentum. These finite-sized local regions yield nonlinear fluctuation constraints, beyond the Boltzmann factor. One such fluctuation constraint applied to simulations of the Ising model lowers the energy, makes the entropy extensive, and greatly improves agreement with the corrections to scaling measured in ferromagnetic materials and critical fluids.

PACS Finite-size systems 64.60.F– Equilibrium properties near critical points, critical exponents 64.60.De Statistical mechanics of model systems (Ising model, Potts model, field-theory models, Monte Carlo techniques, etc) 75.40.Mg Numerical simulation studies 


  1. 1.
    T.L. Hill, Nano Lett. 1, 111 (2001)CrossRefADSGoogle Scholar
  2. 2.
    R. Böhmer, R.V. Chamberlin, G. Diezemann, B. Geil, A. Heuer, G. Hinze, S.C. Kuebler, R. Richert, B. Schiener, H. Sillescu, H.W. Spiess, U. Tracht, M. Wilhelm, J. Non-Cryst. Solids 235–237, 1 (1998)CrossRefGoogle Scholar
  3. 3.
    R. Richert, J. Phys.: Condens. Matter 14, R703 (2002)CrossRefADSGoogle Scholar
  4. 4.
    V. Korenman, J. Appl. Phys. 57, 3000 (1985)CrossRefADSGoogle Scholar
  5. 5.
    O. Steinsvoll, T. Riste, Hyperfine Interactions 31, 267 (1986)CrossRefADSGoogle Scholar
  6. 6.
    R.V. Chamberlin, Phase Transitions 65, 169 (1998)CrossRefGoogle Scholar
  7. 7.
    L.D. Landau, E.M Lifshitz, Statistical Physics, Course of Theoretical Physics 3rd edn. (Pergamon Press, Oxford, 1980), Vol. 5, Chap. XIIGoogle Scholar
  8. 8.
    L. Onsager, Phys. Rev. 37, 405 (1931)MATHCrossRefADSGoogle Scholar
  9. 9.
    E.T. Jaynes, Ann. Rev. Phys. Chem. 31, 579 (1980)CrossRefGoogle Scholar
  10. 10.
    R.V. Chamberlin, G.H. Wolf, Eur. Phys. J. B 67, 495 (2009)CrossRefADSGoogle Scholar
  11. 11.
    B. Schiener, R. Böhmer, A. Loidl, R.V. Chamberlin, Science 274, 752 (1996)CrossRefADSGoogle Scholar
  12. 12.
    R.V. Chamberlin, Phys. Rev. Lett. 83, 5134 (1999)CrossRefADSGoogle Scholar
  13. 13.
    X.F. Shi, G.B. McKenna, Phys. Rev. Lett. 94, 157801 (2005)CrossRefADSGoogle Scholar
  14. 14.
    R.V. Chamberlin, B. Schiener, R. Böhmer, Mat. Res. Soc. Symp. Proc. 455, 117 (1997)Google Scholar
  15. 15.
    R. Richert, S. Weinstein, Phys. Rev. Lett. 97, 095703 (2006)CrossRefADSGoogle Scholar
  16. 16.
    R.V. Chamberlin, Phys. Rev. Lett. 82, 2520 (1999)CrossRefADSGoogle Scholar
  17. 17.
    M.R.H. Javaheri, R.V. Chamberlin, J. Chem. Phys. 125, 154503 (2006)CrossRefADSGoogle Scholar
  18. 18.
    R.V. Chamberlin, Nature 408, 337 (2000)CrossRefADSGoogle Scholar
  19. 19.
    R.V. Chamberlin, Phys. Lett. A 315, 313 (2003)MATHCrossRefADSGoogle Scholar
  20. 20.
    S. Srinath, S.N. Kaul, H. Kronmüller, Phys. Rev. B 59, 1145 (1999)CrossRefADSGoogle Scholar
  21. 21.
    P.M. Shand, J.G. Bohnet, J. Goertzen, J.E. Shield, D. Schmitter, G. Shelburne, D.L. Leslie-Pelecky, Phys. Rev. B 77, 184415 (2008)CrossRefADSGoogle Scholar
  22. 22.
    H.E. Nigh, S. Legvold, F.H. Spedding, Phys. Rev. 132, 1092 (1963)CrossRefADSGoogle Scholar
  23. 23.
    S. Arajs, R.V. Colvin, J. Appl. Phys. 32, 336S (1961)CrossRefADSGoogle Scholar
  24. 24.
    C. Hohenemser, N. Rosov, A. Kleinhammes, Hyperfine Interactions 49, 267 (1989)CrossRefADSGoogle Scholar
  25. 25.
    M.F. Collins, Magnetic critical scattering (Oxford Univ. Press, New York, 1989)Google Scholar
  26. 26.
    M. Seeger, S.N. Kaul, H. Kronmüller, R. Reisser, Phys. Rev. B 51, 12585 (1995)CrossRefADSGoogle Scholar
  27. 27.
    S.S.C. Burnett, S. Gartenhaus, Phys. Rev. B 43, 591 (1991)CrossRefADSGoogle Scholar
  28. 28.
    Y. Garrabos, C. Lecoutre, F. Palencia, B. Le Neindre, C. Erkey, Phys. Rev. E 77, 021116 (2008)CrossRefADSGoogle Scholar
  29. 29.
    K. Ried, Y. Millev, M. Fähnle, H. Kronmüller, Phys. Rev. B 49, 4315 (1994)CrossRefADSGoogle Scholar
  30. 30.
    G. Orkoulas, A.Z. Panagiotopoulos, M.E. Fisher, Phys. Rev. E 61, 5930 (2000)CrossRefADSGoogle Scholar
  31. 31.
    P. Butera, M. Comi, Phys. Rev. B 65, 144431 (2002)CrossRefADSGoogle Scholar
  32. 32.
    N. Menyuk, K. Dwight, T.B. Reed, Phys. Rev. B 3, 1689 (1971)CrossRefADSGoogle Scholar
  33. 33.
    C.C. Huang, J.T. Ho, Phys. Rev. B 12, 5255 (1975)CrossRefADSGoogle Scholar
  34. 34.
    J. Souletie, J.L. Tholence, Sol. St. Comm. 48, 407 (1983)CrossRefADSGoogle Scholar
  35. 35.
    S.N. Kaul, Phase Transitions 47, 23 (1994)CrossRefGoogle Scholar
  36. 36.
    I.A. Campbell, K. Hukushima, H. Takayama, Phys. Rev. B 76, 134421 (2007)CrossRefADSGoogle Scholar
  37. 37.
    M.A. Anisimov, J. Phys.: Condens. Matter 12, A451 (2000)CrossRefADSGoogle Scholar
  38. 38.
    T.R. Thurston, G. Helgesen, J.P. Hill, D. Gibbs, B.D. Gaulin, P.J. Simpson, Phys. Rev. B 49, 15730 (1994)CrossRefADSGoogle Scholar
  39. 39.
    N. Bernhoeft, A. Hiess, S. Langridge, A. Stunault, D. Wermeille, C. Vettier, G.H. Lander, M. Huth, M. Jourdan, H. Adrian, Phys. Rev. Lett. 81, 3419 (1998)CrossRefADSGoogle Scholar
  40. 40.
    P. Böni, J.L. Martinez, J.M. Tranquada, Phys. Rev. B 43, 575 (1991)CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of PhysicsArizona State UniversityTempeUSA
  2. 2.Department of Chemistry and BiochemistryArizona State UniversityTempeUSA

Personalised recommendations