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Applied Mathematics Research eXpress (2009) Vol. 2008 : article ID abn004, 36 pages, doi:10.1093/amrx/abn004 published on January 6, 2009
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Sampling Constraints in Average: The Example of Hugoniot Curves

Jean-Bernard Maillet1 and Gabriel Stoltz

1 CEA/DAM, BP 12, 91680 Bruyères-le-Châtel, France
2 Université Paris Est, CERMICS, Projet MICMAC ENPC-INRIA, 6 & 8 Av. Pascal, 77455 Marne-la-Vallée Cedex 2, France

Correspondence: Correspondence to be sent to: stoltz{at}cermics.enpc.fr

We present a method for sampling microscopic configurations of a physical system distributed according to a canonical (Boltzmann–Gibbs) measure, with a constraint holding in average. Assuming that the constraint can be controlled by the volume and/or the temperature of the system, and considering the control parameter as a dynamical variable, a sampling strategy based on a nonlinear stochastic process is proposed. Convergence results for this dynamics are proved using entropy estimates. As an application, we consider the computation of points along the Hugoniot curve, which are equilibrium states obtained after equilibration of a material heated and compressed by a shock wave.



References

  1. Allen M. P., Tildesley D. J. Computer Simulation of Liquids (1987) Oxford: Oxford University Press.
  2. Arnold A., Markowich P. A., Toscani G., Unterreiter A. On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck-type equations. Communications in Partial Differential Equations (2001) 26(1):43–100.
  3. Bakry D., Emery M. Diffusions hypercontractives. Séminaire de Probabilités (1985) 19. Berlin: Springer. 177–206. Lecture Notes in Mathematics 1123.
  4. Balian R. From Microphysics to Macrophysics: Methods and Applications of Statistical Physics (2007) 1–2. Berlin: Springer.
  5. Brennan J. K., Rice B. M. Efficient determination of Hugoniot states using classical molecular simulation techniques. Molecular Physics (2003) 101(22):3309–22.
  6. Brünger A., Brooks C. L., Karplus M. Stochastic boundary conditions for molecular-dynamics simulations of ST2 water. Chemical Physics Letters (1984) 105(5):495–500.[CrossRef]
  7. Cancès E., Legoll F., Stoltz G. Theoretical and numerical comparison of some sampling methods for molecular dynamics. Mathematical Modelling and Numerical Analysis (2007) 41(2):351–90.
  8. Frenkel D., Smit B. Understanding Molecular Simulation: From Algorithms to Applications (2002) 2nd ed. San Diego, CA: Academic Press.
  9. Gross L. Logarithmic Sobolev inequalities. American Journal of Mathematics (1975) 97(4):1061–83.
  10. Guionnet A., Zegarlinski B. Lectures on logarithmic Sobolev inequalities. Séminaire de Probabilités (2003) 36. Berlin: Springer. 1–134. Lecture Notes in Mathematics 1801.
  11. Hastings W. K. Monte Carlo sampling methods using Markov chains and their applications. Biometrika (1970) 57:97–109.[Abstract/Free Full Text]
  12. Holley R., Stroock D. Logarithmic Sobolev inequalities and stochastic Ising models. Journal of Statistical Physics (1987) 46(5–6):1159–94.
  13. Hoover W. G. Canonical dynamics: Equilibrium phase-space distributions. Physical Review A (1985) 31(3):1695–97.[CrossRef][Medline]
  14. Kato T. Abstract evolution equations of parabolic type in Banach and Hilbert spaces. Nagoya Mathematical Journal (1961) 19:93–125.
  15. Maillet J. B., Mareschal M., Soulard L., Ravelo R., Lomdahl P. S., Germann T. C., Holian B. L. Uniaxial Hugoniot: A method for atomistic simulations of shocked materials. Physical Review E (2000) 63. 016121.
  16. Metropolis N., Rosenbluth A. W., Rosenbluth M. N., Teller A. H., Teller E. Equations of state calculations by fast computing machines. Journal of Chemical Physics (1953) 21(6):1087–91.[CrossRef]
  17. Nosé S. A unified formulation of the constant temperature molecular-dynamics methods. Journal of Chemical Physics (1984) 81(1):511–19.[CrossRef]
  18. Shardlow T. Splitting for dissipative particle dynamics. SIAM Journal on Scientific Computing (2003) 24(4):1267–82.
  19. Skeel R. D., Izaguirre J. A. An impulse integrator for Langevin dynamics. Molecular Physics (2002) 100(24):3885–91.
  20. Wang W., Skeel R. D. Analysis of a few numerical integration methods for the Langevin equation. Molecular Physics (2003) 101(14):2149–56.
  21. Zeidler E. Nonlinear Functional Analysis and its Applications 1: Fixed-Point Theorems (1986) Berlin: Springer.

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This Article
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