Skip to main content

Advertisement

Log in

Controlled dynamics on energy landscapes

The European Physical Journal B Aims and scope Submit manuscript

Abstract

In systems with complex multi-minima energy landscapes, it is often not only the global minimum which is of great importance. For example, in materials science, metastable compounds corresponding to local minima on the landscape play a crucial role in many technological applications. In order to reach such modifications, both in computational and real world situations, it is necessary to optimally control the dynamics of the system on the landscape. We present a general method, how to design optimal temperature schedules for reaching particular basins on a complex landscape, by constructing a coarse-grained transition probability matrix from stochastic global landscape explorations, and subsequently using optimal control techniques on the Master equation describing the dynamics on the simplified energy landscape. As a demonstration example, the landscape of MgF2 is considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D.J. Wales, Energy Landscapes with Applications to Clusters, Biomolecules and Glasses (Cambridge University Press, Cambridge, 2004)

  2. J.C. Schön, M. Jansen, Int. J. Mat. Res. 100, 135 (2009)

    Article  Google Scholar 

  3. S.H. Northrup, J.T. Hynes, J. Chem. Phys. 73, 2700 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  4. R.F. Grote, J.T. Hynes, J. Chem. Phys. 73, 2715 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  5. H. Grubmüller, Phys. Rev. E 52, 2893 (1995)

    Article  ADS  Google Scholar 

  6. J.C. Schön, M. Jansen, Z. Kristallogr. 216, 307 (2001)

    Article  Google Scholar 

  7. J.C. Schön, M. Jansen, Z. Kristallogr. 216, 361 (2001)

    Article  Google Scholar 

  8. J.-P. Aubin, A. Lesne, J. Math. Phys. 46, 043508 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  9. S.R. Williams, D.J. Evans, J. Chem. Phys. 127, 184101 (2007)

    Article  ADS  Google Scholar 

  10. J.C. Schön, M.A.C. Wevers, M. Jansen, J. Phys.: Condens. Matter 15, 5479 (2003)

    Article  ADS  Google Scholar 

  11. K.H. Hoffmann, J.C. Schön, Found. Phys. Lett. 18, 171 (2005)

    Article  MATH  Google Scholar 

  12. J.C. Schön, M. Jansen, Angew. Chem. Int. Ed. 35, 1286 (1996)

    Article  Google Scholar 

  13. S.M. Woodley, R. Catlow, Nat. Mater. 7, 937 (2007)

    Article  ADS  Google Scholar 

  14. A.-C. Garcia, in Proceedings of Les Houches School on Nonlinear Excitations in Biomolecules, Les Houches, 1994, edited by M. Peyrard (Springer, Berlin, 1994), pp. 191–208

  15. S. Govindarajan, R.A. Goldstein, Proteins: Struct. Funct. Gen. 29, 461 (1997)

    Article  Google Scholar 

  16. S. Govindarajan, R.A. Goldstein, Biopolymers 42, 427 (1997)

    Article  Google Scholar 

  17. S.V. Krivov, M. Karplus, Proc. Natl. Acad. Sci. 101, 14766 (2004)

    Article  ADS  Google Scholar 

  18. Y. Fukunishi, Proteins: Struct. Funct. Gen. 33, 408 (1998)

    Article  Google Scholar 

  19. H. Frauenfelder, P.W. Fenimore, R.D. Young, IUBMB Life 59, 506 (2007)

    Article  Google Scholar 

  20. P. Sibani, Physica A 258, 249 (1998)

    Article  ADS  Google Scholar 

  21. T. Komatsuzaki, K. Hoshino, Y. Matsunaga, Regularity in Chaotic Transitions on Multibasin Landscapes, in Advances in Chemical Physics, Part B, edited by M. Toda, T. Komatsuzaki, T. Konishi, R. Stephen Berry, S.A. Rice (Applications to Chemical Reaction Dynamics in Complex Systems, Wiley, New York, 2005), Vol. 130, pp. 257–313

  22. P. Sibani, J.C. Schön, P. Salamon, J.O. Andersson, Europhys. Lett. 22, 479 (1993)

    Article  ADS  Google Scholar 

  23. P. Salamon, P. Sibani, R. Frost, Facts, Conjectures, and Improvements for Simulated Annealing, in Monographs on Mathematical Modeling and Computation, 1st edn. (SIAM, Philadelphia, 2002), Vol. 7

  24. M.A.C. Wevers, J.C. Schön, M. Jansen, J. Phys.: Condens. Matter 11, 6487 (1999)

    Article  ADS  Google Scholar 

  25. C.-B. Li, Y. Matsunaga, M. Toda, T. Komatsuzaki, J. Chem. Phys. 123, 184301 (2005)

    Article  ADS  Google Scholar 

  26. K.H. Hoffmann, P. Sibani, Phys. Rev. A 38, 4261 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  27. P. Sibani, K.H. Hoffmann, Phys. Rev. Lett. 63, 2853 (1989)

    Article  ADS  Google Scholar 

  28. C. de Groot, D. Würtz, K.H. Hoffmann, in Parallel Problem Solving from Nature, edited by H.P. Schwefel, R. Maenner (Springer-Verlag, Berlin, 1991), pp. 445–454

  29. A. Fischer, K.H. Hoffmann, J.C. Schön, J. Phys. A 44, 1 (2011)

    Article  Google Scholar 

  30. M. Jansen, J.C. Schön, Angew. Chem. Int. Ed. 45, 3406 (2006)

    Article  Google Scholar 

  31. B. Andresen, K.H. Hoffmann, K. Mosegaard, J. Nulton, J.M. Pedersen, P. Salamon, J. Phys. 49, 1485 (1988)

    Article  Google Scholar 

  32. P. Salamon, J.D. Nulton, J.R. Harland, J. Pedersen, G. Ruppeiner, L. Liao, Comput. Phys. Commun. 49, 423 (1988)

    Article  ADS  Google Scholar 

  33. K.H. Hoffmann, D. Würtz, C. de Groot, M. Hanf, Concepts in Optimizing Simulated Annealing Schedules: an Adaptive Approach for Parallel and Vector Machines, in Parallel and Distributed Optimization, edited by M. Grauer, D.B. Pressmar (Springer-Verlag, Berlin, Heidelberg, New York, 1991), pp. 154–175

  34. J. Lässig, K.H. Hoffmann, Phys. Rev. E 79, 046702 (2009)

    Article  ADS  Google Scholar 

  35. Finite-Time Thermodynamics and Thermoeconomics, edited by S. Sieniutycz, P. Salamon (Taylor and Francis, New York, 1990)

  36. R.E. Kunz, P. Blaudeck, K.H. Hoffmann, R.S. Berry, J. Chem. Phys. 108, 2576 (1998)

    Article  ADS  Google Scholar 

  37. J.C. Schön, Ber. Bunsenges. 100, 1388 (1996)

    Article  Google Scholar 

  38. J.C. Schön, H. Putz, M. Jansen, J. Phys.: Condens. Matter 8, 143 (1996)

    Article  ADS  Google Scholar 

  39. A.R. West, Solid State Chemistry and Its Applications (Wiley, New York, 1984)

  40. A. Bach, D. Fischer, X. Mu, W. Sigle, P.A. van Aken, M. Jansen, Inorg. Chem. 50, 1563 (2011)

    Article  Google Scholar 

  41. M.A.C. Wevers, J.C. Schön, M. Jansen, J. Solid State Chem. 136, 223 (1998)

    Article  Google Scholar 

  42. M.A.C. Wevers, J.C. Schön, M. Jansen, J. Phys. A 34, 4041 (2001)

    Article  ADS  MATH  Google Scholar 

  43. M.A.C. Wevers, Energetische und entropische Aspekte der Energielandschaften von MgF2, CaF2 und Li x Na6−x N2 ( x = 0,1,...,6) sowie ein Vergleich mit ab-initio-Rechnungen, Ph.D. thesis, University of Bonn, 1999

  44. K.H. Hoffmann, P. Salamon, Appl. Math. Lett. 22, 1471 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. K.H. Hoffmann, P. Salamon, Physica A 390, 3086 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  46. V.F. Krotov, Control and Cybernetics 17, 115 (1988)

    MathSciNet  MATH  Google Scholar 

  47. K. Ergenzinger, K.H. Hoffmann, P. Salamon, J. Appl. Phys. 77, 5501 (1995)

    Article  ADS  Google Scholar 

  48. A. Franz, K.H. Hoffmann, J. Comput. Phys. 176, 196 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. M. Santoro, J.C. Schön, M. Jansen, Phys. Rev. E 76, 1 (2007)

    Article  Google Scholar 

  50. J.C. Schön, Z. Anorg. Allg. Chem. 635, 1794 (2009)

    Article  Google Scholar 

  51. F. Heilmann, K.H. Hoffmann, Europhys. Lett. 70, 155 (2005)

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Karl Heinz Hoffmann.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hoffmann, K.H., Schön, J.C. Controlled dynamics on energy landscapes. Eur. Phys. J. B 86, 220 (2013). https://doi.org/10.1140/epjb/e2013-31042-4

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjb/e2013-31042-4

Keywords

Navigation