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.
Similar content being viewed by others
References
D.J. Wales, Energy Landscapes with Applications to Clusters, Biomolecules and Glasses (Cambridge University Press, Cambridge, 2004)
J.C. Schön, M. Jansen, Int. J. Mat. Res. 100, 135 (2009)
S.H. Northrup, J.T. Hynes, J. Chem. Phys. 73, 2700 (1980)
R.F. Grote, J.T. Hynes, J. Chem. Phys. 73, 2715 (1980)
H. Grubmüller, Phys. Rev. E 52, 2893 (1995)
J.C. Schön, M. Jansen, Z. Kristallogr. 216, 307 (2001)
J.C. Schön, M. Jansen, Z. Kristallogr. 216, 361 (2001)
J.-P. Aubin, A. Lesne, J. Math. Phys. 46, 043508 (2005)
S.R. Williams, D.J. Evans, J. Chem. Phys. 127, 184101 (2007)
J.C. Schön, M.A.C. Wevers, M. Jansen, J. Phys.: Condens. Matter 15, 5479 (2003)
K.H. Hoffmann, J.C. Schön, Found. Phys. Lett. 18, 171 (2005)
J.C. Schön, M. Jansen, Angew. Chem. Int. Ed. 35, 1286 (1996)
S.M. Woodley, R. Catlow, Nat. Mater. 7, 937 (2007)
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
S. Govindarajan, R.A. Goldstein, Proteins: Struct. Funct. Gen. 29, 461 (1997)
S. Govindarajan, R.A. Goldstein, Biopolymers 42, 427 (1997)
S.V. Krivov, M. Karplus, Proc. Natl. Acad. Sci. 101, 14766 (2004)
Y. Fukunishi, Proteins: Struct. Funct. Gen. 33, 408 (1998)
H. Frauenfelder, P.W. Fenimore, R.D. Young, IUBMB Life 59, 506 (2007)
P. Sibani, Physica A 258, 249 (1998)
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
P. Sibani, J.C. Schön, P. Salamon, J.O. Andersson, Europhys. Lett. 22, 479 (1993)
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
M.A.C. Wevers, J.C. Schön, M. Jansen, J. Phys.: Condens. Matter 11, 6487 (1999)
C.-B. Li, Y. Matsunaga, M. Toda, T. Komatsuzaki, J. Chem. Phys. 123, 184301 (2005)
K.H. Hoffmann, P. Sibani, Phys. Rev. A 38, 4261 (1988)
P. Sibani, K.H. Hoffmann, Phys. Rev. Lett. 63, 2853 (1989)
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
A. Fischer, K.H. Hoffmann, J.C. Schön, J. Phys. A 44, 1 (2011)
M. Jansen, J.C. Schön, Angew. Chem. Int. Ed. 45, 3406 (2006)
B. Andresen, K.H. Hoffmann, K. Mosegaard, J. Nulton, J.M. Pedersen, P. Salamon, J. Phys. 49, 1485 (1988)
P. Salamon, J.D. Nulton, J.R. Harland, J. Pedersen, G. Ruppeiner, L. Liao, Comput. Phys. Commun. 49, 423 (1988)
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
J. Lässig, K.H. Hoffmann, Phys. Rev. E 79, 046702 (2009)
Finite-Time Thermodynamics and Thermoeconomics, edited by S. Sieniutycz, P. Salamon (Taylor and Francis, New York, 1990)
R.E. Kunz, P. Blaudeck, K.H. Hoffmann, R.S. Berry, J. Chem. Phys. 108, 2576 (1998)
J.C. Schön, Ber. Bunsenges. 100, 1388 (1996)
J.C. Schön, H. Putz, M. Jansen, J. Phys.: Condens. Matter 8, 143 (1996)
A.R. West, Solid State Chemistry and Its Applications (Wiley, New York, 1984)
A. Bach, D. Fischer, X. Mu, W. Sigle, P.A. van Aken, M. Jansen, Inorg. Chem. 50, 1563 (2011)
M.A.C. Wevers, J.C. Schön, M. Jansen, J. Solid State Chem. 136, 223 (1998)
M.A.C. Wevers, J.C. Schön, M. Jansen, J. Phys. A 34, 4041 (2001)
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
K.H. Hoffmann, P. Salamon, Appl. Math. Lett. 22, 1471 (2009)
K.H. Hoffmann, P. Salamon, Physica A 390, 3086 (2011)
V.F. Krotov, Control and Cybernetics 17, 115 (1988)
K. Ergenzinger, K.H. Hoffmann, P. Salamon, J. Appl. Phys. 77, 5501 (1995)
A. Franz, K.H. Hoffmann, J. Comput. Phys. 176, 196 (2002)
M. Santoro, J.C. Schön, M. Jansen, Phys. Rev. E 76, 1 (2007)
J.C. Schön, Z. Anorg. Allg. Chem. 635, 1794 (2009)
F. Heilmann, K.H. Hoffmann, Europhys. Lett. 70, 155 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights 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
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2013-31042-4