Molecular simulation in the canonical ensemble and beyond
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) no. 2, p. 333-350
In this paper, we discuss advanced thermostatting techniques for sampling molecular systems in the canonical ensemble. We first survey work on dynamical thermostatting methods, including the Nosé-Poincaré method, and generalized bath methods which introduce a more complicated extended model to obtain better ergodicity. We describe a general controlled temperature model, projective thermostatting molecular dynamics (PTMD) and demonstrate that it flexibly accommodates existing alternative thermostatting methods, such as Nosé-Poincaré, Nosé-Hoover (with or without chains), Bulgac-Kusnezov, or recursive Nosé-Poincaré Chains. These schemes offer possible advantages for use in computing thermodynamic quantities, and facilitate the development of multiple time-scale modelling and simulation techniques. In addition, PTMD advances a preliminary step toward the realization of true nonequilibrium motion for selected degrees of freedom, by shielding the variables of interest from the artificial effect of thermostats. We discuss extension of the PTMD method for constant temperature and pressure models. Finally, we demonstrate schemes for simulating systems with an artificial temperature gradient, by enabling the use of two temperature baths within the PTMD framework.
DOI : https://doi.org/10.1051/m2an:2007019
Classification:  74A25,  82C80
@article{M2AN_2007__41_2_333_0,
author = {Jia, Zhidong and Leimkuhler, Ben},
title = {Molecular simulation in the canonical ensemble and beyond},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {41},
number = {2},
year = {2007},
pages = {333-350},
doi = {10.1051/m2an:2007019},
zbl = {1138.82007},
mrnumber = {2339632},
language = {en},
url = {http://www.numdam.org/item/M2AN_2007__41_2_333_0}
}

Jia, Zhidong; Leimkuhler, Ben. Molecular simulation in the canonical ensemble and beyond. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) no. 2, pp. 333-350. doi : 10.1051/m2an:2007019. https://www.numdam.org/item/M2AN_2007__41_2_333_0/

[1] H.C. Andersen, Molecular dynamics simulations at constant pressure and/or temperature. J. Chem. Phys. 72 (1980) 2384-2393.

[2] E.J. Barth, B.B. Laird and B.J. Leimkuhler, Generating generalized distributions from dynamical simulation. J. Chem. Phys. 118 (2003) 5759-5768.

[3] E. Barth, B. Leimkuhler and C. Sweet, Approach to thermal equilibrium in biomolecular simulation, in New Algorithms for Macromolecular Simulation, B. Leimkuhler, C. Chipot, R. Elber, A. Laaksonen, A. Mark, T. Schlick, C. Schütte and R. Skeel Eds., Springer Lecture Notes in Computational Science and Engineering 49 (2006). | Zbl 1094.92004

[4] L. Boltzmann, On certain questions of the theory of gases. Nature 51 (1895) 413-415. | JFM 26.1059.08

[5] S.D. Bond, B.J. Leimkuhler and B.B. Laird, The Nosé-Poincaré method for constant temperature molecular dynamics. J. Comput. Phys. 151 (1999) 114-134. | Zbl 0933.81058

[6] A.C. Branka and K.W. Wojciechowski, Generalization of Nosé and Nosé-Hoover isothermal dynamics. Phys. Rev. E 62 (2000) 3281-3292.

[7] A. Bulgac and D. Kusnezov, Canonical ensemble averages from pseudomicrocanonical dynamics. Phys. Rev. A 42 (1990) 5045-5048.

[8] E. Cancés, F. Legoll and G. Stoltz, Theoretical and numerical comparison of some sampling methods for molecular dynamics. ESAIM: M2AN (to appear). | Numdam | MR 2339633 | Zbl 1138.82341

[9] R. Car and M. Parinello, Unified approach for molecular dynamics and density function theory. Phys. Rev. Lett. 55 (1985) 2471-2475.

[10] C. Chipot, Free energy calculations in biological systems: how useful are they in practice, in New Algorithms for Macromolecular Simulation, B. Leimkuhler, C. Chipot, R. Elber, A. Laaksonen, A. Mark, T. Schlick, C. Schütte and R. Skeel Eds., Springer Lecture Notes in Computational Science and Engineering 49 (2006). | Zbl 1094.92005

[11] J. Delhommelle, Correspondence between configurational temperature and molecular kinetic temperature thermostats. J. Chem. Phys. 117 (2002) 6016-602.

[12] E.G. Flekkøy and P.V. Coveney, From molecular dynamics to dissipative particle dynamics. Phys. Rev. Lett. 83 (1999) 1775-1778.

[13] D. Frenkel and B. Smith, Understanding Molecular Simulation. Academic, London (1996). | Zbl 0889.65132

[14] S.P.A. Gill, Z. Jia, B. Leimkuhler and A.C.F. Cocks, Rapid thermal equilibration in coarse-grained molecular dynamics. Phys. Rev. B 73 (2006) 184304.

[15] E. Hernandez, Metric-tensor flexible-cell for isothermal-isobaric molecular dynamics simulation. J. Chem. Phys. 115 (2001) 10282-10290.

[16] W.G. Hoover, Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A 31 (1985) 1695-1697.

[17] W.G. Hoover, K. Aoki, C.G. Hoover and S.V. De Groot, Time reversible deterministic thermostats. Physica D (2004) 253-267. | Zbl 1054.80009

[18] J.H. Jeans, On the vibrations set up in molecules by collisions. Phil. Mag. 6 (1903) 279-286. | JFM 34.0833.02

[19] J.H. Jeans, On the partition of energy between matter and ether. Phil. Mag. 10 (1905) 91-97. | JFM 36.0842.01

[20] O.G. Jepps, G. Ayton and D.J. Evans, Microscopic expressions for the thermodynamic temperature. Phys. Rev. E, 62 (2000) 4757-4763.

[21] Z. Jia and B.J. Leimkuhler, A projective thermostatting dynamics technique. Multiscale Model. Simul. 4 (2005) 563-583. | Zbl 1100.82008

[22] D. Kusnezov, Diffusive aspects of global demons. Phys. Lett. A 166 (1992) 315-320.

[23] F. Legoll, M. Luskin and R. Moeckel, Non-ergodicity of the Nosé-Hoover thermostatted harmonic oscillator, arXiv preprint (November 2005, math.DS/0511178). ARMA (to appear). | MR 2299758 | Zbl 1122.82002

[24] B.J. Leimkuhler and C.R. Sweet, The canonical ensemble via symplectic integrators using Nosé and Nosé-Poincaré chains. J. Chem. Phys. 121 (2004) 108-116.

[25] B.J. Leimkuhler and C.R. Sweet, A Hamiltonian formulation for recursive multiple thermostats in a common timescale. SIAM J. Appl. Dyn. Sys. 4 (2005) 187-216. | Zbl 1075.92057

[26] W.K. Liu, E.G. Karpov, S. Zhang and H.S. Park, An introduction to computational nano-mechanics and materials. Comput. Meth. Appl. Mech. Eng. 193 (2004) 1529-1578. | Zbl 1079.74506

[27] Y. Liu and M.E. Tuckerman, Generalized Gaussian Moment Thermostatting: A new continuous dynamical approach to the canonical ensemble. J. Chem. Phys. 112 (2000) 1685-1700.

[28] G.J. Martyna, M.E. Tuckerman and M.L. Klein, Nosé-Hoover chains: The canonical ensemble via continuous dynamics. J. Chem. Phys. 97 (1992) 2635-2643.

[29] S. Nosé, A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys. 81 (1984) 511-519.

[30] J. Powles, G. Rickayzen and D.M. Heyes, Temperature: old, new and middle-aged. Mol. Phys. 103 (2005) 1361-1373.

[31] L. Rosso, P. Mináry, Z. Zhu and M.E. Tuckerman, On the use of the adiabatic molecular dynamics technique in the calculation of free energy profiles. J. Chem. Phys. 116 (2002) 4389-4402.

[32] H.H. Rugh, Dynamical approach to temperature. Phys. Rev. Lett. 78 (1997) 772-774.

[33] T. Schneider and E. Stoll, Molecular-dynamics study of a three-dimensional one-component model for distortive phase transitions. Phys. Rev. B 17 (1978) 1302-1322.

[34] J.B. Sturgeon and B.B. Laird, Symplectic algorithm for constant-pressure molecular-dynamics using a Nosé-Poincare thermostat. J. Chem. Phys. 112 (2000) 3474-3482.

[35] E.B. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids. Philos. Mag. A 73 (1996) 1529-1563.

[36] R.G. Winkler, V. Kraus and P. Reineker, Time-reversible and phase-conserving molecular dynamics at constant temperature. J. Chem. Phys. 102 (1995) 9018-9025.