Numerical integration of the extended variable generalized Langevin equation with a positive Prony representable memory kernel.

Generalized Langevin dynamics (GLD) arise in the modeling of a number of systems, ranging from structured fluids that exhibit a viscoelastic mechanical response, to biological systems, and other media that exhibit anomalous diffusive phenomena. Molecular dynamics (MD) simulations that include GLD in conjunction with external and/or pairwise forces require the development of numerical integrators that are efficient, stable, and have known convergence properties. In this article, we derive a family of extended variable integrators for the Generalized Langevin equation with a positive Prony series memory kernel. Using stability and error analysis, we identify a superlative choice of parameters and implement the corresponding numerical algorithm in the LAMMPS MD software package. Salient features of the algorithm include exact conservation of the first and second moments of the equilibrium velocity distribution in some important cases, stable behavior in the limit of conventional Langevin dynamics, and the use of a convolution-free formalism that obviates the need for explicit storage of the time history of particle velocities. Capability is demonstrated with respect to accuracy in numerous canonical examples, stability in certain limits, and an exemplary application in which the effect of a harmonic confining potential is mapped onto a memory kernel.

[1]  H. Mori.  Transport, Collective Motion, and Brownian Motion , 1965 .

[2]  H. Mori.  A Continued-Fraction Representation of the Time-Correlation Functions , 1965 .

[3]  W. Coffey,et al.  The Langevin equation : with applications to stochastic problems in physics, chemistry, and electrical engineering , 2012 .

[4]  C. Brooks.  Computer simulation of liquids , 1989 .

[5]  Shin-Ho Chung,et al.  A generalized Langevin algorithm for studying permeation across biological ion channels , 2008 .

[6]  C. Harris,et al.  Generalized Brownian dynamics. I - Numerical integration of the generalized Langevin equation through autoregressive modeling of the memory function. II - Vibrational relaxation of diatomic molecules in solution , 1990 .

[7]  J. Padró,et al.  A time-saving algorithm for generalized Langevin-dynamics simulations with arbitrary memory kernels , 1990 .

[8]  Mason,et al.  Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids. , 1995, Physical review letters.

[9]  E. Hairer,et al.  Geometric numerical integration illustrated by the Störmer–Verlet method , 2003, Acta Numerica.

[10]  Dan Gordon,et al.  Generalized Langevin models of molecular dynamics simulations with applications to ion channels. , 2009, The Journal of chemical physics.

[11]  Igor M. Sokolov,et al.  Models of anomalous diffusion in crowded environments , 2012 .

[12]  Giovanni Bussi,et al.  Langevin equation with colored noise for constant-temperature molecular dynamics simulations. , 2008, Physical review letters.

[13]  Scott A. McKinley,et al.  Statistical challenges in microrheology , 2012, 1201.5984.

[14]  X. Xie,et al.  Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule. , 2004, Physical review letters.

[15]  Giovanni Bussi,et al.  Nuclear quantum effects in solids using a colored-noise thermostat. , 2009, Physical review letters.

[16]  David A. Weitz,et al.  Diffusing-wave-spectroscopy measurements of viscoelasticity of complex fluids , 1997 .

[17]  R. Kupferman.  Fractional Kinetics in Kac–Zwanzig Heat Bath Models , 2004 .

[18]  Steven J. Plimpton,et al.  Accurate and efficient methods for modeling colloidal mixtures in an explicit solvent using molecular dynamics , 2008, Comput. Phys. Commun..

[19]  H. Callen,et al.  Irreversibility and Generalized Noise , 1951 .

[20]  R. Skeel,et al.  Langevin stabilization of molecular dynamics , 2001 .

[21]  M. Despósito,et al.  Anomalous diffusion: exact solution of the generalized Langevin equation for harmonically bounded particle. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  B. Leimkuhler,et al.  Rational Construction of Stochastic Numerical Methods for Molecular Sampling , 2012, 1203.5428.

[23]  S. Toxvaerd.  Solution of the generalized Langevin equation for a polymer in a solvent , 1987 .

[24]  Steve Plimpton,et al.  Fast parallel algorithms for short-range molecular dynamics , 1993 .

[25]  X. Xie,et al.  Observation of a power-law memory kernel for fluctuations within a single protein molecule. , 2005, Physical review letters.

[26]  B. Berne,et al.  Non‐Markovian activated rate processes: Comparison of current theories with numerical simulation data , 1986 .

[27]  G. Ciccotti,et al.  COMPUTER-SIMULATION OF THE GENERALIZED BROWNIAN-MOTION .1. THE SCALAR CASE , 1980 .

[28]  J. Doll,et al.  Generalized Langevin equation approach for atom/solid–surface scattering: Numerical techniques for Gaussian generalized Langevin dynamics , 1976 .

[29]  R. Skeel,et al.  Analysis of a few numerical integration methods for the Langevin equation , 2003 .

[30]  G. A. Pavliotis,et al.  Asymptotic analysis for the generalized Langevin equation , 2010, 1003.4203.

[31]  J. Andrew McCammon,et al.  Generalized Langevin dynamics simulations with arbitrary time‐dependent memory kernels , 1983 .

[32]  Bruce J. Berne,et al.  On the Calculation of Autocorrelation Functions of Dynamical Variables , 1966 .

[33]  Michele Ceriotti,et al.  Efficient first-principles calculation of the quantum kinetic energy and momentum distribution of nuclei. , 2012, Physical review letters.

[34]  Scott A. McKinley,et al.  Transient anomalous diffusion of tracer particles in soft matter , 2009 .

[35]  R. Kubo.  The fluctuation-dissipation theorem , 1966 .

[36]  T. Schneider,et al.  Molecular-dynamics study of a three-dimensional one-component model for distortive phase transitions , 1978 .

[37]  D. Ermak,et al.  Brownian dynamics with hydrodynamic interactions , 1978 .

[38]  Kenneth S. Schweizer,et al.  Microscopic theory of the dynamics of polymeric liquids: General formulation of a mode–mode‐coupling approach , 1989 .

[39]  E. Guàrdia,et al.  Generalized Langevin dynamics simulation of interacting particles , 1985 .

[40]  John Waldron,et al.  The Langevin Equation , 2004 .

[41]  M. Gregory Forest,et al.  Time-Domain Methods for Diffusive Transport in Soft Matter , 2009, SIAM J. Appl. Math..

[42]  G. Uhlenbeck,et al.  On the Theory of the Brownian Motion , 1930 .

[43]  Abraham Nitzan,et al.  Dynamics of gas–solid interactions: Calculations of energy transfer and sticking , 1977 .

[44]  M. Despósito,et al.  Subdiffusive behavior in a trapping potential: mean square displacement and velocity autocorrelation function. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[45]  Burkhard Dünweg,et al.  BROWNIAN DYNAMICS SIMULATIONS WITHOUT GAUSSIAN RANDOM NUMBERS , 1991 .