Comparing the Efficiencies of Stochastic Isothermal Molecular Dynamics Methods

Molecular dynamics typically incorporates a stochastic-dynamical device, a “thermostat,” in order to drive the system to the Gibbs (canonical) distribution at a prescribed temperature. When molecular dynamics is used to compute time-dependent properties, such as autocorrelation functions or diffusion constants, at a given temperature, there is a conflict between the need for the thermostat to perturb the time evolution of the system as little as possible and the need to establish equilibrium rapidly. In this article we define a quantity called the “efficiency” of a thermostat which relates the perturbation introduced by the thermostat to the rate of convergence of average kinetic energy to its equilibrium value. We show how to estimate this quantity analytically, carrying out the analysis for several thermostats, including the Nosé-Hoover-Langevin thermostat due to Samoletov et al. (J. Stat. Phys. 128:1321–1336, 2007) and a generalization of the “stochastic velocity rescaling” method suggested by Bussi et al. (J. Chem. Phys. 126:014101, 2007). We find efficiency improvements (proportional to the number of degrees of freedom) for the new schemes compared to Langevin Dynamics. Numerical experiments are presented which precisely confirm our theoretical estimates.     

Augmented Langevin approach to fluctuations in nonlinear irreversible processes

A Fokker-Planck equation derived from statistical mechanics by M. S. Green [J. Chem. Phys. 20:1281 (1952)] has been used by Grabertet al. [Phys. Rev. A 21:2136 (1980)] to study fluctuations in nonlinear irreversible processes. These authors remarked that a phenomenological Langevin approach would not have given the correct reversible part of the Fokker-Planck drift flux, from which they concluded that the Langevin approach is untrustworthy for systems with partly reversible fluxes. Here it is shown that a simple modification of the Langevin approach leads to precisely the same covariant Fokker-Planck equation as that of Grabertet al., including the reversible drift terms. The modification consists of augmenting the usual nonlinear Langevin equation by adding to the deterministic flow a correction term which vanishes in the limit of zero fluctuations, and which is self-consistently determined from the assumed form of the equilibrium distribution by imposing the usual potential conditions. This development provides a simple phenomenological route to the Fokker-Planck equation of Green, which has previously appeared to require a more microscopic treatment. It also extends the applicability of the Langevin approach to fluctuations in a wider class of nonlinear systems.     

Stochastic analysis of a Hopf bifurcation: Master equation approach

The effect of inhomogeneous fluctuations in a reaction-diffusion system exhibiting a Hopf bifurcation is analyzed using the master equation approach. A Taylor expansion of the logarithm of the stationary probability, known as the stochastic potential, is calculated. This procedure displays marked analogies with the theory of normal forms. The critical potential, reduced to its local expansion around an arbitrary point of the limit cycle, brings out the essential role played by the phase of the oscillating variables. A comparison with the Langevin analysis of Walgraefet al. [J. Chem. Phys. 78(6):3043 (1983)] is performed.     

A quasi-classical trajectory study of the Cl + H<Subscript>2</Subscript> (D<Subscript>2</Subscript>) reaction on a new BW3 potential energy surface

Molecular reaction dynamics of Cl + H₂ (D₂) has been studied on the latest analytical potential energy surface called BW3 using the Monte Carlo quasi-classical trajectory method. Excitation functions, differential cross sections and angular distributions of HCl and DCl products have been calculated. The excitation functions of the Cl (²P_3/2) + n-H₂ and Cl(²P_3/2) + n-D₂ reactions are also studied. The results are compared with those of quasi-classical trajectory [M. Alagia et al.: Phys. Chem. Chem. Phys. 2 (2000); F. J. Aoiz et al.: J. Phys. Chem. 100 (1996)], quantum mechanical (QM) calculations [F. J. Aoiz et al.:J. Chem. Phys. 115 (2001)] and experimental data [S. H. Lee et al.: J. Chem. Phys. 110 (1999); F. Dong et al.: J. Chem. Phys. 115 (2001)]. Discussions are given to some new results.     

Comparison of modern Langevin integrators for simulations of coarse-grained polymer melts

For a wide range of phenomena, current computational ability does not always allow for atomistic simulations of high-dimensional molecular systems to reach time scales of interest. Coarse-graining (CG) is an established approach to alleviate the impact of computational limits while retaining the same algorithms used in atomistic simulations. It is important to understand how algorithms such as Langevin integrators perform on non-trivial CG molecular systems, and in particular how large of an integration time step can be used without introducing unacceptable amounts of error into averaged quantities of interest. To investigate this, we examined three different Langevin integrators on a CG polymer melt: the recently developed BAOAB method by Leimkuhler and Matthews [J. Chem. Phys. 138 (17), 05B601_1 (2013)], the Grønbech-Jensen and Farago method [Mol. Phys. 111 (8), 983-991 (2013)], or G-JF, and the frequently used Brünger–Brooks–Karplus integrator [Chem. Phys. Lett. 105 (5), 495-500 (1984)], known as BBK. We compute and analyse key statistical properties for each. Our results indicate that the integrators perform similarly for a small friction parameter; however outside this regime, the use of large integration steps produces significant deviations from the predicted diffusivity and steady-state distributions for all methods examined with the exception of G-JF.     

Dynamics of end-to-end loop formation for an isolated chain in viscoelastic fluid

We theoretically investigate the looping dynamics of a linear chain immersed in a viscoelastic fluid. The dynamics of the chain is governed by a Rouse model with a fractional memory kernel recently proposed by Weber et al. [S.C. Weber, J.A. Theriot, A.J. Spakowitz, Phys. Rev. E 82 (2010) 011913]. Using the Wilemski–Fixman [G. Wilemski, M. Fixman, J. Chem. Phys. 60 (1974) 866] formalism we calculate the looping time for a chain in a viscoelastic fluid where the mean square displacement of the center of mass of the chain scales as t^1/2$t^{1/2}$. We observe that the looping time is faster for the chain in a viscoelastic fluid than for a Rouse chain in a Newtonian fluid up to a chain length and above this chain length the trend is reversed. Also no stable scaling of the looping time with the length of the chain seems to exist for the chain in a viscoelastic fluid.     

A Gentle Stochastic Thermostat for Molecular Dynamics

We discuss a dynamical technique for sampling the canonical measure in molecular dynamics. We present a method that generalizes a recently proposed scheme (Samoletov et al., J. Stat. Phys. 128:1321–1336, 2007), and which controls temperature by use of a device similar to that of Nosé dynamics, but adds random noise to improve ergodicity. In contrast to Langevin dynamics, where noise is added directly to each physical degree of freedom, the new scheme relies on an indirect coupling to a single Brownian particle. For a model with harmonic potentials, we show under a mild non-resonance assumption that we can recover the canonical distribution. In spite of its stochastic nature, experiments suggest that it introduces a relatively weak perturbative effect on the physical dynamics, as measured by perturbation of temporal autocorrelation functions. The kinetic energy is well controlled even in the early stages of a simulation.     

A molecular dynamics investigation of dynamical heterogeneity in supercooled water

We investigate the presence of dynamical heterogeneity in supercooled water with molecular dynamics simulations using the new water model proposed by Mahoney and Jorgensen [M.W. Mahoney, W.L. Jorgensen J. Chem. Phys. 112, 8910 (2000)]. Prompted by recent theoretical results [J.P. Garrahan, D. Chandler, Phys. Rev. Lett. 89, 35704 (2002)] we study the dynamical aggregation of the least and the most mobile molecules. We find dynamical heterogeneity in supercooled water and string-like dynamics for the most mobile molecules. We also find the dynamical aggregation of the least mobile molecules. The two kinds of dynamical aggregation appear however to be very different. Characteristic times are different and evolve differently. String-like motions appear only for the most mobile molecules, a result predicted by the facilitation theory. The aggregation of the least mobile molecules is more organized than the bulk while the opposite is observed for the most mobile molecules.     

Shear viscosity of pseudo hard-spheres

We present molecular dynamics simulations of pseudo hard sphere fluid (generalized WCA potential with exponents (50, 49) proposed by Jover et al. [J. Chem. Phys 137, (2012)] using GROMACS package. The equation of state and radial distribution functions at contact are obtained from simulations and compared to the available theory of true hard spheres (HS) and available data on pseudo hard spheres. The comparison shows agreements with data by Jover et al. and the Carnahan–Starling equation of HS. The shear viscosity is obtained from the simulations and compared to the Enskog expression and previous HS simulations. It is demonstrated that the PHS potential reproduces the HS shear viscosity accurately.     

Numerical integration of projective Hamiltonian dynamics

Projective Hamiltonian dynamics [S. Melchionna, J. Chem. Phys. 121, 4534 (2004)] is an approach to reduce mode spreading in complex systems, thus improving the performances of Molecular Dynamics simulations. A numerical integration of the dynamics requires proper treatment of the non-separable form of the Hamiltonian. Due to its symplectic nature, the generalized leapfrog scheme is shown to provide robust and reliable numerical trajectories even in the case of stiff projective forces.