(360f) A Generalized Hamiltonian-Based Algorithm for Rigorous Nonequilibrium Molecular Dynamics Simulation in the Nvt Ensemble

We use a methodical Hamiltonian-based procedure to derive a rigorous algorithm for nonequilibrium molecular dynamics (NEMD) simulation in the canonical (NVT) ensemble. This algorithm combines a generalization [1, 2] of the Nosé-Hoover thermostat [3, 4], which allows for rigorous simulation in the NVT ensemble in the presence of non-zero total system momentum and external forces, with a rigorous NEMD algorithm for steady, homogeneous flow, either the SLLOD [5] or p-SLLOD algorithm [6, 7] (which are identical for the specific case of planar Couette flow). It should be noted that previous derivations and proofs of rigor for the SLLOD and p-SLLOD algorithms have been performed in the microcanonical (NVE) ensemble. Previous versions of SLLOD and p-SLLOD have had the thermostat incorporated in an ad hoc manner. For the first time, we rigorously derive a Hamiltonian-based SLLOD or p-SLLOD algorithm in the NVT ensemble. The procedure to derive the algorithm begins with the Hamiltonian in terms of the peculiar and COM coordinates in the mathematical frame of reference [1, 2]. We implement the Nosé-Hoover Thermostat in a rigorous way such that the thermostat only acts on the peculiar momenta and not on the COM momentum [1, 2]. The Hamiltonian is then expressed in terms of the laboratory coordinates in the mathematical frame of reference. The equations of motion are derived in terms of the laboratory coordinates in the mathematical frame of reference. A non-canonical transformation is identified for the transformation from the peculiar and COM coordinates in the mathematical frame of reference to the peculiar and COM coordinates in the physical frame of reference. The equations of motion are then expressed in terms of the peculiar and COM coordinates in the physical frame of reference. In this work, we prove that the Hamiltonian in the physical frame of reference is a conserved quantity when expressed in terms of peculiar and COM coordinates. We integrated the equations of motion using the reversible reference system propagator algorithm (r-RESPA) developed by Tuckerman et al. [8]. We devised the r-RESPA for shear flow following the procedure by Tuckerman et al. [8] and Cui et al. [9]. We report simulation results of a simple fluid across a range of shear rates, including very high shear rates, where thermostat artifacts of non-rigorous NEMD algorithms are well known to exist. We compare our simulation results with previous simulation results. Specifically, Erpenbeck [10] observed a string phase in nonequilibrium molecular-dynamics simulations of shear flow in hard sphere fluids. Evans and Morris [11] concluded that the string phase is an artifact due to the use of a profile-biased thermostat which assumes a stable linear velocity profile at high shear rates. Subsequently, Delhommelle et al [12] used a configurational thermostat to model shear flow of simple fluids in the NVT ensemble. We compare the results of our NEMD simulations of shear flow for simple flow with each of these published results.

References: 1. Keffer, D.J., C. Baig, P. Adhangale, and B.J. Edwards, A Generalized Hamiltonian-Based Algorithm for Rigorous Equilibrium Molecular Dynamics Simulation in the Isobaric-Isothermal Ensemble. Mol. Sim., 2006. accepted. 2. Keffer, D.J., C. Baig, B.J. Edwards, and P. Adhangale. A Hamiltonian-Based Algorithm for Rigorous Molecular Dynamics Simulation in the NVE, NVT, NpT, and NpH Ensembles. in AIChE Annual Meeting. 2005. Cincinnati, OH. 3. Nosé, S., A molecular dynamics method for simulations in the canonical ensemble. Mol. Phys., 1984. 52(2): p. 255-268. 4. Hoover, W.G., Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A, 1985. 31(3): p. 1695-1697. 5. Evans, D.J. and G.P. Morriss, Statistical Mechanics of Nonequilibrium Liquids. Theoretical Chemistry Monograph Series. 1990, London: Academic Press. 6. Baig, C., B.J. Edwards, D.J. Keffer, and H.D. Cochran, A proper approach for nonequilibrium molecular dynamics simulations of planar elongational flow. J. Chem. Phys., 2005. 122: p. 114103. 7. Edwards, B.J., C. Baig, and D.J. Keffer, Nonlinear Response Theory for Arbitrary Steady-State Flows. J. Chem. Phys., 2005. 123: p. 114106. 8. Tuckerman, M., B.J. Berne, and G.J. Martyna, Reversible Multiple Time Scale Molecular-Dynamics. J. Chem. Phys., 1992. 97: p. 1990-2001. 9. Cui, S.T., P.T. Cummings, and H.D. Cochran, Multiple Time Step Nonequilibrium Molecular Dynamics Simulation of the Rheological Properties of Liquid n-Decane. J. Chem. Phys., 1996. 104: p. 255-262. 10. Erpenbeck, J.J., Shear Viscosity of the Hard-Sphere Fluid Via Nonequilibrium Molecular Dynamics. Phys. Rev. Lett, 1984. 52: p. 1333. 11. Evans, D.J. and G.P. Morriss, Shear Thickening and Turbulence in Simple Fluids. Phys. Rev. Lett, 1986. 56: p. 2172. 12. J. Delhommelle, J. Petravic, and D. J. Evans, Non-Newtonian behavior in simple fluids. J. Chem. Phys., 2004. 120(13): p. 6117-6123.