Nose-Hoover-Langevin

We applied the Langevin temperature control [57] (temperature 300 K, damping coefficient 1/ps) and the Nose-Hoover Langevin piston pressure control [58,59] (target pressure 1 atm, oscillation period 100 fs, and oscillation decay time 50 fs). To ensure stable temperature and pressure, an equilibra-... 

Nose-Hoover-Langevin An Ergodic Weakly Coupled Thermostat... 

The Nose-Hoover-Langevin (NHL) method is based on a simple idea replace the chain in the Nos6-Hoover Chain, whose sole purpose is to maintain a Gaussian distribution in the auxiliary variable, by a stochastic Langevin-type thermostat. The method was first proposed in [323]. The proof of ergodicity (more precisely the confirmation of the Hormander condition), for a problem with harmonic internal interactions, was given in [226] and we roughly follow the treatment from this paper. 

The generator for Nose-Hoover-Langevin dynamics (8.27)-(8.29), acting on a suitable test functionis... 

Numerical Integration of the Nose-Hoover-Langevin Equations... 

For the Nose-Hoover-Langevin method, we observe that, while small values of IX increase the high frequency oscillations observed in the kinetic energy, the rate of convergence is essentially determined by y and die average kinetic energy indeed converges in the timescale of these simulations. The presence of the stochastic... 

Nosi-Hoover Nosi-Hoover-Langevin Nose-Hoover-Langevin Langevin... 

Fig. 8.4 The computed velocity auto-conelation functions are plotted for Nose-Hoover and Nose-Hoover-Langevin dynamics (left), where we overlay plots from simulations using all possible parameter sets where y 0,0.05,0.5,5,50 and /r 20,2,0.2,0.02. We additionally plot the same result for Langevin dynamics (right), overlaying curves for y 0.05,0.1,0.2,..., 51.2, where high to low friction is plotted from red to blue respectively. The salient qualitative features of the results remain qualitatively unchanged when using different values of the parameters for the Nos6-Hoover thermostats (see inset) whereas the Langevin thermostat has a much harsher effect on the dynamics, with significant differences as the friction is increased...

In this section we present numerical methods based on splitting for DPD and for another stochastic momentum-conserving method, the pairwise Nose-Hoover-Langevin (PNHL) method. 

Extended Variable Momentum-Conserving Thermostats Pairwise Nose-Hoover/Nose-Hoover-Langevin Methods... 

Fig. 8.8 We compare the time evolution of the system s total momentum in the x and y components (top row, marked blue and red respectively), and momentum distributions (bottom row) for various methods. In the lower row of figures, the dark, solid curve represents the exact normal density, while the superimposed histograms are constructed from the simulation data. From l to right the PNHL-N thermostat, Langevin dynamics, Nose-Hoover-Langevin (NHL) dynamics and the Beiensen device. The same initial conditions were used with small values of total momenta. The Berendsen device shows a rapid and systematic increase in the momenta, NHL and Langevin exhibit different behavior, obtaining large fluctuating values, whereas the PNHL-N thermostat preserves the total momenta exactly. Standard DPD results are identical to those of PNHL in this respect...

Two of more sophisticated and commonly used approaches are the Nose-Hoover thermostat [79,80] and Langevin method [81]. In the Langevin method, additional terms are added to the equations of motion corresponding to a fiiction term and a random force. The Langevin equation of motion is given by... 

The Berendsen [82] and Gauss [83] thermostats are also among other methods used. The Berendsen thermostat [82] was developed starting from the Langevin formalism by eliminating the random forces and replacing the friction term with one that depends on the ratio of the desired temperature to current kinetic temperature of the system. The resulting equation of motion takes the same form as the Nose-Hoover equation with... 

Consider for example (1) high-friction Langevin processes, and (2) (Nose-Hoover) constant temperature molecular dynamics. For both cases the dynamics is reversible and the transfer operator is self-adjoint. For type (1) examples, conditions (Cl) and (C2) are known to be satisfied under rather weak condition on the potential [2]. For type (2) examples, it is unknown whether or not the conditions are satisfied however, it is normally assumed in molecular dynamics that they are valid for realistically complex systems in solution. 

After energy minimization, the simulations were run at constant pressure using a Langevin piston Nose-Hoover method as implemented in the NAMD software package until the system had reached its equilibrium volume at a pressure of 1 atm and 300 K in the NPT (constant particle number, pressure, and temperature) ensemble. The system s behavior was then simulated for 200 ns (100 million steps) in the A F T (constant particle number, volume, and... 

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