One-dimensional quartic double well coupled to a harmonic oscillator

Consider a simple two-variable system described by the Hamiltonian in Eq. (3) with a potential of the form

 

For this simple problem, can be calculated analytically, leading to a free energy profile in x given by

 

Simulations of this model system were carried out using , a=1, , 2.878, and . With these parameters, the free energy profile in x has two wells located at 1.189 and a barrier at x=0. In addition, the free energy barrier is 10.

In order to ensure efficient barrier crossing, a temperature was chosen for AFED simulations. With this choice of , the convergence of the free energy profile with was tested by performing simulations with . For these mass choices, simulations of steps using a time step 0.25 10 were performed. Canonical sampling is obtained using the recently introduced generalized Gaussian moment thermostat(GGMT) algorithm [13]. In general, the thermostatting method can have an influence on the efficiency of the two free energy methods. A detailed comparison between the GGMT approach and the more standard Nosé-Hoover chain algorithm will be given elsewhere [15].

Figure 1 shows the trajectory of x as a function of time for and corresponding to an ordinary dynamics simulations compared to the AFED parameters ( and ).

  

Figure 1: Trajectory of x as a function of the number of steps for a double well coupled to a harmonic oscillator in Eq.(19). (a) Standard Molecular Dynamics; (b) Adiabatic (AFED) dynamics with and .

The figure shows that without the adiabaticity conditions, barrier crossing is a rare event as would be expected in an ordinary dynamics calculation. In contrast, the AFED dynamics case shows frequent barrier crossing and, hence, efficient sampling of the configuration space available to x. Figure 2 shows the trajectories of x(t) and y(t) over a relatively small number of MD steps.

  

Figure 2: Short-time trajectories of x(t) (a) and y(t) (b) showing the adiabatic character of the motion.

It can be seen from the figure that the evolution of y is very similar to that of x, however, y exhibits, in addition, rapid fluctuations about x, i.e., y possesses two widely disparate time scales. Figure 3 shows the free energy profiles obtained from the AFED simulations for the different choices of together with the analytical result.

  

Figure 3: Adiabatic (AFED) Dynamics free energy profile for a double well coupled to a harmonic oscillator in Eq.(19). The figure shows the convergence of the free energy profile as a function of with . The solid line is the analytical free energy profile, the dotted line corresponds to , the long dashed line corresponds to , and the short dashed line corresponds to . in all cases. For comparison, the bare potential is shown with the dot-dashed line.

The figure shows that when is too small, adiabaticity is not well maintained and the free energy profile is not well reproduced. It can be seen that for the agreement between the AFED and analytical results is very good. In Sec. 4, a general protocol for determining the adiabaticity control parameters, and is discussed.