The Nosé-Hoover Chain

The final thermostat we consider is one based on the extended Lagrangian formalism, which leads to a deterministic trajectory; i.e., there are no random forces or velocities to deal with. The most common and so far most reliable thermostat of this kind is the Nosé-Hoover thermostat. This thermostat can be implemented as a “single” or a “chain”; here, we consider a chain.

The basic idea of the Nosé-Hoover thermostat is to use a friction factor to control particle velocities. This friction factor is actually the scaled velocity, $v_{\xi_1}$, of an additional and dimensionless degree of freedom, $\xi_1$. This degree of freedom has an associated “mass”, $Q_1$, which effectively determines the strength of the thermostat. The equations of motion obeyed by this additional degree of freedom guarantee that the original degrees of freedom (${\bf r}^N$, ${\bf p}^N$) sample a canonical ensemble. This degree of freedom is the terminus of a chain of similar degrees of freedom, each with their own mass. The chain has a total of $M$ “links.” The overall set of equations of motion are:

$$\dot{\bf r}_i = \frac{\dot{\bf p}_i}{m_i}$$ (215)

$$\dot{\bf p}_i = {\bf F}_i - \frac{p_{\xi_1}}{Q_1}{\bf p}_i$$ (216)

$$\dot{\xi}_k = \frac{p_{\xi_k}}{Q_k} k = 1,\dots,M$$ (217)

$$\dot{p_{\xi_1}} = \left(\sum_i\frac{p_i^2}{m_i} - Lk_BT\right)-\frac{p_{\xi,2}}{Q_2}p_{\xi_1}$$ (218)

$$\dot{p_{\xi_k}} = \left(\frac{p^2_{\xi_{k-1}}}{Q_{k-1}} - k_BT\right) - \frac{p_{\xi_{k+1}}}{Q_{k+1}}p_{\xi_k}$$ (219)

$$\dot{p_{\xi_k}} = \left(\frac{p^2_{\xi_{M-1}}}{Q_{M-1}} - k_BT\right)$$ (220)

The main advantage of the Nosé-Hoover chain thermostat is that the dynamics of all degrees of freedom are deterministic and time-reversible. No random numbers are used. The code mdlj_nhc.c implements an $M$ = 2 Nosé-Hoover chain thermostat in an MD simulation of an Lennard-Jones fluid, by implementing Algorithms 30, 31, and 32 from Frenkel & Smit. The relevant parameters are nhcT, the setpoint temperature, and nhcQ, the two masses. Fig. 27 illustrates the use of the NHC thermostat on an N=512, $\rho$ = 0.84 LJ system.

Figure 27: Temperature vs time (output every 10 time steps) for four MD simulations of the LJ fluid at density 0.84 with 512 particles with initial velocities assigned to give an initial temperature of 2.0. A 2-mass Nosé-Hoover chain with masses indicated in the legend is used to maintain the temperature at 2 until $t$ = 2, at which time the setpoint temperature is instantaneously raised to 3. The third number in the legend label is the product of the number of particles and the relative fluctuation in instantaneous temperature measured for the second half of each respective simulation, which in the canonical ensemble should be 2/3.