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7.3 NPT Molecular Dynamics

The $NPT$ dynamics of Morriss and Evans (see end of previous subsection) was actually a $NPE_{kin}$ dynamics, with zero fluctuation of the kinetic energy. A genuine $NPT$ dynamics would constrain $E_{kin}$ to stay near a desired average but with - hopefully gaussian distributed - fluctuations around that value.

Such a method was conceived by Hoover (1985). Extending the Hoover-Nosé idea to keep both the pressure and the kinetic energy near a desired value $P_{0}$, he wrote the equations of motion as
$\displaystyle \dot{\vec{s}_{i}}$ $\textstyle =$ $\displaystyle \vec{v}_{i}   V^{-1/3}$ (7.18)
$\displaystyle \dot{\vec{v}_{i}}$ $\textstyle =$ $\displaystyle \frac{1}{m}\vec{K}_{i}-(\chi+\zeta)  \vec{v}_{i}$ (7.19)
$\displaystyle \dot{\zeta}$ $\textstyle =$ $\displaystyle \frac{2}{Q}
\left[ E_{kin} - E_{0} \right]$ (7.20)
$\displaystyle \chi$ $\textstyle =$ $\displaystyle \frac{\dot{V}}{3V}$ (7.21)
$\displaystyle \dot{\chi}$ $\textstyle =$ $\displaystyle \frac{1}{t_{P}^{2}  kT} \left[ P-P_{0} \right]$ (7.22)

where $t_{P}$ and $Q$ are relaxation times of the pressure and temperature regulators, respectively.
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F. J. Vesely / University of Vienna