7.1.1 Gauss dynamics

Gaussian, or isokinetic, dynamics was introduced by Hoover and Evans in 1982. This is the idea:

Let

$$\sigma\{ \vec{v}\} \equiv \sum_{i=1}^{N}\frac{m}{2}v_{i}^{2}-E_{0}$$ (7.1)

be a (non-holonomic) constraint to the Newtonian motion of the $N$-particle system. This is only one constraint equation for $3N$ variables, thus we have some freedom of choice.

A dynamical constraint may always be put in terms of constraint forces. A particularly economical and ``impartial'' prescription for the definition of these forces was given by C. F. Gauss in 1829:

• From $\sigma\{ \vec{v}\}=0$ for all times we have
  

  $$0=\dot{\sigma} =\sum_{i}\left( \frac{\partial}{\partial \vec... ...cdot \dot{\vec{v}_{i}} \equiv m \vec{v} \cdot \dot{\vec{v}}$$ (7.2)

  
  
  Since $\dot{\vec{v}}=\left( \vec{K}+\vec{Z}\right)/m$ (with the Newtonian forces $\vec{K}$ and the yet unknown constraint forces $\vec{Z}$) we have
  

  $$\vec{v} \cdot \left( \vec{K}+\vec{Z}\right) = 0$$ (7.3)

  
  

• Among all possible sets of constraint forces let us choose the one with the smallest norm. Requiring $\left\vert \vec{Z}^{0}\right\vert^{2}=min$ we find, upon variation, that $\vec{Z}^{0} \cdot \delta \vec{Z}=0$. Since for all physically allowed variations we have $\delta \vec{Z} \cdot \vec{v}=0$ we conclude that $\vec{Z}^{0}=-\lambda \vec{v}$. Inserting this in $\vec{v} \cdot \left( \vec{K}+\vec{Z}^{0}\right) = 0$ we find
  

  $$\vec{Z}_{i}^{0}= - \frac{\sum_{j}\vec{v}_{j}\cdot \vec{K}_{... ... = - \frac{\vec{v}\cdot\vec{K}}{\vert\vec{v}\vert^{2}} \vec{v}$$ (7.4)

  
  

• The equations of motion according to Gauss and Hoover are thus
  

  $$\dot{\vec{r}_{i}} = \vec{v}_{i}$$ (7.5)

  $$\dot{\vec{v}_{i}} = \frac{1}{m} \left[ \vec{K}_{i} - \frac{\sum_{j}\vec{v}_{j}\cdot \vec{K}_{j}}{\sum_{j}v_{j}^{2}} \vec{v}_{i} \right]$$ (7.6)

  
  
  These e.o.m. lead to a deterministic trajectory with $E_{kin}=const$.
  
  Note that $\vec{v}$ and $\vec{K}$ are needed at the same time step; thus the usual Verlet algorithm is not applicable here. Instead, a Predictor-Corrector algorithm or Runge-Kutta should be used.