Lately I have been simulating a system in an NPT-ensemble where $N$ is constant and $P$ and $T$ are coupled to a Berendsen barostat/thermostat. The system is cubic and periodic in all directions.

I am trying to fathom how the pressure affects my system, why it fluctuates so much and how negative pressure should be interpreted in systems such as the one described above.

I have some knowledge of classical MD, and it is my understanding that (in general three dimensional system), the total pressure of the system is computed with the virial equation

\begin{equation} P=Nk_{b}TV^{-1}+\frac{1}{3}\bigg<\sum_{i=1}^{N}r_{i}\cdot F_{i}\bigg>V^{-1}\tag{1} \end{equation} where $N$ is the number of particles, $T$ is temperature and $k_{b}$ is the Boltzmann constant.

By looking at equation 1, it is not fully clear to me what negative pressure would imply, so I looked how the equation is derived.

The derivation starts from the Clausius virial function

\begin{equation} W(r_{1},...,r_{n})=\sum_{i}^{N}r_{i}\cdot F_{i}^{TOT} \tag{2} \end{equation}

where $F_{i}^{TOT}$ is the total force acting on particle $i$.

This can be averaged over the whole MD trajectory, and integrated to obtain

\begin{equation} \big<W\big> = -\lim_{t\rightarrow 0}\frac{1}{t}\int^{t}_{0}d\tau\sum_{i=1}^{N}m_{i}|r'(\tau)|^{2} \tag{3} \end{equation}

from which, by using equipartition of energy, one can obtain

\begin{equation} \big<W\big>=-3Nk_{b}T \tag{4} \end{equation}

where 3 is the dimensionality, $N$ is the number of particles, $T$ is temperature and $k_{b}$ is the Boltzmann constant.

The force exerted by the container walls can be taken in to account by 'splitting' the $F_{i}^{TOT}$ term in to internal and external part, and then evaluate $\big<W^{EXT.}\big>$ using:

\begin{equation} \big<W^{EXT.}\big> = 3L³(-P) = -3PV\tag{5} \end{equation}

Where $L$ is the length of the cubic box (so that $L_{x}~=~L_{y}~=~L_{z}~=~L$)

Now, since $\big<W\big> = -3Nk_{b}T$, one can rewrite it as:

\begin{equation} \bigg<\sum_{i=1}^{N}r_{i}\cdot F_{i}\bigg> - 3PV = -3Nk_{b}T\tag{6} \end{equation}

and get the original virial equation (1).

My questions are:

  1. Is this correct? Mathematics (unfortunately) is not yet my strong suit, and I know that my reasoning (even with great help from different MD-related tutorials and sites) is prone to errors.

  2. Could you explain the negative pressures in MD simulation simply by arguing that the force exerted by the container walls is negative?

  3. If so, can one just argue that because the force is negative, the volume of the container must be increasing?

Thank you for your time.


1 Answer 1

  1. Looks fine. 2. I'd phrase that more generally as the simulation volume would tend to expand. 3. But can only do so if the ensemble permits it (pressure can be negative in NVT).

The fluctuations are a consequence of making an observation of a macroscopic quantity over small numbers of degrees of freedom, or time. See http://www.gromacs.org/Documentation/Terminology/Pressure


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