Questions and problems?

# Constant Pressure

I assume that the pressure should stablize at approximately 1.0 atmosphere.
Bill Ross:

Except 1) the physics of pressure does not lend itself to instantaneous evaluation/correction and 2) the algorithm used by Amber (scaling the box & translating everything, and then applying SHAKE or letting bond, angle, etc. potentials rectify things) is not the best available. Having most values in the range -100..100 is the best I ever hope for w/ my 8-10000 atom dna/water systems.

Dave Case:

...pressure fluctuations are large because the pressure-volume isotherm of a liquid is extremely steep: it takes a large pressure change to yield a small volume change, and conversely: a volume that is instantaneously slightly out of equilibrium yields a large pressure. The fluctuations in pressure are then related to this: in a thermodynamic sense (e.g. Landau & Lifshitz, "Statistical Physics" sec. 114):

```             mean value of (delta p)**2 = -kT(dp/dV)
```
where the derivative is evaluated at constant S (!). Rearranging the above equation, approximating (dp/dV) at constant S by 1/(dV/dp) at constant T, yields
```           (delta p)**2 = kT/(V*beta)  or delta p ~= sqrt(1/beta)
```
where beta is isothermal compressibility and I have set kT/V to 1 bar in the final equation. If we use the macroscopic value for beta for water (roughly 4 x 10**-5 bar**-1) we obtain an rms pressure fluctation of 160 bar. This calculation is based on macroscopic ideas: pressure fluctuations in very small volumes would probably be larger than this.

Having said this, it must be noted that the procedure used in Amber (based on ideas of Berendsen) does not generate a simple thermodynamic ensemble, and (in addition) should not be expected to yield fluctuations that match the thermodynamic formula cited above. It will be interesting to see if anyone reports a systematic study of these numbers as a function of the type of simulation being done (especially on the volume of the simulated system.)

User dialogue with Dave Case:

in Sander output files (standard name: mdout and mden) values of the pressure are reported. I'm wandering in which unit this values are given. A further problem with the pressure is that I have got negative values. What does negative values mean? Is the pressure given as relative pressure?

Pressures are reported in bars; negative pressures indicate that the system would like to contract its volume, i.e. that an outward force on the walls of the container would be required to keep the system at its current volume; positive pressure is the opposite: the system would like to increase its volume, and an inward-directed external force would be required to keep the system at its current volume.

As the reported pressure is the absolute pressure I would suggest to interpret the values in the following way:

```  If the instantaneous pressure is greater than PRES0 the system
wants to increase is volume (overpressure); if the instantaneous pressure
is lower than PRES0 but greater than zero the system wants to contract its
volume (subpressure); an instantaneous pressure  below zero
is an artifact due to the way the virial term is calculated.
```

I must say that I don't agree with your interpretation....

The pressure is computed from the virial in the standard way. The only influence of PRES0 is on how the volume of the system will respond: if the instantaneous pressure is greater than PRES0, the MD algorithm in Amber will slowly increase the volume of the system; vice-versa if the instantaneous pressure is below PRES0. The *interpretation* of the computed pressure is independent of the value of PRES0.

There is nothing "wrong" with negative pressures, and they need not arise from errors or truncations in the virial calculation. The equation of state of a rare gas fluid (for example) has substantial regions of density and temperature where the equilibrium pressure is negative [see, for example, chapter 14 of McQuarrie's "Statistical Mechanics", or other textbooks.] One of the tests I made (many years ago) to verify the correctness of the way Amber calculates pressure terms was to reproduce these equations of state for a Lennard-Jones 6-12 fluid. [One can hope that bugs have not crept in since then!]