Green-Kubo with pppm

Dear Lammps users,

I have been quite reluctant to post this problem on the Lammps email list as it could be considered one which relates more to theory than a Lammps methodology.

I have been performing non-equilibrium simulations of homogenous systems of varying complexity using the “fix heat” command to create a one dimensional non-linear temperature gradient. I then analyze the numerical derivative of the temperature profile in the dimension of the applied heat flux to calculate a “local” thermal conductivity. The temperature gradients produced are non-linear and typically vary by a factor of two over the simulated area. I have done a comparison of these NEMD results with simulations a colleague has ran using GROMACS with small linear linear gradients and the agreement is to within around 1%.

It is my feeling therefore that these NEMD simulations are acceptably accurate.

It was however my initial intention to try and prove the validity of this non-linear temperature gradient technique by performing separate NVE Green-Kubo calculations using the local density and temperature at given points along my NEMD simulations.

The comparison of both the non-linear temperature gradient method and GK method for calculation of the thermal conductivity show very good agreement for two initial systems, namely a pure Lennard-Jones system and a harmonic dumbbell system. The comparison for my latest system (SPC/E water using pppm for the electrostatics) however differs by quite a large factor (~14%). I particular it appears to systematically be higher for the Green-Kubo calculations.

Unfortunately after a significant search I have not been able to find any direct comparisons of the thermal conductivity for SPC/E water by Green-Kubo and NEMD, however there are papers which show good agreement for the viscosity.

I guess simply put, my question is:

Have there been any bench mark tests performed for the stress-tensor/viral K-space components (In particular using pppm)?

Many thanks and apologies for the non-specific nature of the question, Jeff.

Hi Jeff,
Here are my two cents. While I haven’t done any bench marking as you requested. I have performed some simulations (~4 years back) for epoxies with Green-Kubo and NEMD methods. For Green-Kubo, I used ewald (had my own code at that time which included portions of ewald.cpp of lammps; Green-kubo formalism was not incorporated in lammps at that time) as I could understand that better. I also saw about 20% higher thermal conductivity in Green-Kubo approach. I did not try PPPM at that time as I could not understand workings of PPPM to the detail I wanted to. Ewald was on the other hand was much more straightforward and there was some literature explaining the equations.

I hope this help. If you are interested, you can look at Varshney et al. Polymer, 2009, 50, 3378.

Best Regards,

You could try measuring kappa for rigid and flexible water. I think
there are SPC/E variants that allow for flexible bonds. It's possible
that the constraints enforced by SHAKE affect kappa.


Hi Jeff,

We recently completed a project to compare water models using NEMD and GK methods. We see good agreement with LAMMPS using both of the ‘per-atom’ k-space methods (Ewald and PPPM) and with NEMD. As Steve pointed out, the constraints change the TC. There is a recent paper with a good discussion from the group of Muller-Plathe. I also recommend Vikas’ paper (below), it has a nice breakdown of the different contributions to TC.


Thank you for all the replies. I am actually in the middle of carrying out some simulations that I was hoping would provide a good test of the K-space contributions to the virial.

As a first test I have tried to reproduce a Green-Kubo based thermal conductivity calculation from the paper "J. Chem. Phys. 120, 8676 (2004)".

I ran the simulations for approximately the same system conditions ~256 atoms T=1339K D=1.433 g/cm^{3}and the same duration/sampling but get a thermal conductivity which is ~40% larger.

As a secondary test I have started running an NEMD calculation with the same system to see whether the results agree with either the Lammps' Green-Kubo calculations or those of the aforementioned paper in view validating either.

It is good to hear though that you found agreement for water using both NEMD and Green-Kubo. Would it be possible for you to give me an approximate figure for the % agreement you get between both methods and at what temperature and density the simulations were performed.

I have attached an eps image showing the comparison I mentioned earlier for NaCl

Many thanks, Jeff.

lammps_test.eps (28.7 KB)