Dear all,

I have a question reguarding how the temperature is computed in eFF.

While browsing compute_temp_eff.cpp (lammps-14Mar13) it seems that

LAMMPS is adding:

-) the translational energy of each atom (3 degrees of freedom each)

-) the translational energy of each electron (3 degrees of freedom each)

-) the electronic radius kinetic energy of each electron (3 degrees of

freedom each)

but then you equate this quantity to:

3/2 * k_B * T * (3* Number_Atoms)

(for the sake of clarity, I'm assuming a 3 dimensional system and

neglecting the degrees of freedom "lost" for any reason -fixes or

conservation- but it does not change the point).

Doesn't this violate the equipartition theorem?

Since all the kinetic contribution listed above are quadratic terms in

the Hamiltonian shouldn't their sum be equal to:

3/2 * k_B * T * (3* Number_Atoms) + 3 * k_B * T * (3* Number_Electrons)

I'm wondering because by using the actual implementation the average

temperature and temperature variance of either the atoms only or the

electrons only do not correspond to the "system" temperature, while

implementing the second definition of T seems to solve the discrepancy

(at least according to some quick tests).

What am I missing?

Best,

Paolo

Paolo,

Your question is not so much about the eff FF but about the origins of quantum mechanics. In your line of reasoning you have assumed that the electrons follow the Maxwell-Boltzman distribution which is not correct. Actually, once would have to increase the temperature a whole lot to place electrons into the classical limit. Only when the electronic density of the system is huge you might not need so high temperatures but then you are almost at a compression level that is only achieved by objects that have suffered a drastic gravitational collapse. I donâ€™t know myself the details of eff but any model that assumes electrons following a MB distribution would be destined to hit the same sort of paradoxes that physicists encountered at the late eighteen century.

Carlos

Carlos,

thank you for the prompt reply, but I'm not sure it answers my question.

I'm a little bit rusted on the nuances of the proof of the general

equipartition theorem, but I'm quite sure it does not require to

assume the Maxwell-Boltzmann distribution (also because non quadratic

terms of the Hamiltonian can generate a variety of distributions). As

far as I remember, one pretty much needs an ergodic system, the

definition of entropy and the assumption of a continuum of energy

levels (conditions that are clearly not always true). Of course I may

be wrong, but even if the equipartition isn't holding for the

electrons, I have problem understanding why shouldn't hold for the

atoms. Because if it holds, we have that the ensemble average of the

kinetic energy of the atoms (<KE_atoms>) is equal to (again for a 3

dimensional system):

3/2 * k_B * T

but since in the temp/eff the exact same quantity is supposed to be equal to

<KE_atoms> + <KE_electrons>

(KE_electrons includes translation and radial kinetic energy, which

are both positive terms) it would imply that <KE_electrons> = 0...

If the equipartition is not true for the atoms, I strongly suspect

that the ensemble we are sampling is not canonical (or micro-canonical

if there is no thermostat), which may be worth at least a note in the

manual.

In conclusion, I'm not debating on how much the actual eFF model is

able to capture the real physic (including the thermal distributions

of the electrons), but only the definition of the temperature in the

limit of the model and how it should be compared to the temperature we

obtain with different model or experimentally.

Best,

Paolo

PS: in the first email I wrote that the electronic radius kinetic

energy of each electron counts for 3 degrees of freedom, but more

likely is only one, since the Gaussian width is described by one

scalar. In this case -if the equipartition theorem is true- each

electron would add on average only 2 * k_B * T.

Paolo, Carlos,

Temperature in eFF can be indeed a bit confusing. The virial expression essentially defines temperature so that the heat capacity of the system is 3/2 k N T (were only nuclei are excited). That is, we assume only the classical (nuclear) dof contribute to the "specific heat capacity", which is true in the limit of electrons becoming free particles (i.e. s>>) for example, at very high temperatures).

This is valid for temperatures well below the Fermi temperature. At intermediate temperatures, what we have observed to happen is that the kinetic contribution to the pressure from the electrons is recovered indirectly through the Pauli potential -- as temperature increases, the electrons become excited, which causes their average size to increase, which raises the pressure via the Pauli potential. We have not added any corrections at temperatures comparable to or higher than the Fermi temperature, but in principle, a separate ideal gas electron pressure can be added.

One also needs to keep in mind that the user in eFF can define the dynamic electron mass (associated with the KE of motion). Only for the quantum mechanical derived terms (i.e. electronic wavefunction kinetic energy and Pauli repulsion terms) the electron mass is hardwired to 1 (amu).

Best,

Andres

P.D. You can find some additional notes on temperature in some of our papers on eFF, e.g. JCC, 32 (3), 2011 and Phys Rev B 85, (2012) and it should also appear in some of Julius' original publications on eff, e.g. PNAS January 27, 2009 vol. 106.

Paolo,

Yes, I was too loose with my use of the SM terminology. While referring to particles following MB statistics I meant to say particles with a continuum energy spectrum. And now {during daytime } , I realize that your Q was indeed on the eff technicalities.

Guess Andres response served to conclude the debate.

Carlos

Andreas,

sorry for the delay in my answer but I had a busy week.

I realized that I'm raising two distinct issues: the definition of the

temperature in eFF and the consequences of that choice.

About the first, for good measure I checked again the papers you are

mentioning and everything that I have access to that mentions eFF, but

I still don't find any theoretical justification of the formula.

Temperature in eFF can be indeed a bit confusing. The virial expression essentially defines temperature so that the heat capacity of the system is 3/2 k N T (were only nuclei are excited). That is, we assume only the classical (nuclear) dof contribute to the "specific heat capacity", which is true in the limit of electrons becoming free particles (i.e. s>>) for example, at very high temperatures).

I don't know what the "virial expression" is but the virial theorem

alone does not connect the temperature with the kinetic energy. For

that you can use the equipartition theorem -which btw holds for the

electrons too, since their motion is classical- or combine the virial

theorem with the definition of the heat capacity. In both cases

-unsurprisingly- you get the same result: if you consider the kinetic

energy of the electrons you have to include their degrees of freedom.

Of course you can ignore them, but also their kinetic energy should be

neglected too.

As for the electrons not being excited -as far as the model goes-

since they are modeled as classical quantities, which can have

different velocities, they can be translationally and "radially"

excited.

Regarding, the consequences of the standard eFF definition of T, I'd

like only to stress that with this definition one can have a system at

givrn temperature with ions that are completely frozen and very hot

electrons. Which can be wrong or not, but for sure the ions are not in

a canonical ensemble at the temperature of the system.

That said, I came to realize that this definition is not a bug in

LAMMPS, but a precise choice, so this is probably the wrong place to

discuss it. Therefore I will not press the issue, but since I still

have doubts about it, I'd like to suggest instead to use an approach

similar to Car-Parrinello simulations, which shares some similarity

with eFF, and use a double NH chain thermostat, one for the ions and

one for the electrons. Although is not a perfect solution and needs a

small tweak in the code in order to compute the temperature of the

electrons (right now a group containing only electrons will always

have a temp/eff of 0 K and "compute temp" neglects the radial kinetic

energy), is probably the only solution until the colored noise

thermostat is ready.

Best,

Paolo