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 don’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