# Re: [AMBER] How the instantaneous temperature is calculated in Amber ?

From: Marek Maly <marek.maly.ujep.cz>
Date: Mon, 01 Apr 2019 19:37:31 +0200

Thanks Carlos !

it confirms my guess, and e.g. in case of water, the
average kinetic energy per one molecule is then really 9*0.5*k*T.

In case of rigid water model it should be (9-3)*0.5*k*T as the 3 dof are
fixed so shuld be substracted.

If we fix also the momentum of the whole simulated system of rigid water
molecules
we should substract additional 3 degrees of freedom, so should be valid:

E_kin_total = [N*(9-3)-3]*1/2*k*T (1)

where N is the number of water molecules and E_kin_total is the total
kinetic energy
of the simulation box at the given moment.

Am I right ?

Moreover, as I already wrote, in my opinion If I use just translational
part of the total kinetic energy (translational with respect to the whole
molecules i.e. kinetic energy of centers of mass of molecules) I can write
(independently on intramolecular constraints and independently on type of
molecule (it could be heterogenous molecular mixture)):

E_kin_total_translational = [N*3 - 3]*1/2*k*T (2)

(if fixed (at zero) total momentum of the simulated system is used)

Am I right ?

Bill, regarding to connection of the AVERAGE value of the part of
potential energy which is composed
of quadratic contributions (like k_r/2(r-r0)^2 or k_phi/2(phi-phi_0)^2 )
the Equipartition theorem speaks clearly in my opinion i.e.

<k_r/2(r-r0)^2> = 1/2*k*T

<k_phi/2(phi-phi_0)^2> = 1/2*k*T

But, if I understood well this part of the Equipartition theorem (i.e. its
application on vibrational degrees of freedom of molecules) has some
limits in that sense, that for sufficiently high temperatures works well
(as the classical harmonic oscillator approximates here rather well the
quantum oscillator ) but for low temperatures the molecular vibrations
should be described using quantum physics (quantum oscillator) and the
classical physics approximation is poor at least with respect to
sufficiently good estimates of heat capacities.

see for example this section:

https://en.wikipedia.org/wiki/Equipartition_theorem#History

So regarding this limitations I would say, that in practice it should be
probably better to use just the kinetic energy (or their parts) for the
calculation of the instantaneous temperature.

Marek

Dne Mon, 01 Apr 2019 13:16:16 +0200 Carlos Simmerling
<carlos.simmerling.gmail.com> napsal/-a:

> If you go to the amber web page and in the search bar type something like
> "how is temperature calculated" you get some useful links to prior
> discussions on this. It does indeed use 1/2kT for each dof, subtracting
> the
> number of constraints from the total 3N dof (such as shaken bonds).
>
> On Sun, Mar 31, 2019, 11:20 PM Marek Maly <marek.maly.ujep.cz> wrote:
>
>> Hi Bill,
>>
>> simply because all the contributions to the energy of the molecule which
>> are
>>
>> so each contribution to the total energy in the shape like k/2(r-r0)^2
>> contributes as 1/2*k*T
>>
>> see here :
>>
>>
>> https://en.wikipedia.org/wiki/Equipartition_theorem#Potential_energy_and_harmonic_oscillators
>>
>> But as I wrote, the easiest way to calculate instantaneous temperature
>> of
>> the system composed of complex molecules should be probably to apply
>> equipartition theorem just on translational
>> part of the kinetic energy. Then we do not need to care about the
>> internal
>> structure of molecules. Am I right ?
>>
>> Best wishes,
>>
>> Marek
>>
>>
>>
>> Dne Mon, 01 Apr 2019 02:57:02 +0200 Bill Ross <ross.cgl.ucsf.edu>
>> napsal/-a:
>>
>> > Why would bond (potential) energy be part of temperature? Asking for a
>> > friend. :-)
>> >
>> > Bill
>> >
>> > On 3/31/19 5:35 PM, Marek Maly wrote:
>> >> Hello,
>> >>
>> >> I would like to know how exactly the instantaneous temperature is
>> >> calculated in Amber.
>> >>
>> >> I assume that the Equipartition theorem is used but which degrees of
>> >> freedom are taken in account in case of more complicated molecules
>> >> (flexible models) ?
>> >>
>> >> Could be possible to describe it more in detail on relatively simple
>> >> molecular system composed just of water molecules (flexible molecular
>> >> model of course with bond and bond angle harmonic potentials) or
>> >> eventually to provide the relavant reference ?
>> >>
>> >> My guess is, that the averages of kinetic energy <E_kin> or bond
>> energy
>> >> (if harmonic approximation is used) <E_bond> or the average of both
>> >> energies <E_kin+E_bond> of such molecule could be connected with
>> the
>> >> instantaneous temperature using Equipartition theorem this way.
>> >>
>> >> <E_kin> = 9*0.5*k*T
>> >> <E_bond> = 3*0.5*k*T
>> >> <E_kin+E_bond> = 12*0.5*k*T
>> >>
>> >> but I am not sure.
>> >>
>> >> Thank you in advance,
>> >>
>> >> Best wishes,
>> >>
>> >> Marek
>> >>
>> >>
>> >>
>> >>
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>>
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Received on Mon Apr 01 2019 - 11:00:02 PDT
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