From: Marcos Sotomayor (sotomayo_at_ks.uiuc.edu)
Date: Fri Feb 09 2007 - 16:49:13 CST
Sterling,
This discussion is getting interesting and, actually, I think I did not
express myself correctly below (only about the time-dependent part). The
work should be indeed \int F*dd, however your integration step in the F
\times d plot should be as small as possible when your force is changing on time.
The difference between the work done and the energy change should be only
numerical when using an NVE ensemble. Good thing, we can test it! I
suggest to perform a small simulation in which you output force and
distance every time step to check it.
Marcos
On Fri, 9 Feb 2007, Sterling Paramore wrote:
> Marcos,
>
> Well, I'd like to keep the First Law no matter what. So I'd say that for
> adiabatic extension, however the energy changed should be exactly equal to
> the work. I would think that it would end up being \int F*d, but I could be
> wrong.
>
> -Sterling
>
> Marcos Sotomayor wrote:
>
>>
>> Hi Sterling,
>>
>> Let me be more specific. First, when the external force applied to your
>> system is time-dependent, the Hamiltonian of your system is not conserved
>> anymore and does not represent the total energy. Moreover, the expression F
>> \times d, as far as I can tell, is no longer the work done on the system
>> (usually one assumes a time independent hamiltonian/force to derive that
>> expression). So my sentence should have read like: " The quantity F \times
>> d (previously work done) should only be similar to the energy change".
>> Please correct me if I am wrong on that.
>>
>> I think you are right in that the total amount of work done by the external
>> force does match the energy change, but you cannot use Fd to measure it.
>> Therefore, the first law of thermodynamics is not violated. There is a nice
>> article on a similar topic that you may want to check:
>>
>> "Invariants for time-dependent Hamiltonian systems"
>> Jurgen Struckmeier and Claus Riedel
>> Physical Review E 64 026503
>>
>> Regards,
>> Marcos
>>
>>
>>
>
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