From: Thomas A. Witten <t-witten@uchicago.edu>
Date: January 29, 2006 1:54:23 PM CST
To: Felipe Barra <felipe.bdlg@gmail.com>
Subject: Re: jarzynski

Many thanks, Felipe -- I am glad you took the time to tell me this.  I 
will keep it for the day when I have to report on all this.  Best, Tom
On Jan 29, 2006, at 10:58 AM, Felipe Barra wrote:

> Hi Tom,
>
> Probably you want to know what we did with Joe Bolte.
> I should have write you long time ago, sorry.
> I will tell you briefly.
>
> We did a very simple simulation of a particle in a one dimensional
> 'box' where the right wall was moving with speed v. The wall has 
> infinite mass.
> Then using the standard collision rule we computed the change of 
> energy of
> the particle after each collision (i.e.the work).
> The fix wall (the left wall) was a thermostat of temperature = 1/beta 
> and therefore at
> each collision the speed changed accordingly, but this energy change 
> is not part of
> the work, is the heat.
> We did many realisation of these process with the piston moving for a 
> given time T
> Êand for each we computed the work
> done on the systems, we did a histogram of this work (i.e. we obtained 
> the
> probability of doing some work W).
>
> With that we checked the Jarzynski relation \int dW P(W) exp (-beta W) 
> = exp (-beta Delta F).
> Ê
> We computed analytically the probability of doing work W during the 
> interval of
> time T if there were N collisions during that time. The number of 
> collision is a random variable
> due to collision with the thermal wall.
> To compute the probability P(N) of having N collision was more 
> difficult and we did not
> succeed to get something useful. We obtained P(N) form the simulations 
> and as far as
> I remember we noticed that for high piston speeds it decreased very 
> fast so we could
> at least obtain approximately P(W) for high values of the piston speed.
> Ê
> Ê
> So the idea of all this was just to illustrate in a simple case the 
> Jarzynski relation
> but in the mean while of our work we find two papers (from 2005!!) 
> where similar things were
> reported (I know the idea was not very original but I was attracted by 
> the result and wanted
> to learn a bit more). Any way in these papers the relation was tested 
> in a case simpler that ours.
> They did not have the thermal left wall, instead they had an infinite 
> (and therefore isolated)system.
> Ê
> I don't know if you are familiar with Jarzynski derivation of his 
> relation (there are others).
> It has two parts: First the relation is proved for an isolated system 
> and there he put the contact with the heat reservoir. The interesting 
> part (at least for me) is the second, but these papers test it for 
> isolated (infinite) systems.
> Ê
> So ours was different but more difficult and we could not get it in 
> his whole range of validity.
> So, that is it. For a while I thought that with more work we could 
> have tried to published now I think differently.
> Ê
> Ê
> But know I propose a subject related to fractures and dislocations. I 
> have been
> working with Fernando Lund in a problem related to the interaction of 
> sound with
> dislocations.
> Well, best regards
> Ê
> Felipe
> Ê
> Ê
>
>
> Ê