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Comp. Physics: Finite time Carnot cycle 
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#1
Apr213, 06:59 AM

P: 46

Hi,
For my Bachelor's thesis I've been working on a finite time Carnot cycle. I've finished my numerical analysis using the differential equations governing the time evolution. My next step should be a simulation. First I should stick to a 1 dimensional system. This system consists of a piston and a thermalising wall. This thermalising wall acts as a heat reservoir. Every particle colliding with this wall is absorbed. The wall than ejects a 'new' particle with a certain velocity. This velocity is governed by the stochastic distribution [itex]f(\vec{v},T_i)=C v \exp\left(\frac{m v^2}{2kT_i}\right)[/itex] with [itex]i=h[/itex] while expanding and [itex]i=c[/itex] while compressing. C is the normalisation constant. Since I use a 1D system in this first approximating step, the MaxwellBoltzmann distribution isn't necessary. The piston has a constant velocity u. This is chosen because of the fact that the article I base my calculations on is targeting a system that is easy to control. The article is "Molecular Kinetic analysis of a finitetime Carnot cycle" by Y. Izumida and K. Okuda published in september 2008. I reckon I have to use some sort of MonteCarlo method because of the stochastic nature of the reservoir. I have however not a clue on how to start. But the paper talks about Molecular Dynamics. Am I thinking about it in a wrong way? My previous experience with computational physics is small. I've only worked with a driven pendulum using the GSL library and the Ising model using the metropolis algorithm. Sincerely, Joris 


#2
Apr213, 07:17 AM

Sci Advisor
PF Gold
P: 1,344

First of all, I don't think you'll be able to do it in 1D. Onedimension problems are peculiar because the particles can't overtake each other, and this can create strange effects. Since you have a purely classical system, I see no point in not doing it in 3D from the start.
To find out how to do it, you have to look up molecular dynamics. A book about computational physics would be a good start, for instance, J. M. Thijssen, i]Computational Physics[/i] (Cambridge University Press, 1999). 


#3
Apr213, 08:16 AM

P: 46

Thanks for the quick response.
I get there as well that 1D is not really useful. Further more I understood the teacher wrong. We decided to move the thermalising wall to make the problem in essence 1D. The change was that I could neglect collisions in the ydirection because they became perfectly elastic. Thanks for the book reference. 


#4
Apr213, 08:38 AM

Sci Advisor
PF Gold
P: 1,344

Comp. Physics: Finite time Carnot cycle



#5
Apr213, 12:09 PM

P: 46

You're absolutely right. Forgot that initially.



#6
Apr213, 03:23 PM

P: 46

Ok, I'm stuck again.
The book was useful for grasping the basic ideas. I found how I could initialize the system. And understood the idea from appendix B.3 (boxmuller transform). However I believe I have to find something similar for collisions with the thermalising wall. The distribution now (in the expanding stage) is [itex]f( \vec{v} ,T_h )=C v_x \exp\left( \frac{m v^2}{2kT_h} \right) [/itex]. If I try to use the boxmuller transform I'd get following expression [itex]P(v_x,v_y)dv_xdv_y= C v_x\exp \left(\frac{v^2}{2}\right) dv_xdv_y=C v^2\exp \left(\frac{v^2}{2}\right) \cos (\phi )dvd\phi=P(v,\phi)dvd\phi[/itex] After that I would find (in analogy of the appendix mentioned before) [itex]\left( g^{1}\right) ^\prime (v,\phi)=C v^2\exp \left(\frac{v^2}{2}\right) \cos (\phi )[/itex] (1) I used g for to stress the difference with the original distribution I don't see how I can get similar expressions as for the MaxwellBoltzmann distribution, to get random numbers distributed like [itex]f(\vec{v},T_h)[/itex]. I suspect I need another way to do this. BoxMuller doesn't seem to work at first sight. Furthermore does the derivative of the inverse function in (1) even make sense? Does it mean [itex]g^\prime(x,y)=\frac{\partial g(x,y)}{\partial x}+\frac{\partial g(x,y)}{\partial y}[/itex]? 


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