# Comp. Physics: Finite time Carnot cycle

by JorisL
Tags: carnot, comp, cycle, finite, physics, time
 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 $f(\vec{v},T_i)=C v \exp\left(-\frac{m v^2}{2kT_i}\right)$ with $i=h$ while expanding and $i=c$ while compressing. C is the normalisation constant. Since I use a 1D system in this first approximating step, the Maxwell-Boltzmann 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 finite-time Carnot cycle" by Y. Izumida and K. Okuda published in september 2008. I reckon I have to use some sort of Monte-Carlo 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
 Sci Advisor PF Gold P: 1,344 First of all, I don't think you'll be able to do it in 1D. One-dimension 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).
 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 y-direction because they became perfectly elastic. Thanks for the book reference.
 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 (box-muller transform). However I believe I have to find something similar for collisions with the thermalising wall. The distribution now (in the expanding stage) is $f( \vec{v} ,T_h )=C v_x \exp\left( -\frac{m v^2}{2kT_h} \right)$. If I try to use the box-muller transform I'd get following expression $P(v_x,v_y)dv_xdv_y= C v_x\exp \left(\frac{-v^2}{2}\right) dv_xdv_y=C v^2\exp \left(\frac{-v^2}{2}\right) \cos (\phi )dvd\phi=P(v,\phi)dvd\phi$ After that I would find (in analogy of the appendix mentioned before) $\left( g^{-1}\right) ^\prime (v,\phi)=C v^2\exp \left(\frac{-v^2}{2}\right) \cos (\phi )$ (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 Maxwell-Boltzmann distribution, to get random numbers distributed like $f(\vec{v},T_h)$. I suspect I need another way to do this. Box-Muller doesn't seem to work at first sight. Furthermore does the derivative of the inverse function in (1) even make sense? Does it mean $g^\prime(x,y)=\frac{\partial g(x,y)}{\partial x}+\frac{\partial g(x,y)}{\partial y}$?