Simulation of the fission as a stochastic process

[Heimdall]

Hello


I'm a french student, I'm actually not sure this is the good place to ask my question but as it deals with the nuclear fission I try here... don't hesitate to tell me if there is a better forum... thx..


well, I'm trying to solve numerically the Langevin equation, initially for the brownian motion, it is also used in nuclear physics as you probably know, to describe the fission of nuclei...

the Langevin equation is :

$$\ddot{x}+\beta \dot{x} + \frac{1}{m}\frac{\partial U}{\partial x} = \Gamma(t)$$

where $$\beta=\frac{\gamma}{m}$$ is a friction term (from a stocke's law) divided by the mass of the brownian particle. (so $$\beta$$ is $$s^{-1}$$)

$$U(x)$$ is an extern potential

and $$\Gamma(t)$$ is a stochastic force divided my the mass.


well, now I decide, as Kramers did, to tell that my potential U is represented by two parabolas as you can see on http://nicolas.aunai.free.fr/courb.htm


I call B the height of the maximum, and $$x_b$$ the correspondant X-coordonnate (assuming that the minimum will be at x=0)

I decide that B will be my caracteristic energy for the problem, and $$x_b$$ my caracteristc length.


I want, in order to solve my equation numerically, to write it without dimension, so I introduce the following new variables :

$$Q = \frac{x}{x_b}$$

$$\Pi = \frac{U}{B}$$

$$t'=\omega t$$ where $$\omega$$ is the caracteristic pulsation of my parabolas

i.e. the equations of the parabolas are :

$$U_1(x)=\frac{1}{2}m\omega^2 x^2$$ for $$x<x_0$$

$$U_2(x)=-\frac{1}{2}m\omega^2(x-x_b)^2 + B$$ for $$x>x_0$$

assuming the continuity of the potential, the functions and the derivatives shoud be equal between them at $$x_0$$ the junction point. these conditions give us the height B which is equals to :

$$\frac{1}{4}m\omega^2 x_b^2$$


well now we have to re-write the langevin equation, replacing by the new variables... and here began my problems...


$$\ddot{x}+\beta \dot{x} + \frac{1}{m}\frac{\partial U}{\partial x} = \Gamma(t)$$

$$\dot{x}$$ means $$\frac{dx}{dt}$$, so since we have $$t'=\omega t$$, the derivation by t' gives us an $$\omega$$



$$\omega^2 x_b\ddot{Q}+\beta \omega x_b \dot{Q} + \frac{B}{m}\frac{\partial \Pi}{\partial Q} \frac{\partial Q}{\partial x} = \Gamma(t)$$

we know that $$\frac{\partial Q}{\partial x} = \frac{1}{x_b}$$

so we have :

$$\omega^2 x_b\ddot{Q}+\beta \omega x_b \dot{Q} + \frac{B}{mx_b}\frac{\partial \Pi}{\partial Q}= \Gamma(t)$$


and with the B value we have :

$$\omega^2 x_b\ddot{Q}+\beta \omega x_b \dot{Q} + \frac{\omega^2 x_b}{4}\frac{\partial \Pi}{\partial Q}= \Gamma(t)$$

finally, dividing by $$\omega^2 x_b$$ we obtain :

$$\ddot{Q}+\frac{\beta}{\omega}\dot{Q} + \frac{1}{4}\frac{\partial \Pi}{\partial Q}= \frac{1}{\omega^2 x_b}\Gamma(t)$$

which I think is almost the result that I'm looking for.. since the first term is without dimention, $$\frac{\beta}{\omega}$$ is without dimension too, and the third term too..

my problem is the stochastic term...

now I don't know how to do with it... if you have an idea...

I now, that the autocorrelation function of the Langevin Force is :

$$<\Gamma(t)\Gamma(t')> = 2\beta T \delta(t_1-t_2)$$

since $$t_i = \frac{t_i'}{\omega}$$, can I write the following equation :

$$<\Gamma(t)\Gamma(t')> = 2\beta T \delta(\frac{t_1'}{\omega}-\frac{t_2'}{\omega})$$

and with $$\delta(a x)=\frac{1}{a}\delta(x)$$

I find :

$$<\Gamma(t)\Gamma(t')> = 2\omega \beta T \delta(t_1'-t_2')$$


so $$\Gamma(t)$$ should be proportionnal to $$\sqrt{(2\omega\beta T \delta(t_1'-t_2'))}$$

in the nuclear system, k (the Boltzmann constant) is equal to 1, so we can also introduce the new variable :

$$\Theta=\frac{T}{B}$$

so we can put it in our relation :

$$\Gamma(t)\ \ \ \alpha\ \ \ \sqrt{(2\omega\beta B\Theta\delta(t_1'-t_2'))}$$


well... I'm not sure where I'm going with this... can someone help me ?


thanks !

 

[AndyW]

Hello there, it's great to see that you are interested in nuclear physics and are trying to solve the Langevin equation numerically. This is definitely the right place to ask your question and I will do my best to help you out.

First of all, it is important to note that the Langevin equation is a stochastic equation, meaning that it involves random variables and cannot be solved deterministically. This is why you are having trouble with the stochastic term in your equation. The Langevin equation is typically solved using numerical methods, such as Monte Carlo simulations, to approximate the solution.

As for the rest of your calculations, they seem to be on the right track. However, I would suggest consulting with a nuclear physicist or a mathematician to ensure that your calculations are correct. Also, make sure to double check your units and ensure that they are consistent throughout your equations.

I hope this helps and good luck with your research! Don't hesitate to reach out if you have any further questions.

 

[sir-pinski]

Hello,

Thank you for your question and for sharing your work on simulating the fission process as a stochastic process. It seems like you have a good understanding of the Langevin equation and have made some progress in solving it numerically. However, I am not an expert in nuclear physics, so I cannot provide specific guidance on your equations. I would suggest seeking help from a professor or a more specialized forum for this topic.

One thing I can suggest is to check your units and make sure they are consistent throughout your equations. Also, it might be helpful to look at some existing literature on simulating fission as a stochastic process to see how others have approached it. This could give you some insight and potentially help you with your equations.

Overall, it seems like you are on the right track and have a good understanding of the concepts involved. Keep working on it and seeking help when needed, and I'm sure you will make progress. Good luck!