Ex_04

Abstract

•Using Python to draw function pictures.
•Using Python to solve differential equation.

Background

•Learning to solve physics problems by using computer.
•Learning how to use class function.

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The Main Body

● Problem 1.5

Consider again a decay problem with two types of nuclei A and B, but now suppose that nuclei of type A decay into ones of type B, while nucleiof type B decay into ones of type A. Strictly speaking, this is not a "decay" process, since it is possible for the type B nuclei to turn back into type A nuclei. A better analogy would be a reasonance in which a system can tunnel or move back and forth between two states A and B which have equal energies. The corresponding rate equations are

$${dN_A\over dt}={N_B \over τ}-{N_A \over τ} \qquad\qquad {dN_B\over dt}={N_A \over τ}-{N_B \over τ}$$
where for simplicity we have assumed that the two types of decay are characterized by the same time constant, . Solve this system of equations for the numbers of nuclei,$N_A$and$N_B$,as functions of time .Consider different initial conditions, such as$N_A=100$,$N_B=0$etc., and take$τ=1s$. Show that your numerical results are consistent with the idea that the system reaches a steady state which$N_AandN_B$are constant. In such a steady state, the time derivatives ${dN_B\over dt}and{dN_A \over dt}$should vanish.

mathematical formula

$$f(x_0+△x)=f(x_0)+\frac{f'(x_0)}{1!} \ △x+\frac{f''(x_0)}{2!}△x^2+......+\frac{f^{(n)}(x_0)}{n!}△x^n+Rn(x)$$
$$f(x_0+△x)≈f(x_0)+\frac{ df(x_0)}{dx} |_{x=x_0} △x$$

● solution

For problem 1.5,it dives us two relevant $N_A$ and $N_B$ which are characterised by two first-order differential equations.We can raplace $dN_A \over dt$with former equation,so,

$$N_A(T+△t)≈N_A(t)+({dN_B\over τ}-{dN_A \over τ})△t$$
It's the same with $N_B$
Taylor expansion for $N_A$：
$$N_A(T+△t)≈N_A(t)+({dN_B\over τ}-{dN_A \over τ})△t$$
Similarly，
$$N_B(T+△t)≈N_B(t)+({dN_A\over τ}-{dN_B \over τ})△t$$

Code.1 Shows the picture when initial NA=100,NB=20

import pylab as pl
class uranium_decay:
    def __init__(self, number_of_NA = 100,number_of_NB = 0, time_constant = 1, time_of_duration = 5, time_step = 0.05):
        self.na_uranium = [number_of_NA]
        self.nb_uranium = [number_of_NB]
        self.t = [0]
        self.tau = time_constant
        self.dt = time_step
        self.time = time_of_duration
        self.nsteps = int(time_of_duration//time_step + 1)
        print("Initial number of NA ->", number_of_NA)
        print("Initial number of NB ->", number_of_NB)
        print("Time constant ->", time_constant)
        print("time step -> ", time_step)
        print("total time -> ", time_of_duration)
    def calculate(self):
        for i in range(self.nsteps):
            tmpa = self.na_uranium[i] + (self.nb_uranium[i]-self.na_uranium[i])/self.tau * self.dt
            tmpb = self.nb_uranium[i] + (self.na_uranium[i]-self.nb_uranium[i])/self.tau * self.dt
            self.nb_uranium.append(tmpb)            self.na_uranium.append(tmpa)
            self.t.append(self.t[i] + self.dt)
    def show_results(self):
        pl.plot(self.t, self.na_uranium)
        pl.plot(self.t, self.nb_uranium)
        pl.xlabel('time ($s$)')
        pl.ylabel('Number of Nuclei')
        pl.show()
a = uranium_decay()
a.calculate()
a.show_results()

result: NA=100 NB=0

now we change the initial conditions：
result: NA=100 NB=50
result: NA=20 NB=80

● Conclusion

Euler mothed is powerful to solve simple first order ordinary differential equations.draw figures by Python is a little difficult as well as interesting.It is essential to master the basic Python syntax，for me,there is still a long way to go.

Thanks to

Hu Shengyong