Computational Physics Chapter 1 problem 1.5

by defeng kong(2014301020081)

未分类

---

1.Abstract

Consider a decay problem with types of nuclei A and B.The corresponding rate equations are

$$\frac{dN_{A}}{dt}=\frac{N_{B}}{\tau }-\frac{N_{A}}{\tau }$$
$$\frac{dN_{B}}{dt}=\frac{N_{A}}{\tau }-\frac{N_{B}}{\tau }$$
question:take$\tau$=1s,consider different initial conditions,such as$N_{A}$=100,$N_{B}$=0,Show that your numerical results are consistent with the idea that the system reaches a steady state in which $N_{A}$and$N_{B}$are constant.In such a steady state,the time derivatives $\frac{dN_{A}}{dt}$and$\frac{dN_{B}}{dt}$should vanish.

2.Background

In the class of Computational Physics,we learn the algorithm '$N{_{U}}\left ( t+\Delta t \right )\approx N{_{U}}\left ( t \right )-\frac{N{_{U}}\left ( t \right )}{\tau }\Delta t$ to solve the decay problem and how to design and run the program.Now,we will use the way to solve it too.

3.the Main body

Gain the algorithm:

we have

$$\frac{dN_{A}}{dt}=\frac{N_{B}}{\tau }-\frac{N_{A}}{\tau }$$
$$\frac{dN_{B}}{dt}=\frac{N_{A}}{\tau }-\frac{N_{B}}{\tau }$$ $$\Rightarrow$$
$$\frac{d\left ( N_{A}-N_{B} \right )}{dt}=-2\frac{\left ( N_{A}-N_{B} \right )}{\tau }$$ we order NA_NB=$\left ( N_{A}-N_{B} \right )$ have $$\left ( NA_{-}NB \right )\left ( t+\Delta t \right )\approx \left ( NA_{-}NB \right )\left ( t \right )-\frac{\left ( NA_{-}NB \right )\left ( t \right )}{\tau }\Delta t$$

initialize:

if$\tau$=1s, $N_{A}$=100,$N_{B}$=0,we can also set that the time step is 0.05s；

t=0 
NA_NB=100
dt=0.05
T=1

calculate:

t=t+dt
NA_NB=NA_NB-(NA_NB/T)*dt

store:

x=[]
y=[] 
x.append(t)
y.append(NA_NB)

plot:

plt.figure()
plt.xlabel('t') 
plt.ylabel('NA-NB') 
plt.title('decay_NANB')                    
plt.plot(x,y)
plt.text(2,80,'Time Constant=1s' 'Time Step=0.05s')
plt.show()

the whole code:

import matplotlib.pyplot as plt
x=[]
y=[]  
t=0 
NA_NB=100
dt=0.05
T=1  
i=0 
while i<=100:
    x.append(t)
    y.append(NA_NB)
    t=t+dt
    NA_NB=NA_NB-(NA_NB/T)*dt
    i=i+1
plt.figure()
plt.xlabel('t') 
plt.ylabel('NA-NB') 
plt.title('decay_NANB')                    
plt.plot(x,y)
plt.text(2,80,'Time Constant=1s' 'Time Step=0.05s')
plt.show()

4.Conclusion

the graph:

From the graph,we can see when t=5,NA-NB$\approx$0,it can be regarded as the steady state.we also think the NA-NB=0 when t is large.so we have NA=NB when in the steady state.so

$$\frac{dN_{A}}{dt}=\frac{N_{B}}{\tau }-\frac{N_{A}}{\tau }=0$$
$$\frac{dN_{B}}{dt}=\frac{N_{A}}{\tau }-\frac{N_{B}}{\tau }=0$$
because$\frac{dN_{A}}{dt}$=0,$\frac{dN_{B}}{dt}$=0,we will have
$N_{A}$=constant=$N_{B}$

5.Thanks