Interact Exercise 6


Put the standard imports for Matplotlib, Numpy and the IPython widgets in the following cell.

In [13]:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np

In [14]:
from IPython.display import Image
from IPython.html.widgets import interact, interactive, fixed

Exploring the Fermi distribution

In quantum statistics, the Fermi-Dirac distribution is related to the probability that a particle will be in a quantum state with energy $\epsilon$. The equation for the distribution $F(\epsilon)$ is:

In [15]:


In this equation:

  • $\epsilon$ is the single particle energy.
  • $\mu$ is the chemical potential, which is related to the total number of particles.
  • $k$ is the Boltzmann constant.
  • $T$ is the temperature in Kelvin.

In the cell below, typeset this equation using LaTeX:

The Fermi-Dirac equation is given by: $$\large F(\epsilon)=\frac{1}{e^{\frac{(\epsilon-\mu)}{kT}}+1}$$ Where:

$\epsilon $ is the single particle energy. $\mu $ is the chemical potential, which is related to the total number of particles. $k $ is the Boltzmann constant. $T $ is the temperature in Kelvin.

Define a function fermidist(energy, mu, kT) that computes the distribution function for a given value of energy, chemical potential mu and temperature kT. Note here, kT is a single variable with units of energy. Make sure your function works with an array and don't use any for or while loops in your code.

In [16]:
def fermidist(energy, mu, kT):
    """Compute the Fermi distribution at energy, mu and kT."""
    return 1/(np.exp((energy-mu)/kT)+1)

In [17]:
assert np.allclose(fermidist(0.5, 1.0, 10.0), 0.51249739648421033)
assert np.allclose(fermidist(np.linspace(0.0,1.0,10), 1.0, 10.0),
    np.array([ 0.52497919,  0.5222076 ,  0.51943465,  0.5166605 ,  0.51388532,
               0.51110928,  0.50833256,  0.50555533,  0.50277775,  0.5       ]))

Write a function plot_fermidist(mu, kT) that plots the Fermi distribution $F(\epsilon)$ as a function of $\epsilon$ as a line plot for the parameters mu and kT.

  • Use enegies over the range $[0,10.0]$ and a suitable number of points.
  • Choose an appropriate x and y limit for your visualization.
  • Label your x and y axis and the overall visualization.
  • Customize your plot in 3 other ways to make it effective and beautiful.

In [43]:
def plot_fermidist(mu, kT):
    fermi = fermidist(np.linspace(0,10.0, 100), mu, kT)
    f = plt.figure(figsize=(9,6))
    ax = plt.subplot(111)
    plt.plot(np.linspace(0,10.0, 100), fermi)
    plt.xlabel(r"Energy $\epsilon$")
    plt.ylabel(r"Probability $F(\epsilon)$")
    plt.title(r"Probability that a particle will have energy $\epsilon$")

In [44]:
plot_fermidist(4.0, 1.0)

In [45]:
assert True # leave this for grading the plot_fermidist function

Use interact with plot_fermidist to explore the distribution:

  • For mu use a floating point slider over the range $[0.0,5.0]$.
  • for kT use a floating point slider over the range $[0.1,10.0]$.

In [46]:
interact(plot_fermidist, mu=(0,5.0, .1), kT=(0.1,10.0, .1));

Provide complete sentence answers to the following questions in the cell below:

  • What happens when the temperature $kT$ is low?
  • What happens when the temperature $kT$ is high?
  • What is the effect of changing the chemical potential $\mu$?
  • The number of particles in the system are related to the area under this curve. How does the chemical potential affect the number of particles.

Use LaTeX to typeset any mathematical symbols in your answer.

a) When kT is low, a particle has a higher probability of having a lower energy $\epsilon < \mu$ and a lower probability of having an energy $\epsilon > \mu$. b) When kT is high, a particle has a more uniform probability of having any energy $\epsilon$. Though, it does still tend to favor lower $\epsilon$. c) The chemical potential $\mu$ affects where the curve $F(\epsilon)$ hits its inflection point. d) A higher chemical potential $\mu$ gives us more area under this curve and thus more particles.

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