Fix Small Typo

.gitignore

@@ -1,4 +1,3 @@
-problems
 _build
-*solutions
+*solutions*
 *ai

_update.sh

@@ -0,0 +1,7 @@
+lecture_file=lecture_files/Lecture$1.md
+git add $lecture_file
+git commit -m "Adding lecture notes for Lecture $1"
+git push
+rm -rf _build
+jupyter-book build .
+ghp-import -n -p -f _build/html

lecture_files/Lecture5.md

@@ -229,7 +229,7 @@ referred to as the [**Gibbs entropy**](https://en.wikipedia.org/wiki/Entropy_(st
 
 $$S = -k_B \sum_j p_j \ln p_j$$
 
-You will shows this expression's equivalence to the Boltzmann definition
+You will show this expression's equivalence to the Boltzmann definition
 of the entropy on [Problem Set 2](../../problems/ps_2/problem_set_2). We now have the expressions:
 
 $$\begin{aligned}

problems/ps_1/problem_set_1.md

@@ -0,0 +1,222 @@
+# Problem Set 1 (Due Wednesday, September 20, 2023) 
+
+## Question 1: Polymer dimensions
+
+Ideal polymer chains are often described as undergoing *random walks* on
+a lattice. Like ideal gases, the monomers in an ideal polymer chain do
+not interact with each other; that is, the segments of the chain do not
+exclude volume and can overlap. Consider a **one-dimensional** ideal
+polymer chain composed of $N$ independent segments.
+
+One end of the chain is placed at the origin. A single chain conformation can then be
+generated by iteratively placing a single segment of the chain a
+distance $b$ in either the positive or negative $x$ dimension from the
+current chain end - *i.e.*, the chain elongates by taking "steps" along
+the one-dimensional coordinate. The end-to-end distance of the chain,
+$x$, is the distance between the origin and the end of the last segment
+placed. 
+
+![image](pset_1_random_walk_fig.png)
+Figure 1 shows an example for $N = 8$ in which the end-to-end distance is after all 8 steps are taken.
+
+```{admonition} **(a)**
+
+For a one-dimensional ideal chain with $N$ segments and segment
+size $b$, calculate the probability, $p(x)$, that the end-to-end
+distance of the polymer is equal to $x$. Assume that there is an **equal
+likelihood** of taking a step in either the positive or negative
+direction for each chain segment.
+
+<details>
+  <summary>Hints</summary>
+  First compute the probability that we take n steps to the right.
+</details>
+
+```
+
+
+```{admonition} **(b)**
+
+For the chain described in part **a**, using Stirling's
+approximation, show that in the large
+$N$ limit the probability distribution $p(x)$ can be approximated by:
+
+$$p(x) = C  \exp \left ( -\frac{x^2}{2Nb^2}\right )$$
+
+where $C=\frac{1}{\sqrt{2\pi Nb^2}}$ is a normalization constant
+to enforce that $\int_{-\infty}^{\infty} p(x)  dx = 1$. Make sure
+to show how you obtain $C$.
+
+<details>
+  <summary>Hints</summary>
+    Define the quantity a = x/Nb, where a << 1 in the large N limit,
+    and write a Taylor series expansion for ln(1-a) when appropriate.
+
+</details>
+
+```
+
+
+```{admonition} **(c)**
+
+Show that the entropy of an ideal one-dimensional chain in the
+large $N$ limit is given by:
+
+$S(N, x) = -\frac{k_Bx^2}{2Nb^2} + S(N) + k_B \ln{C}$.
+
+where $C$ is our normalizing constant from above, and S(N) is the x-independent entropy term.
+
+```
+
+## Question 2: Magnetization of a paramagnet
+
+Consider a system of $N$ distinguishable, non-interacting atoms with
+magnetic dipole moments (spins) in a magnetic field $H$ at constant
+temperature. Each spin $s_i$ has a magnetic moment $\mu$ and can be in
+one of two states: parallel to the field ($s_i = 1$) or anti-parallel to
+the field ($s_i = -1$). The energy of each microstate is due only to
+interactions between the spins and the magnetic field, and is given by:
+
+$$E = -\sum_{i=1}^N s_i \mu H$$
+
+The magnetization of the material is defined as:
+
+$$\begin{aligned}
+M = \sum_i^N s_i \mu = -\frac{E}{H}
+\end{aligned}$$
+
+This model is commonly used to describe paramagnetic materials. In this
+problem, we will derive an expression for the ensemble-average
+magnetization of a paramagnet in a magnetic field.
+
+
+```{admonition} **(a)**
+
+Assuming that the paramagnet has a fixed energy $E$, write an
+expression for the entropy as a function of the number of spins aligned
+with the field, $N_+$, and the total number of spins, $N$.
+
+
+<details>
+  <summary>Hints</summary>
+    Use the [Boltzmann formulation](https://en.m.wikipedia.org/wiki/Boltzmann's_entropy_formula) for entropy.
+
+</details>
+```
+
+```{admonition} **(b)**
+
+Show that the expression for $N_+$ as a function of the number of
+spins $N$, the magnetic field strength $H$, the magnetic moment $\mu$,
+and the temperature $T$ (or as a function of $\beta \equiv 1/k_B T$) is
+
+
+$$ N_+ = \frac{N}{1+exp(-2H\beta\mu)}$$
+
+
+<details>
+  <summary>Hints</summary>
+  Procedurally, this is almost identical to our previous lecture.
+
+</details>
+```
+
+
+
+```{admonition} **(c)**
+Show that the magnetization of a paramagnet is given by:
+
+$$\begin{aligned}
+M = N\mu \tanh (\beta \mu H) 
+\end{aligned}$$
+
+```
+
+## Question 3: Mixing entropy
+
+Consider two different ideal fluids containing $N_1$ and $N_2$
+molecules, respectively, for a total of $N$ molecules. All molecules
+exclude the same molar volume and are assumed to interact weakly with
+each other and with themselves, such that the potential energy of the
+system is negligible. The two fluids are initially completely demixed
+due to an impermeable partition; the partition is then removed and the
+fluids are allowed to mix at constant volume as illustrated in Figure 2.
+Assume also that the molecules of each fluid are **indistinguishable**
+from the molecules of the same fluid.
+
+![image](pset_1_mixing_entropy_fig.png)
+
+```{admonition} **(a)**
+
+Assume that the two fluids occupy a fictitious lattice that
+spans the available volume. Each molecule occupies a single lattice site
+and all lattice sites are occupied; thus, there are are $N_1$ lattice
+sites occupied by molecule 1 and $N_2$ lattice sites occupied by
+molecule 2. Using this approximation, calculate $\Omega(N_1, N_2)$,
+which is defined as the number of microstates for the mixture of fluids.
+
+
+<details>
+  <summary>Hints</summary>
+  Identical arrangements of the fluids are indistinguishable.
+
+</details>
+
+```
+
+```{admonition} **(b)**
+
+Derive an expression for the entropy change associated with
+mixing the two ideal fluids in terms of the mole fractions,
+$x_1 = N_1/N$ and $x_2 = N_2/N$, of the two components. That is,
+determine an expression for:
+
+$$\Delta S_{\textrm{mix}} = S_{\textrm{mixed}} - S_{\textrm{demixed}}$$
+
+<details>
+  <summary>Hints</summary>
+  How many distinguishable configurations are there for totally demixed systems?
+
+</details>
+
+```
+
+```{admonition} **(c)**
+
+Is the lattice model assumption reasonable for molecules
+occupying a continuous set of positions (rather than discrete points)?
+Why or why not?
+
+```
+
+## Question 4: Stirling's approximation
+
+
+```{admonition} Python Exercise
+
+Write a Python program that calculates the percent error of Stirling's
+approximation as a function of $N$, where Stirling's approximation is
+defined as:
+
+$$\ln N! \approx N \ln N - N$$
+
+Include with your solution a copy of your Python code (including
+comments as necessary), a plot of $\ln N!$ and Stirling's approximation
+as a function of $N$ up to $N=100$, and a plot of the error of
+Stirling's approximation as a function of $N$ up to $N=100$. Your grade
+for this problem will be based in part on the readability of your code
+and plots in addition to the accuracy of the solution.
+```
+
+**Note: this problem is intended to provide you with practice in Python
+programming prior to the assignment of the simulation project. If you
+need resources for learning to code in Python, see [here](https://sts.doit.wisc.edu/).**
+
+The two requested plots are provided below. 
+
+![Comparison of Stirling's approximation vs. $\ln N!$ as a function of
+$N$.](pset_1_plot_stirling.png)
+
+![Relative error of Stirling's approximation vs. $N$, which approaches
+zero as $N$ exceeds
+$\approx 50$.](pset_1_plot_stirling_error.png)