Mathematics – Understanding Physics and Astronomy

A Very Brief Introduction to Measure Theory and the Lebesgue Integral (Part I)

In the next few posts, I shall be discussing recent topics of study that, to me at least, have been very intruiging. In previous posts, I have talked about Hilbert spaces. I have of late been considering the mathematics necessary to formally understand in a pure mathematical sense what a Hilbert space is. This post, like the others on this site, serves as a reference of newly learned topics that are of interest (to me, at least; such a comment is subjective, of course).

The purpose of this post is two-fold: (1.) to provide an update with what I’ve been up to; (2.) introduce some interesting mathematics that have expanded my understanding to the “size” of a set as well as operations such as differentiation and integration.

Here is a quick summary of what I plan to cover in the next few posts (to a brief extent):

Elementary Sets and their Measure: Here I will discuss the concept of length and try extend length in greater dimensions to that of a measure of a set. Much of this topic will rely on geometric intuition.
Lebesgue Measure: This section will dicuss the concept of Lebesgue measure and distinguish it from the elementary measure. Also brief mention will be made of measurability of sets and functions.
General Measure: Discussion will be made of a general measure as a function as well as measurable spaces and measure spaces.
Lebesgue Integral: This topic will introduce the concept of the Lebesgue integral as compared to the Riemann integral.
$L^{p}$ and $l^{p}$ spaces: This section will discuss the concept of a norm as it relates to the spaces $L^{p}$ and $l^{p}$ , and will define each space. We will also introduce the concept of Banach spaces.
Proof that $l^{p}$ space is a Banach space.

Section 1: Elementary Sets and their Measure:

The question that we want to answer is this: Given an arbitrary set, how do we go about measuring it?

In order to understand the difficulties present in this question we must first consider what are called elementary sets and the elementary measure. Elementary sets are those sets which are intuitively easy to measure; that is, intervals, rectangles, and boxes. We now give the formal definition of an elementary set:

Definition. (Interval; Elementary Set) We define an interval to be a subset of the real line $\mathbb{R}$ which take one of the following forms:

$[a,b] := \{x\in \mathbb{R}|a\leq x \leq b\} \label{(1.1)}$ ;

$[a,b) := \{x\in \mathbb{R}|a\leq x < b\} \label{(1.2)}$ ;

$(a,b] := \{x\in \mathbb{R}|a< x\leq b\} \label{(1.3)}$ ;

$(a,b):= \{x\in \mathbb{R}|a< x< b\} \label{(1.4)}$ ,

The length of an interval $I=[a,b]$ denoted $l(I):= b-a$ . For dimensions $d\geq 2$ , we define the measure of such sets as equalling the $d$ -times Cartesian product of intervals $I_{d}$ ; that is,

$\displaystyle m(B) := \prod_{i=1}^{d}l(I_{i}); \label{(2)}$

we sometimes call sets of dimension 2 or greater as “boxes.” Thus, elementary sets are those subsets of $\mathbb{R}^{d}$ such that

$\displaystyle m(E)= \bigcup_{i=1}^{d}m(B_{i}), \label{(3)}$

where $B$ is $i$ -th $d$ -dimensional box contained in $\mathbb{R}^{d}$ .

What this definition is doing is the following: first it introduces the concept of an interval and establishes the well-understood concept of its length as being the difference between the two endpoints provided one is less than the other. The definition then generalizes the idea of a length to 2 and 3 dimensions and beyond. Note that in 2-dimensions the interval then becomes a rectangle in the plane. Thus, the measure of the length of an interval then becomes the measure of the area of a rectangle. Similarly, for $d=3$ we replace rectangles with cubes and the area with the volume. For dimensions $d>3$ , we replace cubes with boxes of $d$ -dimension. Therefore, elementary sets are those subsets of $d$ -dimensional real space that are unions of finitely-many boxes.

Section 2: Lebesgue Measure

In the last section, we discussed sets for which we can measure quite easily. Though ideally we would like to be able to measure more general sets; that is, sets that are more general than elementary sets. Therefore, we require a different way of measurement. Thus, we come to need the Lebesgue measure.

In order to introduce the Lebesgue measure we need to first introduce the concept of the outer measure, which we now define

Definition. (Outer Measure) We define the outer measure of a set $E\subset \mathbb{R}$ , denoted $m^{*}(E)$ to be

$\displaystyle m^{*}(E) = \inf\bigg\{\sum_{k=1}^{\infty}l(I_{k})|\forall k\in \mathbb{N}, I_{k} \text{ is open such that } E \subset \bigcup_{k=1}^{\infty}I_{k}\bigg\}.$

The outer measure of a set in a sense “overestimates” the size of a given set and then takes the smallest such overestimate to within a specified tolerance. Thus, it estimates the size of the given set “from the outside,” and is used in lieu of the elementary measure when we are dealing with sets that we cannot easily measure the set in a geometrically-intuitive way.

We conclude this post with the definition of the Lebesgue measure given in two forms; the first will be in terms of what we have defined so far, and the second will be defined in terms that will be covered in the next post.

Definition. (Lebesgue Measure.) We define the Lebesgue measure of the set $E$ to be a set whose measure $m(E)=m^{*}(E)$ ; that is, its measure is equal to the outer measure.

The second way of defining this is as follows:

Definition. (Lebesgue Measure V.2) The Lebesgue measure is the measure on the measureable space $(\mathbb{R},\mathcal{L})$ where $\mathcal{L}$ is the $\sigma$ -algebra of Lebesgue measurable subsets of $\mathbb{R}$ that assigns to each Lebesgue measurable set its outer measure.

The next post will discuss measures in general, as well as measurable sets, measureable spaces, Borel sets, and $\sigma$ -algebras.

Until then, clear skies!

Open covers, Finite Subcovers, and COMPACTNESS

A second topological concept that is introduced in analysis is compactness. It is a concept that is associated with the Bolzano-Weierstrass Theorem which is as follows

THM. (Bolzano-Weierstrass). Let $A$ be any infinite bounded set of $\mathbb{R}$ . Then there is at least one $x\in \mathbb{R}$ such that every open ball centered on $x$ will contain at least one point in $A$ .

The idea of the proof of this statement is to show that the intersection $B_{x}(\epsilon)\cap A \neq \emptyset$ .

Insofar as compactness is concerned, there are a few different ways to introduce the concept. I will present the various definitions and show that they are all equivalent.

Method 1: Open Covers and Finite Subcovers.
In order to define compactness in this way, we need to define a few things; the first of which is an open cover.

Definition. [Open Cover.] Let $(X,d)$ be a metric space with the defined metric $d$ . Let $A\subset (X,d)$ . Then an open cover for $A$ is a collection of open sets $\{O_{\alpha}|\alpha \in \mathbb{N}\}$ such that
$\displaystyle A \subset \bigcup_{\alpha\in \mathbb{N}}O_{\alpha}$ .

N.B. The collection of open subsets $O_{\alpha}$ may be of infinite cardinality.

Another definition that we will need is the following:

Definition. [Limit Point/Cluster Point.] Let $(X,d)$ be a metric space and let $B\subset (X,d)$ , and let $x_{0}\in X$ . Then $x_{0}$ is a limit point or a cluster point of $A$ if any open ball of center $x_{0}$ contains an infinite number of points from $A$ .

We need one more definition before we define compactness:

Definition. [Finite Subcover.] Given an open cover $\{O_{\alpha}\}$ , a finite subcover is a finite subcollection of open sets from $\{O_{\alpha}\}$ such that
$\displaystyle \bigcup_{\alpha = 1}^{n}O_{\alpha}$ .

Therefore, we can now definite compactness as follows:

Definition. [Compact Set.] Let $(X,d)$ be a metric space with the defined metric $d$ , and let $A\subset (X,d)$ . Then we say that $A$ is compact if every open cover for $A$ has a finite subcover.

To make this more concrete, consider the following example:
Example: Let $X= \mathbb{R}$ and let $d:\mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R} \triangleq d(p,q)=|p-q|.$ Then the open interval $(0,1)$ is not a compact set. To see why consider the set of open subsets $(1/n,1)$ for $n\in \mathbb{N}$ . Note that $(0,1)\subset \bigcup_{n \in \mathbb{N}}(1/n,1)$ . However,
$\displaystyle (0,1) \not\subset \bigcup_{n=1}^{m}(1/n,1)$ . In other words, (or rather in words) what this says is that if we consider all of the open sets of the form $(1/n,1)$ (e.g. (1,1), (1/2,1), (1/3,1)…). We see that for each n =1,2,3,…, the open set increases in size*. Thus, if we consider all the elements that are in at least one of these increasing intervals, then the union $(1,1)\cup (1/2,1) \cup (1/3,1) \cup ... \cup (1/n,1)$ contains the interval $(0.1)$ . However, note that if we take only a finite number $m$ , for simplicity say $m=3$ , then we have that the union $(1,1) \cup (1/2,1) \cup (1/3,1)$ does not contain all of the points that are contained in $(0,1)$ . Therefore, what this says is that while we can form an open cover for $(0,1)$ we cannot find a finite subcover for that set. Therefore, $(0,1)$ is not a compact set.
*: There is a concept related to the size of an interval which lends itself to a field of study in analysis called measure theory (may post on this topic at a later time).

Method 2: Sequences and Subsequences:
This approach has the benefit that we can just state the definition outright:

Definition. [Compact Set.] Let $A\subset \mathbb{R}$ is compact if every sequence in $A$ has a subsequence that converges to a limit that is also in $A$ .

There is one other type of “definition” used to understand compactness. Some books call this the Characterization of Compactness on the Real Line.
Theorem. [Compact Set.] Let $A\subset \mathbb{R}$ . Then $A$ is compact if and only if $A$ is closed and bounded.

The following theorem states that each of these different ways that are used to define compactness are in fact equivalent:

Theorem. Let $A\subset \mathbb{R}$ . Then each of the following statements are equivalent:
(1.) $A$ is compact;
(2.) $A$ is closed and bounded;
(3.) Every open cover $\{O_{\alpha}\}$ of $A$ has a finite subcover.

The implication of (2.) implies (1.) is what is referred to as the Heine-Borel Theorem. Furthermore, to circle back to the Bolzano-Weierstrass Theorem we can rewrite this statement in terms of compactness:

Theorem. [Bolzano-Weierstrass Theorem.] Let $(X,d)$ be a compact metric space, and let $A$ be an infinite subset of $(X,d)$ . Then $A$ has at least one cluster point.

The next post will discuss the proofs of the theorems in this post. Further posts will most likely be on astrophysics and/or cosmology. Until then, clear skies!

CONVERGENT SEQUENCES, CAUCHY SEQUENCES, COMPLETENESS

If one takes quantum mechanics, when they first encounter the wavefunction which is a complex-valued function, they learn that the arena in which quantum mechanics is a Hilbert space. If one goes further in order to understand what a Hilbert space is they find that it is a complete inner product space. While many physicists take advantage of this fact, they do not really interest themselves with what this means in a rigorous mathematical sense. When I first encountered this, I was unsatisfied with the so-called “definition” of a Hilbert space. So I found that I had to learn more advanced mathematics; more specifically, real analysis. To that end, the purpose of this post is to understand what the term “complete” means. To remedy any confusion of what an inner product space is, an inner product space is a vector space $V$ that equipped with an inner product $\langle u, v \rangle$ .

In order to understand what completeness is, we require a couple of definitions:

Definition. Let $\{p_{n}\}_{n=1}^{\infty}$ where $n\in \mathbb{N}$ be a sequence of points in the metric space $(E,d)$ . A point $p\in \mathbb{E}$ is called a limit of the sequence of points if for any $\epsilon>0$ , there exists $N\in \mathbb{N}$ such that if $n>N$ , $d(p,p_{n})< \epsilon$ . If such a limit exists, then we say that the sequence of points $\{p_{n}\}_{n=1}^{\infty}$ converges to the point $p\in E$ .

What this says intuitively, is that in the sequence of points above there exists a term for which $n=N$ which corresponds to the point $p_{N}$ in the metric space $E$ , beyond which any later terms in the sequence will be contained in what we call an open ball which is defined to be the set given by $B_{p}(\epsilon)= \{q\in E|d(q,p)<\epsilon\}$ . We can regard the term $p_{N}$ as a “boundary point”.

Definition. A sequence of points $\{p_{n}\}_{n=1}^{\infty}$ in a metric space $(E,d)$ is said to be a Cauchy sequence, if for any $\epsilon>0$ , there exists $N\in \mathbb{N}$ such that whenever $n,m>N, d(p_{n},p_{m})< \epsilon$ .

The intuitive idea behind this concept is that suppose we take two terms in the sequence of points $\{p_{n}\}_{n=1}^{\infty}$ , we say that it is a Cauchy sequence if whenever these two chosen terms are “beyond the boundary” the distance between these two terms are within $\epsilon$ of each other in the metric space $(E,d)$ .

One important result that I am not going to prove is the following:

Theorem. If $\{p_{n}\}_{n=1}^{\infty}$ is a convergent sequence of points in the metric space $(E,d)$ , then such a sequence is Cauchy.

An important note: the converse of this theorem is not necessarily true. If the converse is indeed true, we get the following definition:

Definition. A metric space $(E,d)$ is said to be complete if every Cauchy sequence of points in the metric space $(E,d)$ converges to a point $p\in E$ .

An example of this is that $\mathbb{R}$ with the metric $d(p,q)=|p-q|$ is a complete metric space. Intuitively, what this means is that given a Cauchy sequence that converges in $\mathbb{R}$ to a real number. In other words, any possible Cauchy sequence will converge to some real number $p$ .

The next post will discuss compactness in the context of metric spaces, covers, and open covers.

Clear Skies!

Introduction to Metric Spaces

Metric Spaces are one of those mathematical topics that everyone intuitively understands. The best example of this is that of three dimensional Euclidean space $E^{3}$ . This serves as the basis for the intuitive concept of a “space”, and our ability to ascribe a distance between to points in three-dimensional space can be described by a distance function $d: E\times E \rightarrow E$ , or a metric. The underlying set $E$ together with the metric $d$ form what is called a metric space $(E,d)$ .

To be more mathematically precise, we make the following definition.

Definition. A metric space is a set $E$ together with a rule which associates with pair $p,q\in E$ a real number $d(p,q)$ such that
$\displaystyle d(p,q)\geq 0, \forall p,q\in E$
$\displaystyle d(p,q) = 0 \iff p=q$
$\displaystyle d(p,q)=d(q,p) \forall p,q\in E$
$\displaystyle d(p,r) \leq d(p,q)+ d(q,r)$

As an example, suppose that the underlying set $E = \mathbb{R}$ and the metric coupled with this set is defined by $d(p,q)= |p-q|$ . To verify that this indeed a metric space we must show that the four axioms are satisfied.

Claim. The mathematical structure $(\mathbb{R},d)$ in which $d:\mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ is defined by $d(p,q)=|p-q|$ is a metric space.
Proof. Let $p,q\in \mathbb{R}$ . Then by definition of $d(p,q)$ and by definition of the absolute value function, we have that $|p-q|\geq 0$ , so that axiom 1 is satisfied. Suppose now that the points in $\mathbb{R}$ are equal, i.e. that $p=q\in \mathbb{R}$ . Then by definition of $d(p,q)$ , we have that $|p-q|=|0|=0.$ Conversely, suppose that $|p-q|= 0.$ By the triangle inequality we have that $|p-q|\geq |p|-|q|=0$ This implies that $|p|= |q|$ . Thus, condition (2.) is satisfied and hence the distance between two points in $\mathbb{R}$ is zero if and only if the two points are the same. To prove condition (3.), let $p,q\in \mathbb{R}$ , so that $d(p,q) = |p-q|$ . By virtue of the definition of the absolute value, we can say that
$\displaystyle d(p,q) = |p-q| = |-(p+q)|= |-1||q-p|= |q-p|= d(q,p).$
Thus, we see that the arguments of the proposed distance function is symmetric with respect to its arguments, namely any real numbers $p,q\in \mathbb{R}$ . To prove condition (4.), consider any three points $p,q,r\in \mathbb{R}$ , then the distance function between the points $p,r \in \mathbb{R}$ becomes
$\displaystyle d(p,r) = |p-r| = |p-q+q-r|,$
wherein we add zero in the form of adding and subtracting the point q (a very common trick in analysis). Then by the properties of the absolute value it follows that
$\displaystyle d(p,r) = |p-q+q+r| \leq |p-q|+|q-r| = d(p,q)+d(q,r)$ ,
where by the last equality follows from the definition of the distance function. Therefore, all four conditions have been satisfied, and hence by our definition of a metric space, it follows that $(\mathbb{R},d)$ whose distance function $d:\mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ is defined by $d(p,q) = |p-q|$ is indeed a metric space. $\square$

Introduction to Groups

IMAGE OBTAINED FROM pixabay.com.

Abstract algebra can be broken into about two or three sections: (1) Groups; (2) Rings and Fields; (3) Vector Spaces. (A fourth topic that could be considered its own is Galois Theory.) The typical way this version of algebra is introduced is to start by covering groups, then to introduce the concepts of rings and fields in the context of group theory. Rings are interesting, but they are not of interest to us right now. In this post, I attempt to define what a group is and to explain what the definition means. I then follow this up with examples of groups using the real numbers, integers, and complex numbers. In the next post, I will attempt to prove that the real numbers and integers are groups under addition and that the complex numbers form a group under complex multiplication. To that end we must introduce the concept of a group.

Definition. A group $G$ is a non-empty set equipped with a single binary operation $\ast$ that satisfies the following four axioms:

For all $g_{1}, g_{2}\in G$ , $g_{1}\ast g_{2}\in G$ ;
For all $g_{1}, g_{2}, g_{3}\in G$ , $g_{1}\ast (g_{2}\ast g_{3}) = (g_{1}\ast g_{2})\ast g_{3}$ ;
For all $g\in G$ , there exists an element $g^{-1}\in G : g \ast g^{-1}=e_{G}=g^{-1} \ast g$ ;
For all $g\in G$ , there exists an element $e_{G}\in G$ such that $g \ast e_{g}=g=e_{g} \ast g$

This definition might not seem very helpful, but it’s the reason as to why we are allowed to use the properties of numbers that we used in high school. What this definition says is that we have a collection of objects that we call elements that is endowed with a binary operation. An example of this would be traditional addition or multiplication. The word binary simply means that it requires two elements to produce another element. For instance, consider the following group $G=(\mathbb{Z},+)$ (we’ll discuss what this notation means later on in the post). Let $a=2\in \mathbb{Z}$ and let $b=3\in \mathbb{Z}$ . We can add these two integers to get another integer, call it $c=5\in \mathbb{Z}$ . This is what we mean by a binary operation.

Now that we understand what the operation is, we can get into what it takes for a set to be a group. Statement 1 simply says that given any two elements in the set $a,b$ , if $a \ast b$ is also in the set, then we say that the set is closed under the operation $\ast$ . Statement 2 says that given any three elements in the set, the order in which the operation is performed, as dictated by the parentheses, is immaterial. This statement ensures that the elements of the set are associative. To clarify this a bit more, consider the following sum in $(\mathbb{Z},+)$ :

$2 + (3 + 4)$

The second statement says that I can remove the parentheses from the sum without it changing value. Indeed, in each case one gets 9, and so that’s what statement 2 ensures: associativity. Statement 3 says that there is an inverse element to each element in the set. For addition, this inverse element is the additive inverse, by which I mean that for every $a\in G, a+(-a)=e_{G}$ . For multiplication, the inverse element is the multiplicative inverse, in which case we have that for every element $a\in G, a\cdot a^{-1}=e_{G}$ . In the definition, I was careful to include both the right and left inverse. The reason for this is not all sets of elements commute, that is it is not always true that $ab=ba$ . As an example, let $M(2,\mathbb{R})$ be the set of all $2\times 2$ matrices whose determinant is non-zero. Matrices are known to non-commutative, and so $\textbf{AB}\neq\textbf{BA}$ . If it is true that every element in the group commutes, then $G$ is referred to as an abelian group.

Finally, statement 4 ensures that there is an identity element. In the definition and explanation of statement 3, I denoted this as $e_{G}$ . For addition, the identity element is $0$ , since given any element of $G$ , if we add $0$ to it we get the element again. Furthermore, for multiplication, the identity is 1 since anything multiplied by 1 is itself. Note that for addition and multiplication we have different forms for the inverse and identity elements. Thus, we can write these sets in one of two ways: in additive notation $a+b$ , and in multplicative notation $ab$ , depending on the operation involved. If all four of these axioms, as we call them, are satisfied, then the set is a group under the prescribed operation.

The typical examples of groups include the real numbers $\mathbb{R}$ , the integers $\mathbb{Z}$ , and the complex numbers $\mathbb{C}$ to name a few. We denote the structure by stating what set we are considering, followed by a the binary operation, written as $(G, \ast)$ .

In the next post, I will look at $\mathbb{R}, \mathbb{Z}, \mathbb{C}$ in detail and I will show how to prove that the reals and integers are groups under addition and that the complex numbers are a group under complex multiplication.

A PROBLEM IN THERMODYNAMICS AND STATISTICAL MECHANICS: ANALYTICAL AND NUMERICAL STUDY OF AN EINSTEIN SOLID-Analytical Solution

IMAGE CREDIT/OBTAINED FROM: https://mappingignorance.org/2015/12/17/einstein-and-quantum-solids/

Quite some time ago, I had posted a numerical study of an Einstein solid and I now present the analytical study of an Einstein solid. As this was one problem in one of my problem sets while studying thermodynamics and statistical mechanics, one may find this exact problem in the following text:

Schroeder D.V., An Introduction to Thermal Physics, (2000). Addison Wesley Longman. Chapter 3: Interactions and Implications: Problem 3.25. pp. 108.

What I will be presenting is my solution to this problem and I will be offering my interpretation of the problem statement and the implications of the solution.

We begin with the provided approximation,

$\displaystyle \Omega(N,q)\approx \bigg(\frac{q+N}{q}\bigg)^{q}\bigg(\frac{q+N}{N}\bigg)^{N}. (1)$

This expression represents the multiplicity of an Einstein solid with $N$ oscillators and $q$ energy units. Recall that the equation to find the entropy is the following

$\displaystyle S=k\ln{(\Omega(N,q))}. (2)$

Thus, upon substitution of the multiplicity (Eq.(1)) into the equation for entropy (Eq.(2)), one arrives at the equation

$\displaystyle S=k\ln{\bigg\{\bigg(\frac{q+N}{q}\bigg)^{q}\bigg(\frac{q+N}{N}\bigg)^{N}\bigg\}}. (3)$

Upon making use of the properties of logarithms, we may write the equation equivalently as

$\displaystyle S = k\ln{\bigg\{\bigg(\frac{q+N}{q}\bigg)^{q}\bigg\}}+k\ln{\bigg\{(\frac{q+N}{N}\bigg)^{N}\bigg\}}, (4)$

and again using the well-known property that $\ln{x}^{a}=a\ln{x}$ , we may further simplify Eq.(4) such that we arrive at the expression for the entropy of an Einstein solid:

$\displaystyle S = kq\ln{\bigg\{\bigg(\frac{q+N}{q}\bigg)\bigg\}}+Nk\ln{\bigg\{\bigg(\frac{q+N}{N}\bigg)\bigg\}}. (5)$

We may omit the factor of $(2\pi q(q+N))^{1/2}N^{-1/2}$ from Stirling’s approximation owing to the fact that if $N$ and $q$ are very large numbers, then $\sqrt{q}<<q$ and $\sqrt{N}<<N$ . Hence, it follows that $\sqrt{q+N}<<(q+N)$ . So we see that the aforementioned factor is of no consequence provided that $q$ and $N$ (i.e. the number of energy units and oscillators) is very large.

The second part of this problem asks to take the expression we derived for the entropy and compute the temperature. Recall that the definition for temperature is given by the equation

$\displaystyle \frac{1}{T}=\bigg(\frac{\partial S}{\partial U}\bigg)^{-1}. (6)$

From substitution we may write the following

$\displaystyle \frac{1}{T}=\bigg\{\frac{\partial}{\partial U}\bigg(k\bigg(\frac{U}{\epsilon}+N\bigg)\ln{\bigg(\frac{U}{\epsilon}+N\bigg)}\bigg)-\frac{\partial}{\partial U}\bigg(\frac{kU}{\epsilon}\ln{\bigg(\frac{U}{\epsilon}\bigg)}\bigg)-\frac{\partial}{\partial U} (N\ln{(N))}\bigg\}^{-1}, (7)$

where I have made use of the properties of logarithms and made the substitution $q=U/\epsilon$ . Differentiating and simplifying yields,

$\displaystyle \frac{1}{T}=\bigg\{\bigg(\frac{k}{\epsilon}\bigg)\ln{\bigg(\frac{U}{\epsilon}+N\bigg)}-\ln{\bigg(\frac{U}{\epsilon}\bigg)}\bigg\}^{-1}. (8)$

Further simplification yields the equation for temperature $T$ of an Einstein solid

$\displaystyle T = \frac{\epsilon}{k\ln{\bigg(\frac{U+\epsilon N}{U}}\bigg)}. (9)$

Part three asks us to find the equation for the heat capacity from our temperature equation. Recall that the equation for heat capacity is of the form

$\displaystyle C = \bigg(\frac{\partial U}{\partial T}\bigg). (10)$

However, we need an equation for the internal energy $U$ as a function of temperature $T$ . We actually have the opposite. So we must solve Eq.(9) for the internal energy, and doing so yields

$\displaystyle U(T)=\frac{\epsilon N}{\exp{(\epsilon/kT)}-1}. (11)$

Substituting Eq.(11) into Eq.(10) gives

$\displaystyle C = \bigg(\frac{\partial U}{\partial T}\bigg)=\frac{\partial}{\partial T}\bigg(\frac{\epsilon N}{\exp{(\epsilon/kT)}-1}\bigg). (12)$

Using the quotient rule for derivatives gives the equation for the heat capacity

$\displaystyle C =\bigg(\frac{\epsilon^{2}N}{kT^{2}}\bigg) \bigg(\frac{\exp{(\epsilon/kT)}}{{(\exp{(\epsilon/kT)}-1)}^{2}}\bigg). (13)$

The next part asked to show that in the limit $T \rightarrow \infty$ , the heat capacity $C =Nk$ . Recall that the Taylor series expansion for the exponential function $\exp{(x)}$ is given by

$\displaystyle \exp{(x)}=\sum_{j=0}^{\infty}\frac{x^{j}}{j!}. (14)$

For small values of $x$ we may make the approximation $\exp{(x)}\approx 1+ x$ . Then we have the approximate relation

$\displaystyle C \approx \frac{\epsilon^{2}N}{kT^{2}}\frac{(1+\epsilon/kT)}{(1+(\epsilon/kT)-1)^{2}}=\frac{\epsilon^{2}N(1+(\epsilon/kT))}{kT^{2}(\epsilon^{2}/k^2 T^2)}= Nk(1+(\epsilon/kT)). (15)$

Then considering the aforementioned limit yields

$\displaystyle \lim_{T \rightarrow \infty} C = \lim_{T \rightarrow \infty} \bigg(Nk(1+(\epsilon/kT))\bigg)= Nk. (16)$

The final part of my solution to this problem (I did not complete the last portion of the problem see the aforementioned reference for the problem) asks us to graph the resultant equation relating the heat capacity and the temperature. Below is a plot of the function in the technical computing software Maple.

Fig.1 Heat Capacity vs. Temperature

For low temperatures, the heat capacity initially starts at $0$ . However, when $t=0.097$ , there appears to be a dramatic increase in the heat capacity in the dimensionless quantity $C/Nk$ . If the heat capacity $C$ is graphed as a function of temperature $T$ , and one uses the $\epsilon$ values for each of the lead and aluminum curves produced in Fig. 1.14 of Schroeder’s An Introduction to Thermal Physics.

Let me know if I made any mistakes anywhere, and I will do my best to correct them.

Clear skies!

Basics of Tensor Calculus & General Relativity|A Digression into Special Relativity

So far in this series I have given the definitions of vectors, scalars, tensors, and manifolds. As a result, much of this series has been mostly mathematics and not necessarily physics. To that end, the purpose of this post is to develop the salient points of special relativity. Namely, the intention of this post is to cover the following:

Definition of Inertial Reference Frames: Standard Configuration and Einstein’s Postulates.
Development of the Lorentz Transformation Matrix
Discussion of the Newtonian geometry of spacetime
Discussion of the Minkowski geometry of spacetime (i.e. no curvature)
Finally I will show that the quantity $\delta s^{2}$ is invariant with respect to Lorentz transformations. This is a pretty standard problem in most GR textbooks and in fact in some introductory books on SR.

Continue reading Basics of Tensor Calculus & General Relativity|A Digression into Special Relativity

Basics of Tensor Calculus and General Relativity: An Introduction to Manifolds and Coordinates

SOURCE FOR CONTENT: General Relativity: An Introduction for Physicists, Hobson, M.P., Efsttathiou, G., and Lasenby, A.N., 2006. Cambridge University Press.

D-Dimensional Hypersphere and Gamma Function: Introduction to Thermal Physics, Schroeder D.V. 2000. Addison-Wesley-Longmann.

IMAGE CREDIT: NASA/JPL.

The intended purpose of the post is to introduce the concept of manifolds in the context of physics (mathematicians beware!). Furthermore, I will discuss the concepts of Riemannian and pseudo-Riemannian manifolds before moving on towards tensors. This will be the first post in this topic of the series. In order to properly discuss the concepts of general relativity, I will have to break up this part of the series into smaller posts.

Part I. In this part of the series, I will discuss the concept of a tensor, and then discuss the introductory topics of manifolds.

A topic that has long eluded a conceptual understanding on my part is a tensor. In the first post of the series we saw a very technical and quite frustrating definition of a tensor. I have read numerous treatments and watched a number of lectures and videos and based on everything I have encountered, here is my understanding of a tensor as of writing this post…

Tensors are geometric objects that can be viewed in a similar way as one views matrices, whose elements are components of the tensor, and will have an overall value. More specifically, a tensor will take two geometric objects as inputs and will give you a scalar (a real number). Furthermore, under a transformation or in a different reference frame, the scalar that the tensor outputs will remain the same in all frames. The objects that change are, in fact, the components of the tensor. These components must obey specific transformation equations so as to preserve the scalar quantity of produced by the tensor. In more mathematical sense, a tensor is a mapping of a number of vectors (including 1-forms and the like) into the real number ordered field.

(**This is the best definition that I could come up with in order to define in a more satisfactory way a tensor.**)

Manifolds

According to the aforementioned reference, a manifold in its most general sense, is any set that one can describe by specifying parameters continuously. In the context of physics, we deal with differentiable manifolds.

A differentiable manifold is a continuous collection of points where each point is differentiable. This definition isn’t any better than the initial definition of a tensor. So, to elaborate a bit, we shall define the concept of continuity: a manifold is continuous if in the local region of a point $n_{1}$ , there exists points whose difference relative to $n_{1}$ is $dn$ .

From this, we can say that a differentiable manifold is a manifold for which we can ascribe to it a scalar field containing points at which it is possible to take derivatives of all orders.

Some examples include: 3-Dimensional Euclidean Space and Phase Space.

3-Dimensional Space

This differentiable manifold requires 3 coordinates (parameters) to specify a single point in the space. Since it requires three parameters the dimension of this particular manifold is 3. Mathematicians sometimes call this 3-space.

Phase Space

This is a manifold that one encounters more often in physics. I came across this manifold (although I did not refer to it as such) while taking my thermodynamics and statistical mechanics course. I found that this manifold requires 6 parameters in order to specify any point. Typically, these parameters include positions (or one radius vector) and velocities or momenta.

A submanifold or surface that I found applicable to phase space would be the $D$ -dimensional hypersphere whose surface area is given by

$\displaystyle A_{d}(r)=\frac{2\pi^{d/2}}{(\frac{d}{2}-1)!}r^{d-1}=\frac{2\pi^{d/2}}{\Gamma(\frac{d}{2})}r^{d-1}, (1)$

where $\Gamma(\frac{d}{2})$ is the gamma function given by

$\displaystyle \Gamma(d+1)\equiv \int_{0}^{\infty}x^{d}\exp{(-x)}dx. (2)$

To be more precise, the surface area (Eq.1) of this hypersphere is technically the volume of momentum space, but I am including to present a more concrete example of a manifold.

The next post will make the concepts mentioned here a bit more quantitative. This post was really just to introduce a more conceptual understanding.

Clear skies!

Astrophysics Series: Derivation of the Total Energy of a Binary Orbit

SOURCE FOR CONTENT: An Introduction to Modern Astrophysics, Carroll & Ostlie, Cambridge University Press. Ch.2 Celestial Mechanics

Here is my solution to one of the problems in the aforementioned text. I derive the total energy of a binary system making use of center-of-mass coordinates. In order to conceptualize it I have used the binary Alpha Centauri A and Alpha Centauri B. While writing this I stumbled upon the Kepler problem, the two-body problem, and the N-body problem. Leave a comment if you would like me to consider that in another post.

Clear Skies!

Derivation of the Total Energy of a Binary Orbit:

Setup: Consider the nearest binary star system to our solar system: Alpha Centauri A and Alpha Centauri B. These two stars orbit each other about a common center of mass; a point called a barycenter. The orbital radius vector of Alpha Centauri A is $\textbf{r}_{1}$ and the orbital radius vector of Alpha Centauri B is $\textbf{r}_{2}$ . The masses of Alpha Centauri A and B are $m_{1}$ , and $m_{2}$ , respectively. The total mass of the binary orbit $M$ is the sum of the individual masses of each component. In the context of this system, we encounter what is called the two-body problem of which there exists a special case known as the Kepler Problem (by the way let me know if that would be something that you guys would want to see…). We can simplify this two-body problem by making use of center-of-mass coordinates wherein we define the reduced mass $\mu$ . Therefore, the derivation of the total energy of the binary system of Alpha Centauri A and B will be carried out in such a coordinate system.

To derive this energy equation, one would typically make use of center-of-mass coordinates in which

$\displaystyle \textbf{r}_{1}=-\frac{\mu}{m_{1}}r, (0.1)$

and

$\displaystyle \textbf{r}_{2}=\frac{\mu}{m_{2}}r, (0.2)$

where $\mu$ represents the reduced mass given by

$\displaystyle \mu\equiv \frac{m_{1}m_{2}}{m_{1}+m_{2}}=\frac{m_{1}m_{2}}{M}. (0.3)$

Recall from conservation of energy that

$\displaystyle E = \frac{1}{2}m_{1}\dot{r}_{1}^{2}+\frac{1}{2}m_{2}\dot{r}_{2}^{2}-G\frac{m_{1}m_{2}}{|\mathcal{R}|}, (1)$

where $|\mathcal{R}|$ represents the separation distance between the two components. Let us take the derivative of Eqs.(0.1) and (0.2) to get

$\displaystyle \dot{r}_{1}=-\frac{\mu}{m_{1}}v, (2.1)$

and

$\displaystyle \dot{r}_{2}= \frac{\mu}{m_{2}}v. (2.2)$

Substitution yields

$\displaystyle E = \frac{1}{2}\frac{\mu^{2}}{m_{1}}v^{2}+\frac{1}{2}\frac{\mu^{2}}{m_{2}}v^{2}-G\frac{m_{1}m_{2}}{|\mathcal{R}|}. (3)$

Upon making use of the definition of the reduced mass (Eq. (0.3)) we arrive at

$\displaystyle E = \frac{1}{2}\mu v^{2}-G\frac{M \mu}{|\mathcal{R}|}. (4)$

If we solve for $m_{1}m_{2}$ in Eq.(0.3) we get the total energy of the binary Alpha Centauri A and B. This is true for any binary system assuming center-of-mass coordinates.

A Problem in Thermodynamics and Statistical Mechanics: Analytical and Numerical Study of an Einstein Solid

Every physics major at some point in their undergraduate career takes a course in thermodynamics and statistical mechanics. One of my problem sets included a problem that considers an Einstein solid with 50 oscillators and 100 units of energy and then increases the number of oscillators to 5000. I will be presenting my solution to the numerical side of the problem. An Einstein solid can be regarded as

“… a collection of microscopic systems which can store any number of energy ‘units’ of equal size which occur for any quantum-mechanical harmonic oscillator whose potential energy function has the form $\displaystyle \frac{1}{2}k_{s}x^{2}$ …The model of a solid as a collection of identical oscillators with quantized energy units…”

described (defined) by Schroeder in his text Introduction to Thermal Physics. Figure 1 represents the Einstein solid as a whole (in a lattice) and Figure 2 depicts the quantum-mechanical harmonic oscillator interpretation of an Einstein solid.

The problem statement is:

“Use a computer to study the entropy, temperature, and heat capacity of an Einstein solid, as follows. Let the solid contain 50 oscillators (initially), and from 0 to 100 units of energy. Make a table, analogous to Table 3.2, in which each row represents a different value for the energy…Make a graph of entropy vs. energy, and a graph of the heat capacity vs. temperature. Then change the number of oscillators to 5000, and again make a graph of the heat capacity and temperature and entropy and energy, and discuss the predictions and compare it to the predictions to the data for lead, aluminum, and diamond. Estimate the numerical value of $\displaystyle \epsilon$ for each of those solids.”

This problem can be found in the aforementioned text.

Figure 1. Einstein Solid (Lattice); Image Credit/Obtained from https://mappingignorance.org/2015/12/17/einstein-and-quantum-solids/

Figure 2. Quantum-Mechanical Harmonic Oscillator interpretation of an Einstein solid as a collection of these oscillators. Image Credit: http://hyperphysics.phy-astr.gsu.edu/hbase/Therm/einsol.html

Part I: Let $q = 100$ units, and let $N = 50$. The corresponding data table for this Einstein solid follows. The following set of equations were used to determine the multiplicity and entropy.

$\displaystyle \Omega(N,q) = {q+N-1 \choose q} = \frac{(q+N-1)!}{q! (N-1)!}, (1)$

and

$\displaystyle S = Nk \ln{\Omega}, (2)$

where $\Omega$ is the multiplicity. The remaining quantities of temperature were obtained using a simplified form of the central difference equations for the first order derivative. The respective definitions of temperature and heat capacity are

$\displaystyle T = \frac{\partial U}{\partial S}, (3)$

and

$\displaystyle C_{V} = \frac{\partial U}{\partial T}, (4)$

where $U$ represents the internal energy of the Einstein solid, and $S$ is the entropy. The generalized from of the first order central difference approximation has the form

$\displaystyle \frac{dy_{j}}{dx}\approx \frac{y_{j+1}-y_{j-1}}{2h} + \mathcal{O}(h^{2}), (5)$

where $\mathcal{O}(h^{2})$ represents the higher order terms, in this case, the quadratic, cubic, quartic, and so on, and $h$ is the step size for each iteration. For the final iteration (when $q = 100$ units), instead of using the central difference approximation, a backward difference approximation was employed since there does not exist data for $q = 101$ units of energy. The backward difference approximation has the form

$\displaystyle \frac{dy_{j}}{dx}\approx \frac{y_{j}-y_{j-1}}{h}+\mathcal{O}(h). (6)$

Table I (Dimensionless Parameters):

*Energy q*	Ω	S/k	kT/ε	C/Nk
0	1	0	0	N/A
1	50	3.912023005	0.27969284	0.121826198
2	1275	7.150701458	0.328336604	0.453606383
3	22100	10.00333289	0.367875021	0.536183525
4	292825	12.58733044	0.402937926	0.593741905
5	3162510	14.96687657	0.43524436	0.637773801
6	28989675	17.18245029	0.465656087	0.673043377
7	231917400	19.26189183	0.494675894	0.702124659
8	1652411475	21.22550156	0.522626028	0.72660015
9	10648873950	23.08871999	0.549726805	0.747522024
10	62828356305	24.86367234	0.576136157	0.765628174
11	3.427E+11	26.56012163	0.601971486	0.781456694
12	1.74206E+12	28.18608885	0.627322615	0.795411957
13	8.30828E+12	29.74827387	0.652259893	0.807805226
14	3.73873E+13	31.25235127	0.676839501	0.818880855
15	1.59519E+14	32.70318415	0.701107048	0.828833859
16	6.48046E+14	34.1049827	0.725100078	0.837822083
17	2.51594E+15	35.4614241	0.748849881	0.845974847
18	9.3649E+15	36.77574496	0.772382808	0.853399232
19	3.35165E+16	38.05081369	0.795721261	0.860184741
20	1.15632E+17	39.28918792	0.818884446	0.866406816
21	3.8544E+17	40.49316072	0.84188895	0.872129523
22	1.24392E+18	41.66479814	0.864749193	0.877407641
23	3.89401E+18	42.80597005	0.887477794	0.882288296
24	1.18443E+19	43.91837566	0.910085848	0.88681226
25	3.5059E+19	45.00356493	0.932583169	0.891014994
26	1.01132E+20	46.0629565	0.954978471	0.89492748
27	2.84667E+20	47.09785298	0.977279528	0.898576916
28	7.82835E+20	48.10945389	0.999493303	0.901987268
29	2.10556E+21	49.09886689	1.021626052	0.905179739
30	5.54463E+21	50.06711736	1.043683421	0.908173155
31	1.43087E+22	51.01515679	1.065670516	0.910984284
32	3.6219E+22	51.94387004	1.087591972	0.913628113
33	8.99987E+22	52.85408172	1.109452006	0.916118071
34	2.19703E+23	53.74656181	1.131254467	0.918466228
35	5.27286E+23	54.62203054	1.153002873	0.920683463
36	1.24498E+24	55.48116286	1.174700452	0.9227796
37	2.89374E+24	56.32459225	1.196350168	0.924763536
38	6.62514E+24	57.1529142	1.217954752	0.926643346
39	1.4949E+25	57.96668937	1.239516722	0.928426373
40	3.32616E+25	58.76644629	1.261038406	0.930119309
41	7.30133E+25	59.55268389	1.282521958	0.931728264
42	1.58195E+26	60.32587378	1.303969378	0.933258829
43	3.38465E+26	61.08646224	1.325382522	0.934716125
44	7.15391E+26	61.8348721	1.346763119	0.936104855
45	1.49437E+27	62.57150439	1.368112778	0.937429341
46	3.08621E+27	63.29673989	1.389433002	0.938693566
47	6.30374E+27	64.01094048	1.410725193	0.9399012
48	1.27388E+28	64.71445045	1.431990666	0.941055635
49	2.54776E+28	65.40759763	1.45323065	0.942160007
50	5.04457E+28	66.09069447	1.4744463	0.943217222
51	9.89131E+28	66.76403902	1.495638697	0.944229971
52	1.9212E+29	67.42791582	1.516808861	0.945200757
53	3.6974E+29	68.08259672	1.537957749	0.946131903
54	7.05244E+29	68.72834166	1.559086264	0.947025573
55	1.33355E+30	69.36539938	1.580195257	0.947883783
56	2.50041E+30	69.99400804	1.60128553	0.948708411
57	4.64989E+30	70.61439586	1.622357843	0.949501213
58	8.57824E+30	71.22678169	1.643412912	0.95026383
59	1.57025E+31	71.83137547	1.664451416	0.950997796
60	2.85263E+31	72.42837879	1.685473998	0.951704548
61	5.14408E+31	73.01798529	1.706481267	0.952385432
62	9.20957E+31	73.60038111	1.7274738	0.953041713
63	1.63726E+32	74.17574525	1.748452147	0.953674575
64	2.89078E+32	74.74424999	1.769416829	0.954285135
65	5.06999E+32	75.30606117	1.79036834	0.95487444
66	8.83407E+32	75.86133855	1.811307153	0.955443478
67	1.52948E+33	76.41023613	1.832233716	0.955993177
68	2.63161E+33	76.95290236	1.853148456	0.956524415
69	4.50043E+33	77.48948048	1.874051781	0.957038017
70	7.65073E+33	78.02010873	1.894944079	0.957534764
71	1.29308E+34	78.54492059	1.915825721	0.958015394
72	2.17309E+34	79.06404502	1.936697061	0.958480604
73	3.63175E+34	79.57760662	1.957558438	0.958931052
74	6.03655E+34	80.08572588	1.978410175	0.959367363
75	9.98043E+34	80.58851934	1.999252581	0.959790129
76	1.64152E+35	81.08609973	2.020085953	0.960199908
77	2.68612E+35	81.57857622	2.040910573	0.960597234
78	4.37356E+35	82.06605448	2.061726714	0.960982609
79	7.08627E+35	82.54863689	2.082534635	0.961356514
80	1.14266E+36	83.02642266	2.103334587	0.961719401
81	1.8339E+36	83.49950796	2.124126809	0.962071705
82	2.92977E+36	83.96798603	2.14491153	0.962413835
83	4.65939E+36	84.43194735	2.165688971	0.962746183
84	7.37736E+36	84.89147968	2.186459344	0.963069122
85	1.16302E+37	85.34666822	2.207222854	0.963383006
86	1.82567E+37	85.7975957	2.227979695	0.963688173
87	2.85392E+37	86.24434247	2.248730056	0.963984945
88	4.44304E+37	86.68698658	2.269474119	0.964273631
89	6.8892E+37	87.12560389	2.290212057	0.964554522
90	1.064E+38	87.56026816	2.31094404	0.9648279
91	1.63692E+38	87.99105107	2.331670228	0.965094032
92	2.50876E+38	88.41802239	2.352390779	0.965353173
93	3.83058E+38	88.84124995	2.373105841	0.965605568
94	5.82737E+38	89.2607998	2.39381556	0.965851451
95	8.83307E+38	89.67673621	2.414520077	0.966091045
96	1.33416E+39	90.08912176	2.435219526	0.966324564
97	2.00812E+39	90.4980174	2.455914037	0.966552213
98	3.01218E+39	90.90348251	2.476603736	0.966774189
99	4.50306E+39	91.30557493	2.497288745	1.287457337
100	6.70955E+39	91.70435105	2.507672727	1.926043463

Graphing the entropy vs. energy, and the heat capacity vs. temperature gives the following:

Graphs I & II

Part II: Let $q = 100$ units and let $N = 5000$ . Using this in the calculation yields the following table for this Einstein solid. This “dilutes” the system and lowers the temperature:

Table II (Dimensionless Parameters):

*Energy q*	Ω	S/k	kT/ε	C/Nk
0	1	0	0	N/A
1	5000	8.517193	0.122388	0.003049
2	12502500	16.34144	0.131206	0.026553
3	2.08E+10	23.76042	0.137453	0.035575
4	2.61E+13	30.89192	0.14245	0.043342
5	2.61E+16	37.80047	0.146681	0.050387
6	2.18E+19	44.52691	0.150388	0.056922
7	1.56E+22	51.09939	0.153709	0.063064
8	9.74E+24	57.53854	0.156731	0.068885
9	5.42E+27	63.86011	0.159515	0.074436
10	2.72E+30	70.07651	0.162105	0.079755
11	1.24E+33	76.19781	0.164531	0.08487
12	5.16E+35	82.23229	0.166818	0.089804
13	1.99E+38	88.18693	0.168985	0.094575
14	7.13E+40	94.06767	0.171047	0.099198
15	2.38E+43	99.87961	0.173017	0.103687
16	7.47E+45	105.6272	0.174905	0.108052
17	2.2E+48	111.3144	0.176719	0.112303
18	6.14E+50	116.9446	0.178467	0.116448
19	1.62E+53	122.5209	0.180154	0.120494
20	4.07E+55	128.0462	0.181787	0.124447
21	9.73E+57	133.5229	0.183368	0.128314
22	2.22E+60	138.9532	0.184904	0.132098
23	4.85E+62	144.3393	0.186396	0.135806
24	1.02E+65	149.683	0.187849	0.13944
25	2.04E+67	154.9861	0.189265	0.143004
26	3.94E+69	160.2502	0.190646	0.146503
27	7.34E+71	165.4768	0.191995	0.149939
28	1.32E+74	170.6671	0.193314	0.153316
29	2.28E+76	175.8226	0.194604	0.156635
30	3.83E+78	180.9444	0.195868	0.159899
31	6.21E+80	186.0336	0.197106	0.16311
32	9.77E+82	191.0912	0.19832	0.166272
33	1.49E+85	196.1183	0.199512	0.169384
34	2.21E+87	201.1157	0.200682	0.172451
35	3.17E+89	206.0843	0.201831	0.175472
36	4.44E+91	211.025	0.202961	0.17845
37	6.04E+93	215.9384	0.204073	0.181387
38	8E+95	220.8254	0.205166	0.184283
39	1.03E+98	225.6866	0.206243	0.18714
40	1.3E+100	230.5227	0.207304	0.18996
41	1.6E+102	235.3343	0.208349	0.192743
42	1.9E+104	240.122	0.209379	0.195491
43	2.3E+106	244.8863	0.210395	0.198205
44	2.6E+108	249.6279	0.211397	0.200885
45	2.9E+110	254.3472	0.212386	0.203533
46	3.2E+112	259.0447	0.213363	0.20615
47	3.4E+114	263.7209	0.214327	0.208736
48	3.6E+116	268.3762	0.215279	0.211293
49	3.7E+118	273.0112	0.21622	0.213821
50	3.7E+120	277.6261	0.21715	0.21632
51	3.7E+122	282.2214	0.218069	0.218793
52	3.6E+124	286.7975	0.218978	0.221238
53	3.4E+126	291.3548	0.219877	0.223658
54	3.2E+128	295.8935	0.220766	0.226052
55	2.9E+130	300.4141	0.221646	0.228422
56	2.7E+132	304.9169	0.222517	0.230768
57	2.4E+134	309.4022	0.22338	0.23309
58	2.1E+136	313.8703	0.224233	0.235388
59	1.8E+138	318.3214	0.225079	0.237665
60	1.5E+140	322.756	0.225917	0.239919
61	1.2E+142	327.1743	0.226746	0.242152
62	1E+144	331.5765	0.227568	0.244363
63	8.1E+145	335.9628	0.228383	0.246555
64	6.4E+147	340.3337	0.229191	0.248725
65	5E+149	344.6892	0.229991	0.250876
66	3.8E+151	349.0296	0.230785	0.253008
67	2.9E+153	353.3553	0.231572	0.255121
68	2.2E+155	357.6663	0.232353	0.257215
69	1.6E+157	361.9629	0.233127	0.259291
70	1.1E+159	366.2453	0.233896	0.261349
71	8.2E+160	370.5137	0.234658	0.263389
72	5.8E+162	374.7683	0.235414	0.265413
73	4E+164	379.0093	0.236165	0.267419
74	2.7E+166	383.237	0.23691	0.269409
75	1.9E+168	387.4514	0.23765	0.271382
76	1.2E+170	391.6527	0.238384	0.273339
77	8.2E+171	395.8412	0.239113	0.275281
78	5.3E+173	400.0169	0.239837	0.277207
79	3.4E+175	404.1802	0.240556	0.279118
80	2.2E+177	408.331	0.24127	0.281015
81	1.4E+179	412.4696	0.24198	0.282896
82	8.4E+180	416.5962	0.242684	0.284763
83	5.2E+182	420.7108	0.243384	0.286616
84	3.1E+184	424.8136	0.24408	0.288455
85	1.9E+186	428.9048	0.244771	0.29028
86	1.1E+188	432.9845	0.245458	0.292092
87	6.5E+189	437.0529	0.24614	0.29389
88	3.7E+191	441.11	0.246819	0.295676
89	2.1E+193	445.156	0.247493	0.297448
90	1.2E+195	449.191	0.248164	0.299208
91	6.7E+196	453.2152	0.24883	0.300955
92	3.7E+198	457.2286	0.249493	0.30269
93	2E+200	461.2315	0.250152	0.304413
94	1.1E+202	465.2238	0.250807	0.306124
95	5.9E+203	469.2057	0.251458	0.307823
96	3.2E+205	473.1774	0.252106	0.30951
97	1.7E+207	477.1389	0.252751	0.311187
98	8.6E+208	481.0903	0.253392	0.312851
99	4.4E+210	485.0318	0.254029	0.418439
100	2.3E+212	488.9634	0.254345	0.628243

Thus the graphs of the entropy vs. energy and heat capacity vs. temperature follow:

Figure 2. Graphs III and IV.

Figure 3. (Figure 1.14 of Schroeder’s Thermal Physics) Heat Capacity curves for Lead (Pb), Aluminum (Al), and Diamond, respectively as a function of temperature in Kelvin.

Graph II shows the prediction for heat capacity as a function of temperature of an Einstein solid for which there are 100 units of energy and 50 oscillators. The data exhibits a trend that appears to reach an asymptote quickly, then when the temperature reaches T ≈ 2.5, there is a sudden increase in the value of the heat capacity. The approach to determining the final data points was switched from a central difference approximation to a backward difference approximation of the last two entries corresponding to energies q = 99 and q = 100 units. If we ignore the last two, the curve approaches an asymptote at C_V = 1. However, the graphs produced are of the dimensionless quantities involved. The overall curve appears to be logarithmic and resembles the heat capacity curve for lead. The initial increase is almost immediate and its slope appears to be slightly less than lead but greater than aluminum.

Graphs III and IV show the prediction for heat capacity in terms of temperature of an Einstein solid for which the energy is the same, but the number of oscillators is now 5000. The temperature has been reduced and the heat capacity vs. temperature yields a graph that shows a trendline that appears linear. Comparing to Figure 3(Fig. 1.14 in the text), this graph resembles the heat capacity curve for diamond. In Figure 3, the diamond curve is linear throughout. The only discrepancies among Graph IV and Figure 3 are the final two data points in Graph IV. Again, a backward difference approximation was used to determine the final data points for this Einstein solid as well. The value for the constant ε was determined by finding the quotient of the entropy and temperature columns and taking the average value of ε for each energy.

This was the numerical analysis of an Einstein solid’s temperature, energy, entropy, and heat capacity. In the next post, I shall discuss the analytical version of this analysis.

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