Lecture 2: Temperature and the Boltzmann factor

Note

Nature prefers the more probable states to the less probable because in nature processes take place in the direction of greater probability. Heat goes from a body at higher temperature to a body at lower temperature because the state of equal temperature distribution is more probable than a state of unequal temperature distribution. — Max Planck

Warning

This lecture corresponds to Chapter 4 of the textbook.

Summary

Attention

In this lecture, we decide to take a humble look at the meaning of “temperature”. Everyone knows what temperature is… really? The difficulty comes from the fact that the layman definition of temperature is always a relative one; that is: it is presented as a number comparing a hot and a cold object. But here, we want to do more: we want to define it quantitatively. We know that heat (as we’ve seen in the previous lecture) is a form of energy that is transferred from a hot to a cold body until thermal equilibrium is reached. So, starting from there, we can say that temperature is “some number that describes the flow of heat”. This is our starting point.

In our quest to understand thermodynamics, we introduce a very important concept of microstate (to be contrasted with the concept of macrostate). A macrostate is what we can measure while a microstate is all the information require to describe a system. In the lecture notes, we introduce the experiment consisting of looking at the number of heads and tails in a bag of 100 quarters. The macrostate is the number of heads and the number of tails observed in the collection of coins. The microstate is the specific state (heads or tails) of each individual coin. Clearly, there are many ways (=microstates) to realize a 50/50 macrostate. Conversely, there is only one way to realize a 100/0 macrostate.

This last observation leads us to introduce a key concept of thermodynamics and statistical mechanics: a system will always tend to adopt the macrostate that corresponds to the largest number of microstate. In other words, a system will try to maximize the number of microstates $\Omega(E)$ .

Going back to our problem of two bodies (1 and 2) at different initial temperatures that are placed in contact, we can use the principle above to determine how the system will evolve. From the principle above, we conclude that it will evolve so as to maximize the number of microstates! This leads to a key equality:

$\frac{\mathrm{d} \ln \Omega_{1}}{\mathrm{~d} E_{1}}=\frac{\mathrm{d} \ln \Omega_{2}}{\mathrm{~d} E_{2}}$

This prompts us to provide a definition of temperature:

$\frac{1}{k_{\mathrm{B}} T}=\frac{\mathrm{d} \ln \Omega}{\mathrm{d} E}$

which is built in such a way that two systems in thermal equilibrium have the same temperature, according to the equation above.

We also understand that all this is related to probability amplitude: the system will adopt a macrostate with a probability proportional to the number of microstates that yield the macrostate. This can be formalized straightforwardly to find that the probability of a certain macrostate defined by energy E is (this is known as the canonical distribution, valid when a body is placed in contact with a temperature reservoir):

$P(\epsilon) \propto \mathrm{e}^{-\epsilon / k_{\mathrm{B}} T}$

where the term on the right is called the Boltzmann factor.

Learning Material

Copy of Slides

The slides for Lecture 2 are available in pdf format here: pdf

Screencast

Key Definitions

Note

Ergodicity:

a system is ergodic if, given enough time, it will explore all possible microstates.

Ensembles:

A thermodynamic ensemble can be of different types:

The microcanonical ensemble: an ensemble of systems that each have the same fixed energy.
The canonical ensemble: an ensemble of systems, each of which can exchange its energy with a large reservoir that keeps a constant temperature.
The grand canonical ensemble: an ensemble of systems, each of which can exchange both energy and particles with a large reservoir.

Partition function:

The partition function is a normalization factor introduced to make sure that the sum of probabilities is equal to one. It can also be seen as the sum of all possibly Boltzmann factors for a given system in all possible microstates $\{E_i\}$ :

$Z=\sum_{i} \mathrm{e}^{-E_{i} / k_{\mathrm{B}} T}$

We will see in this course that the partition function has all the information needed to describe a given system in contact with a reservoir at temperature T.

Heat reservoir:

a heat reservoir is a body with a given temperature. Its mass is sufficiently large that its temperature is unchanged by the absorption or ejection of heat. In other words, a heat reservoir has a very high heat capacity.

Thermometer:

A thermometer is an object used to measure temperature of another object put in contact with it. It must have a very small heat capacity so that putting it in contact with the other object will not change the other object’s temperature.

A full list of terms, including the ones provided here, can be found in the Index.

Test your knowledge

What is the most accurate claim involving temperature among the assertions below?
1. Temperature is a microscopic measure of the hotness of a system.
2. Temperature is inversely proportional to $d\ln \Omega/dE$ where $\Omega$ is the number of microstates and $E$ is the energy
3. Temperature and heat are synonymous.
4. Temperature is an extensive variable.
5. None of the other options is correct.
Take a macroscopic container with an ideal gas at equilibrium. What assertion is correct?
1. Temperature, volume, and pressure are all intensive variables
2. Pressure, temperature, and energy are all extensive variables
3. Pressure, temperature, and gas density ( $\rho$ , measured in $kg/m^3$ ) are all intensive variables
4. Gas density ( $\rho$ , measured in $kg/m^3$ ), volume, and number of particles are extensive variables
5. None of the other options is correct
Thermodynamics & Statistical Mechanics has 22 registered students. Dr. Meunier is only allowed to give 5 final “A” grades in total. If all students deserve an “A”, how many possibilities will Dr. Meunier have to award that grade?
1. 22!/(17! 5!)
2. 22!/17!
3. 17!/5!
4. 24!/5!
A physicist prepares a 3-state system with energies 0, $\epsilon$ , and $2\epsilon$ ( $\epsilon>0$ ). At what temperatures can you be certain that the expectation value of the energy ( $\expval{E}$ ) is (a) 0 and (b) $\epsilon$ ?
1. 1. $T=0$ and (b) $T=\infty$
2. 1. $T=\infty$ and (b) $T=0$
3. 1. $T=0$ and (b) $T=3\epsilon/k_B$
4. 1. $T=1$ and (b) $T=3\epsilon/k_B$
5. None of the other options is correct
You need to build a thermometer, what should you use (here, the terms small and large are used with reference to the size of the system you are measuring)?
1. A large system made up of a material with small heat capacity
2. A small system made up of a material with small heat capacity
3. A small system made up of a material with extremely high heat capacity
4. A large system made up of a material with very high heat capacity
5. None of the other options is correct

Hint

Find the answer keys on this page: Answers to selected test your knowledge questions. Don’t cheat! Try solving the problems on your own first!

Homework Assignment

Solve the following problems from the textbook: 2.5, 3.4, 4.2, 4.3, 4.4