Boltzmann Distribution in the Canonical Ensemble

Thermodynamics reveals how physical systems reach equilibrium and how energy and probability determine their behavior. In this lesson, we will unravel the canonical ensemble and the Boltzmann Distribution, fundamental tools to understand phenomena like chemical reactions and equilibrium in complex systems. You will discover how these ideas connect temperature with order and chaos, enabling the prediction of seemingly unpredictable behavior.

Learning Objectives:
By the end of this lesson, students will be able to:

1. Identify the types of thermodynamic ensembles (microcanonical, canonical, and grand canonical).
2. Derive the Boltzmann Distribution from thermodynamic principles.
3. Calculate probabilities associated with microstates using the partition function.

CONTENT INDEX:
Thermodynamic Ensembles
The Boltzmann Distribution
Applications of the Boltzmann Distribution
Exercises

One of the most useful conceptual tools in thermodynamics is the concept of the “ensemble.” Among the variety of ensembles, one of the most widely used is the canonical ensemble, from which the Boltzmann Distribution is derived. Both concepts will be reviewed below.

Thermodynamic Ensembles

So far, we have used probabilities to describe thermodynamic systems. Our approach centers on imagining that we can repeat the experiment and measurements an infinite number of times as a way to compensate for our inability to control their microscopic properties (described through microstates). Inspired by these ideas, Gibbs introduced the concept of “ensemble” in 1878: an idealization in which a large number of “copies of the system” are considered, each representing one of its possible states. In thermodynamics, three main types of ensembles are defined:

1. Microcanonical Ensemble: A collection of systems, all with the same fixed energy.
2. Canonical Ensemble: A collection of systems, each of which can exchange energy with a large heat reservoir. As we will see later, this establishes (and defines) the temperature of the system.
3. Grand Canonical Ensemble: A collection of systems where each can exchange both matter (particles) and energy with a large reservoir. Through this, the temperature and chemical potential of the system are defined.

The Canonical Ensemble

Let us consider two coupled systems such that they can exchange energy. This time, however, one of them is enormous compared to the other, and we call it a reservoir, source, or heat bath. This reservoir is so vast that we can take large amounts of energy from it without changing its temperature. The number of ways in which the energy quanta rearrange within a reservoir is, consequently, gigantic. The other system is small in comparison, and we simply call it the system.

We will assume that for each allowed energy of the system, there is a unique microstate, and therefore, the system will always have a value of \Omega = 1. Additionally, we will keep the total energy of the coupled systems fixed at a value of E.

At this point, we can observe that the system and the reservoir form a microcanonical ensemble, where the energy remains constant and all microstates are equally probable.

In this scenario, if the system’s energy is \varepsilon, then the reservoir’s energy will be E - \varepsilon. This situation, where a system is in thermal contact with a large energy reservoir, is what is known as the Canonical Ensemble.

The Boltzmann Distribution

The probability P(\varepsilon) that the system has energy \varepsilon is proportional to the number of microstates accessible to the reservoir multiplied by the number of microstates accessible to the system. That is:

P(\varepsilon)\propto \Omega(E-\varepsilon)\cdot 1.

As we have seen before, temperature can be expressed in terms of the logarithm of \Omega through

\displaystyle \frac{1}{k_B T} = \frac{d\ln\Omega}{dE}

And since \varepsilon \ll E, it is possible to perform a Taylor series expansion of \ln\Omega(E-\varepsilon) around \varepsilon = 0. With this, we obtain:

\displaystyle \ln\Omega(E-\varepsilon) = \ln\Omega(E) - \frac{d\ln\Omega(E)}{dE}\varepsilon + \cdots

Using the expressions above, we get:

\displaystyle \ln\Omega(E-\varepsilon) = \ln\Omega(E) - \frac{\varepsilon}{k_B T} + \cdots

Where T is the temperature of the reservoir. At this point, we can neglect higher-order terms in the Taylor series expansion and state the relationship

\ln \Omega(E-\varepsilon) \approx \ln\Omega(E) - \displaystyle \frac{\varepsilon}{k_B T}

Developing this expression further, we find:

\Omega(E-\varepsilon) \approx \Omega(E) e^{-\displaystyle \frac{\varepsilon}{k_B T}}

Now, comparing this result with the probability P(\varepsilon), we conclude:

P(\varepsilon)\propto e^{-\varepsilon/(k_B T)}

Since the system is in thermodynamic equilibrium with the reservoir, they share the same temperature. However, although the temperature T remains constant, the energy \varepsilon is not; instead, it follows a probability distribution, which we have just derived. This is known as the Boltzmann Distribution or the Canonical Distribution for the canonical ensemble. The term e^{-\varepsilon/(k_B T) } is known as the Boltzmann Factor.

Normalization of the Boltzmann Distribution and the Partition Function

With these developments, we have begun constructing a probability distribution that describes how a small system behaves when coupled to a large reservoir at temperature T. The system has a reasonable chance of attaining an energy \varepsilon less than k_B T, but the exponential term in the Boltzmann distribution decreases rapidly when higher energies are considered. However, we must note that the distribution, as it currently stands, is not strictly a probability distribution; it needs to be normalized. If a system is in contact with a reservoir and has a microstate r with energy E_r, then we have:

P({microstate\;}r)= \displaystyle \frac{e^{-E_r/(k_B T)}}{\displaystyle \sum_{i}e^{-E_i/(k_B T)}}

The summation in the denominator acts as a normalizing factor that ensures P is a probability distribution. This summation is also known as the Partition Function and is denoted by Z

Z = \displaystyle \sum_i e^{-E_i/(k_B T)}

Applications of the Boltzmann Distribution

To illustrate some applications of the canonical ensemble and the Boltzmann Distribution, we will examine how they appear in certain examples. However, before starting, let us introduce a notation for a quantity that frequently appears and can be useful in the future. The factor \beta is defined through the equality:

\beta =\displaystyle \frac{1}{k_B T},

so that, based on this, we can write:

\beta = \displaystyle \frac{d\ln\Omega}{dE},

The Two-State System Problem

Imagine the simplest case of all: a system that can only exist in two states—one with energy 0 and the other with energy \varepsilon\gt 0. What is the average energy of the system?

The Isothermal Atmosphere Problem

A simplified way to study the atmosphere is under the assumption that it is isothermal. Although this assumption is not true, it serves as a first approximation for drawing certain conclusions. For instance, under this assumption, it is possible to estimate the number of particles in the atmosphere as a function of height. How might one make that deduction?

Explosion Hazard! Relationship Between Chemical Reactions and Temperature

Many chemical reactions have an activation energy E_{act} that is approximately 1/2 [eV]. At a temperature of T=300[K], which corresponds roughly to room temperature, the probability of a reaction occurring is proportional to:

e^{-E_{act}/(k_B T)}

What happens to the reaction probability if the temperature increases by 10[K]?

Exercises

1. A system has N states, which can have energy 0 or \Delta. Show that the number of configurations \Omega(E) of the total system with energy E=r\Delta (where r is an integer) is given by:

   \Omega(E) =\displaystyle \frac{N!}{r!(N-r)!}

   Now remove a small amount of energy s\Delta from the system, where s\ll r. Show that:

   \Omega(E-\epsilon) \approx \Omega(E)\displaystyle \frac{r^s}{(N-r)^s}

   and as a consequence, the system has a temperature that can be obtained from the relationship:

   \displaystyle \frac{1}{k_B T} = \frac{1}{\Delta}\ln \left(\frac{N-r}{r} \right)

   Sketch a graph of k_B T as a function of r from r=0 to r=N and explain your results.




2. A visible light photon with energy 2[eV] is absorbed by a macroscopic body that remains at room temperature.

   a) By what factor does \Omega change for a macroscopic body?

   b) Consider a photon emitted by a radio antenna in the FM range (with a typical frequency of 100[MHz]). Based on this, repeat the calculations from the previous part when the absorbed photon comes from an FM source. Use the relation E=hf, where f is the wave frequency and h=4,135\;667\;696 \cdot 10^{-15}[eV \cdot s] is Planck’s constant.




3. Find the average energy \lt{E}\gt for:

   a) A system with n states, where each state can have energies 0, \varepsilon, 2\varepsilon, 3\varepsilon, \cdots , n\varepsilon.

   b) A harmonic oscillator, where a state can have energies 0, \varepsilon, 2\varepsilon, 3\varepsilon, \cdots (with no upper limit).

Whiteboard with all calculations