Some helpful definitions:

Equilibrium – a system in which there are no changes in any of its macroscopic properties with time

Ensemble – a large collection of equivalent macroscopic systems (Example: a whole room full of glasses of water)

Microstate – one particular state in one member of the ensemble (Example: one possible arrangement of molecules in one of the glasses of water)

Configuration (W) – the collection of all equivalent microstates (Example: all of the possible configurations of water molecules in the glass of water are represented by one of the glasses of water in the room)

What are fluctuations?

When we talk about fluctuations in physical chemistry, we are referring to how the value of an observable property in a macroscopic system varies about its average value. Fluctuation occurs because macroscopic systems are in dynamic equilibrium, meaning the properties of the system vary slightly from moment to moment even though the system as a whole is in equilibrium. This happens as a result of the many equivalent microstates that are possible for the system.

Probability and Statistics

In order to discuss the behavior of fluctuations in physical chemistry, it is necessary to understand some basic statistical concepts.

Combinations: The formula used for determining the total number of combinations for a random system in which the particles can be in only one of two different states is:

There are 210 different ways that you can toss the coins so that the total number of heads is 6. This formula does not take into account the order in which the measurements occurred.

 Probability: To find the probability of getting any one combination divide C(N,p) by 2^N, which is the total of all possible combinations. For our coin flipping example,

the probability is 0.21, or 21% of the time 6 out of 10 of the coins tossed will be heads.

Permutations: Permutations give the number of microstates that exist in an ensemble. They are used when the order of the outcomes is important and there are more than two different states possible. The formula used is:

An Illustration using Probability and Dice

The number of ways to roll any number on a single die is one. If the sum of two dice is taken, the amount of ways to get certain numbers changes with the number. For example, the probability of getting one is zero because there is no possible combination of the two dice that would yield a sum of one. There are six ways to get a sum of seven between the two dice. These are 1 and 6, 6 and 1, 2 and 5, 5 and 2, 3 and 4, 4 and 3. If the sum of three dice is taken, we find that there are 27 ways in which to roll a sum of 10 and 27 ways to roll a sum of 11. The results of this experiment are illustrated in the following graph.
 
 

The left part of the graph shows that for one die, the number of combinations for rolling a particular number does not change. The combinations for the sum of two dice are represented by the middle curve, and the combinations for the sum of three dice are represented by the far right curve. The peak of each of the curves is located at the average sum of the dice. For two dice, the average takes place at 7 (possible outcomes from 2 to 12, so the average is (2+12)/2=7). For three dice, the average takes place at 10.5 (possible outcomes from 3 to 18, so the average is (3+18)/2=10.5). From the progression of these curves, it is possible to predict that as more dice are used, the curve will become steeper and more tightly fitted about the average value. This means that using more dice creates more combinations at the average value of the sums, and the deviation from the average value becomes less and less in comparison to the total number of dice used.

Applications to Statistical Mechanics

The standard deviation of an observed property about the average value increases proportional to the square root of N. This can be shown with a comparison of the equations for a binomial distribution and a standard Gaussian curve.

The N in the denominator in the binomial distribution equation corresponds to the s ² (standard deviation) in the Gaussian Curve equation. This shows that the standard deviation increases as the square root of N.

As the number of particles increases, so does the variation about the average value. Because there are more possible microstates, there is a larger range of values. As a result, there are more possible ways to deviate from the average value, thus increasing the standard deviation. This graph illustrates the relation between the standard deviation and N.

Although the standard deviation, represented by the lower line, is always increasing as the square root of N, the number of particles increases as N, a faster rate. The result of this is a curve that appears to be narrower and more tightly fitted about the average value with a large number of particles. This is illustrated by the following set of graphs. The more particles there are, the more observations appear at or close to the average value, narrowing the curve. Looking at the x-axis, however, reveals that the standard deviation is increasing, but at a slower rate than N.

 
 
 
 

The experiment with dice can be related to the configuration of an ensemble. The more dice there are, the more possible configurations of numbers there will be. With a larger number of dice, two different events can be observed. There will be an increasing variance about the average value because there is a greater range of possible values. But there will also be a larger number of ways to get the average value (example: six ways to roll a seven for two dice and 27 ways to roll both a ten and eleven for three dice). Because the number of ways to get the average value increases at a faster rate than the standard deviation, the curve gets thinner and more tightly centered about the average value. So it is with an ensemble. The more particles there are in a macroscopic system, the greater the number of microstates there will be. The range of possible values and standard deviation will therefore increase, but the number of ways to get a configuration close to or at the average value increases at a faster rate. With an increasing number of particles, the curve representing the range of an observable property becomes more tightly fitted around the average value of the property and a smaller percentage of particles deviate significantly from this value.

Relation to Thermodynamics

Second Law:

The more configurations that are possible in an ensemble, the higher the entropy is. This can be illustrated by Boltzmann's equation,

where W is the number of possible configurations. The Second Law of Thermodynamics also says that the entropy of the universe is always increasing. At first glance, the graphs above seem to contradict this statement. As N increases and the number of configurations increases, more and more configurations lie at the average value. This would indicate that entropy is decreasing. The error in this assumption can be discovered with a more careful observation. With increasing N, entropy will always increase, but since the number of particles in the system, N, increases at a faster rate than the entropy, the curve makes entropy to appear to decrease. The ratio S/N can increase or decrease, but S by itself will always increase with a higher value of N.

Third Law:

The Third Law of Thermodynamics says that absolute zero is unattainable. By what we have learned about fluctuations, we can see how this is true. Since observable properties of a macroscopic system are in dynamic equilibrium and the microstates fluctuate slightly about the average value, it is impossible to reach absolute zero. The energy of the system will be an average, and individual microstates will deviate slightly from this value. If energy is never absolutely zero, then temperature can also not reach absolute zero.
 
 

References:

McQuarrie, Donald A. Statistical Mechanics. Harper Collins Publishers, 1976.
 
 

Ulness, Darin. Physical Chemistry Lecture Notes. Concordia College.

Special Thanks to Dr. Ulness for assistance with Mathematica and answering many questions.