# Fundamentals of statistical thermodynamics

## Introduction

Statistical thermodynamics provides a framework for relating the microscopic properties of individual molecules to the measurable macroscopic thermodynamic properties. The molecular-level interpretations of macroscopic thermodynamic quantities are provided by relating them to the positions and momenta of individual molecules. The ability of predicting macroscopic properties based on microscopic properties is the major advantage of statistical thermodynamics over the classical thermodynamics. Although both classical and statistical thermodynamics are governed by the second law of thermodynamics through the thermodynamic property of entropy, the entropy in the classical thermodynamics can only be defined empirically through analysis of irreversible thermodynamic process and cycles. Both Kelvin-Planck and Clusius statements of the second law of thermodynamics are expressed negative statements that cannot be rigorously proved from more basic principles within the frame work of the classic thermodynamics. Although the second law of thermodynamics is a phenomenological law based on the experimental observation, the irreversibility resulted from the second law of thermodynamics can, however, be easily explained based on the random motion of the molecules in the system.

## Microstate and Macrostate

Since the number of molecules in a thermodynamic system is too large, tracking the positions and velocity of each individual molecule in the system is not practical. While the location and velocity of a particular molecule in the system does not have direct correlation with the macroscopic properties (e.g., pressure, temperature, and specific volume, etc.), the collective behavior of the molecules in the system dictates the macroscopic properties of the system (Tien and Lienhard, 1985; Hill, 1987; Fitzpatrick, 2005). From microscale point of view, the entropy can be considered as a measure of the number of possible microstates (i.e., microscopic states) of a system in thermodynamic equilibrium, at which the state of the system, i.e., the macrostate can be defined and described by macroscopic thermodynamic properties. Considering a rigid vessel filled with gas that consists of a huge number of gas molecules, the gas molecules can freely move in the vessel, and collide with each other and with the walls of the vessel. The description of the locations and velocities of all gas molecules in the container constitutes the microstate of the system. On the contrary, if the system is in thermodynamic equilibrium, the system can be adequately described by two independent thermodynamic variables such as pressure and temperature. The description of a thermodynamic system with thermodynamic variables constitutes the state, or macrostate of the system. Theoretically, the thermodynamic properties of a system are determined by its microstate, i.e., the positions and momenta of the molecules in the system. However, the motion of any individual molecules in the system cannot affect the behavior of the system as a whole due to the large number of molecules in the system. To describe the microstate of the system, it is necessary to give positions and momenta of every molecule in the system. On the other hand, when the system is at thermodynamic equilibrium, the macrostate of the system can be uniquely defined using only a few thermodynamic variables. As the result, one macrostate corresponds to many different microstates, i.e., a given set of values of pressure and temperature is consistent with a huge number of microstates. This difference and relations between the microstate and macrostate can be illustrated by considering a system that contains N coins, each of them is either heads up or tails up. To define the microstate of the system, we must know exactly which side of the coin is facing up for all coins. On the contrary, the macrostates of the system can be defined by specifying the total number of the coins that are heads up and tails up. In this example, the property of each coin only has two possible values: heads up or tails up. For a system with gas filled in a tank, each molecule must be described by its spatial coordinate and velocity. Both spatial coordinate and velocity have three components and each component can vary continuously. Therefore, the number of microstates for a thermodynamic system is in fact infinite. When a system is in thermodynamic equilibrium, the macrostate of the system is well defined by the thermodynamic properties but the microstate of the system is constantly changing because the molecules occupy different positions every time and their momenta are also constantly changing as they collide with each other or with the container walls.

## Statistical Definition of Entropy

The entropy in thermodynamics can be interpreted in two distinct but some what related ways: (1) a macroscopic point of view (i.e., classical thermodynamics), and (2) microscopic point of view (i.e., statistical thermodynamics). The statistical definition of entropy is considered to be the fundamental definition because the classical definition can be mathematically derived from it. In statistical thermodynamics, entropy is defined as (Tien and Lienhard, 1985; Laurendeau, 2005)

$S = - k\sum\limits_i {{P_i}\ln ({P_i})} \qquad \qquad(1)$

where i represents a possible precisely defined microstate of the system, the Pi is the probability for the system to be in the state i, and k is a constant of proportionality whose value depends on the unit for entropy. Equation (1) can be viewed as the fundamental definition of entropy since all other formulas for S can be mathematically derived from it, but not vice versa. It gives a measure of entropy for systems of atoms and molecules in the gas phase, thus providing a measure for the entropy of classical thermodynamics. For an isolated thermodynamic system with the given volume and internal energy, there are a huge number Ω of microstates the system can be in. For a system where all accessible microstates have the same probability – referred to as microcanonical system – the probability of every state becomes

${P_i} = \frac{1}{\Omega }\qquad \qquad(2)$

Substituting eq. (2) into eq. (1) leads to

$S = {k_b}\ln \left( \Omega \right)\qquad \qquad(3)$

where kb is the Boltzmann constant, $1.3806 \times {10^{ - 23}}{\rm{J/K}}$, which is the the standard choice for the proportional constant in thermodynamics. The above statistical entropy can be viewed as the amount of uncertainty which remains about a system with a given set of macroscopic variables, such as temperature and volume. The entropy is a measure of the degree to which the probability of the system is spread out over different possible microstates, i.e., the entropy increases with increasing states available to the system with appreciable probability. The entropy of a system at thermodynamic equilibrium is maximized since all information about the initial conditions of the system is lost. Even for an isolated system that is at thermodynamic equilibrium, the microstate is constantly changing. To illustrate this process, let us consider a container with a partition (see Fig. 1) and placing a gas on one side of the partition (A side), with a vacuum on the other side (B side). To simplify our discussion, let us consider the situation that there are only four gas molecules on the A side. Before the partition is removed, molecule a can only stay on the A side. After the partition is removed, the molecule a can either on the A side or B side, i.e., the probability for the molecule a to stay in A side is 1/2. Similarly, the probabilities for molecules b, c, or d to stay on A side is also 1/2. The possible distribution of molecules in A and B side can be summarized in Table 1. There are 16 microstates for this simple system and uniform distribution of molecules (6 microstates) has highest probability. Although it is possible that all molecules stay on the A side, its probability is only 1/16 or ½4. If we consider a system with one mole of gas, the number of molecules will be N = $6.02 \times {10^{23}}$ and the probability for all molecules to return to the A side is $1/{2^N} = 1/{2^{6.02 \times {{10}^{23}}}}$, which is impossible. The probability for the gas to spread out to fill the container evenly is the highest and it represents the new equilibrium macrostate of the system, at which the entropy of the system is at its maximum. It follows from this simple example that the second

Figure 1 Free expansion of gas.

law of thermodynamics can be obtained: the total entropy of any isolated thermodynamic system tends to increase over time, approaching a maximum value.

Table 1 Distribution of molecules in a container
Macrostate Number of molecules on A side 4 1 2 3 0
Number of molecules on B side 0 3 2 1 4
MicrostateMolecules on A sidea
b
c
d
abcda
b
a
c
a
d
b
c
b
d
c
d
b
c
d
a
c
d
a
b
d
a
b
c
0
Molecules on B side0b
c
d
a
c
d
a
b
d
a
b
c
c
d
b
d
b
c
a
b
a
c
a
b
abcda
b
c
d
Number of microstate1 4 6 4 1

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Fitzpatrick, R., 2005, Thermodynamics and Statistical Mechanics: An Intermediate Level Course, Lulu.com, 2005.

Hill, T.L. 1987, An Introduction to Statistical Thermodynamics, Dover Publications, Inc., Mineola, NY.

Laurendeau, N. M., 2005, Statistical Thermodynamics: Fundamentals and Applications, Cambridge University Press, New York.

Tien, C.L., Lienhard, J.H., 1985, Statistical Thermodynamics, Revised Sub. Edition, Hemisphere Publishing Corp., New York, NY.