Statistical Mechanics

Introduction

Statistical mechanics provides a bridge between the macroscopic realm of classical thermodynamics and the microscopic realm of atoms and molecules. We are able to use computational methods to calculate the thermodynamic parameters of a system by applying statistical mechanics. Of particular interest in biochemistry is the ability to calculate free energies associated with a variety of processes such as ligand-receptor interaction and protein stability. This review of some basic principles of statistical mechanics serves as a prelude to discussions of free energy simulations.

We must remember that the energies of molecules, atoms, or electrons are quantized. To describe chemical systems we must know the energies of the quantum states and the distribution of particles (i.e., molecules, atoms, or electrons) among the quantum states. The Schrodinger equation that we discussed in the section on quantum mechanics provides a method for calculating the allowed energies. The Boltzmann distribution law that is a fundamental principle in statistical mechanics enables us to determine how a large number of particles distribute themselves throughout a set of allowed energy levels. Presented below are two derivations of the Boltzmann distribution law, one based on a treatment by Nash [1], and the second, by Barrow [2]. Both approaches are given as a means to illustrate that multiple ways are sometimes available to develop a concept.

Boltzmann Distribution Law (adapted from Nash)

Microstates and Configurations

Consider a simple assembly of three localized (i.e., distinguishable) particles that share three identical quanta of energy. These quanta can be distributed among the particles in ten possible distributions as illustrated in the figure below.

Each of the ten distributions is called a microstate; in our example the ten microstates fall into three groups, or configurations, denoted A, B, and C. When dealing with only three particles, we can count the number of microstates and configurations, but for larger numbers of particles, we need to calculate them. Our simple example above can be used to understand how to calculate the number of microstates and configurations.

For configuration B: The two-quanta parcel of energy can be assigned to any of the three particles, the one-quantum parcel to either of the two remaining particles, and the zero-quantum parcel to the remaining particle. Thus, the total number of ways in which assignments can be made is

For configuration A: We again have three choices in assigning the first (three-quanta) parcel of energy, two choices when we assign the second (zero-quantum), and one choice when we assign the third (zero-quantum). But because the last two parcels are the same, the final distribution is independent of the order in which they are assigned. That is,

distinguishable assignments collapse into one microstate because the two particles end up in the same quantum level. Thus, the total number of microstates associated with configuration A is

For configuration C: We have triple occupancy of the first quantum level and the final distribution is independent of the order in which the particles are assigned. Thus, we have one microstate.


This approach can be generalized to a larger number (N) of localized particles. We have N choices of the particle to which we assign the first parcel of energy, (N-1) choices in assigning the second, etc., to give a total of

distinguishable possibilities if no two energy parcels are the same. If some number (n_a) of the parcels are the same, we can have only N!/n_a! distinct microstates. Symbolizing by W the total number of microstates in a given configuration, we can write

where n_a represents the number of units occupying some quantum level, n_b represents the number of units occupying some other quantum level, etc.

This equation can be written in the general form

For our simple example above, we can calculate the number of microstates in configuration A as

For very large N, it is necessary to deal with natural logarithms. That is,

We can now apply Stirling's approximation in the form

We can then approximate the expression for the number of microstates as

Because , we can simplify to

Predominant Configuration

Note that in our simple example, one of the configurations (B) had a greater number of microstates associated with it than the others. As N increases, a predominant configuration becomes even more apparent.

We of course want to be able to identify this predominant configuration. For large N, it is not practical to do this by the approach we have illustrated above. However, it can be shown that the distribution of microstates for large N can be described by a smooth curve, the sharpness of which increases as the number of particles increases.

In the graph above, the configuration index number represents the fraction describing the number of times a microstate corresponding to a given configuration is observed divided by the total number of microstates. The predominant configuration corresponds to the very peak of the curve. That is,

where dX denotes a change from the predominant configuration to another configuration only infinitesimally different from it. Using this criterion, we can now develop a simple formula that describes the predominant configuration.

The Boltzmann distribution law

Consider an isolated macroscopic assembly of N particles, identical but distinguishable by spatial localization, which share a large number of energy quanta, each of which suffices to promote a particle from one quantum level to the next level above it. Because the assembly is isolated, both N and the energy will remain constant. We now want to determine for which of the enormous number of configurations that can be assumed by the assembly will the number of associated microstates realize its maximum value.

Consider any three successive quantum levels l, m, n, with associated energies e_l, e_m,e_n and numbers of particles n_l, n_m, n_n, respectively. For any given configuration, the number of microstates can be calculated from

Now make a change in the initial configuration such that one particle is shifted from each of levels l and n into level m. This change maintains a constant number of particles and energy but creates a new configuration that differs from the first in that

Hence,

Because we want to calculate the properties of the predominant configuration, let us assume that the original configuration was in fact the predominant configuration for which the number of associated microstates reaches its maximum value, where

Because the change we have made in the configuration is extremely small, we would expect essentially no change in W. Therefore,

from which we can write


Canceling terms that are common to both sides of the equation and inverting gives

Rearranging gives

Expanding some of the factorial terms gives

Cancellation gives

But numbers such as 1 and 2 are small in comparison to the large populations being studied. Hence,

or

This relationship also holds for other successive energy levels. Thus, for a macroscopic assembly of particles with uniform energy spacing between their quantum states, we can describe the predominant configuration as the one for which the following geometric series applies:

A similar approach can be taken when considering a system in which the energy spacing is not uniform. Without going through the derivation, the appropriate equation is

The values of p and q are small, positive integers selected so that the following equation holds:

Thus,

Since

Then,

Note that l, m, and n are any three quantum levels. Thus, for any two quantum levels, the indicated function must have exactly the same value. That is, this function is a constant; it is denoted by beta.
For any two quantum states, i and n, we can write

If i is taken to be the ground state, with population n₀ and energy e₀ = 0, this equation reduces to

or


This equation is the Boltzmann distribution law. It defines the predominant configuration for an isolated macroscopic assembly of identical but distinguishable particles, with any kind of energy spacing between their quantum states. When a system is said to obey a Boltzmann distribution, it will be consistent with the above equation. At equilibrium, the configuration of an isolated macroscopic assembly is typically that described by the Boltzmann distribution law.

The Meaning of Beta
The constant beta has a significant physical meaning. We will not show the derivation, but beta is inversely related to temperature. That is,

where T is in Kelvin, k has unit of energy/K.

Degeneracy

The Boltzmann distribution law, as derived above, considers the population of each distinct quantum state. Some of these quantum states may have the same energy, in which case these states are said to be degenerate. To account for degeneracy, one simply multiplies the energy level by the number of quantum states that have that energy. Degeneracy will be included explicitly in the derivation that follows.

Boltzmann distribution law (adapted from Barrow)

In this derivation of the Boltzmann distribution law, the multiplicity, or degeneracy, of the quantum states is taken into account during the derivation. Again, the basis of the derivation is the search for the most probable distribution.

Consider how four particles could distribute themselves in a box containing one large and one small compartment. Consider the box to be four units in size, with the larger compartment comprising three units and the smaller compartment, one unit.

Figure adapted from G.M. Barrow, Physical Chemistry, 2nd ed. (1966), 98. Copyright 1966 by McGraw-Hill, Inc. McGraw-Hill makes no representations or warranties as to the accuracy of any information contained in the McGraw-Hill material, including any warranties of merchantability or fitness for a particular purpose. In no event shall McGraw-Hill have any liability to any party for special, incidental, tort, or consequential damages arising out of or in connection with the McGraw-Hill Material, even if McGraw-Hill has been advised of the possibility of such damages.

Two factors determine the probabilities of the different arrangements. The first factor involves the relative size of the compartments, which corresponds to different degeneracies. It is calculated as the size of the compartment (the degeneracy) raised to a power corresponding to the number of particles in that compartment. This factor alone would suggest that the most probable distribution is the one in which all of the particles are in the largest compartment.

The second factor in the probability expression involves the total number of ways the particles can be rearranged without altering the distribution. This concept was addressed in the previous derivation of the Boltzmann distribution law. The probability of each distribution is the product of these two factors.

Consider now how an Avogadro's number (N) of molecules will distribute themselves throughout their allowed energy levels.

where W is the probability of a given distribution. We want to find the distribution such that W is a maximum. It is more convenient mathematically to seek a maximum in ln W. Taking the natural logarithm of the above equation, and applying Stirling's approximation to the n_i terms gives

In solving for a maximum in ln W, we must impose two conditions, a constant number of molecules and a constant total energy of the system. That is,

and

These conditions are imposed by applying Lagrange's method of undetermined multipliers. Two parameters, a and b, are introduced. Rather than looking for a maximum in ln W, we seek a maximum in

Thus, we can assure constant N and constant E, and the maximum we find is still a maximum in ln W. To find the maximum with respect to n_i, we solve

Recall our previous expression for ln W. That is,

N is a constant that is independent of the individual n_i's, so we can write

Substituting, we obtain

Considering the relative populations of two energy levels, i and j, we can write

This again is the Boltzmann distribution law, in this case with the degeneracy (g_i's) explicitly shown.

Printed References

[1] Nash, L.K., Elements of Statistical Thermodynamics, 2nd ed., Addison-Wesley Publishing Co. Inc., Reading, MA, 1974.

[2] Barrow, G.M., Physical Chemistry, 2nd ed., McGraw-Hill Book Co., New York, 1966.



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Héctor A. Chacón C.