The Boltzmann Factor

Imagine a container filled with a monatomic gas, such as helium. We conceive the helium atoms as hard spheres, like minuscule billiard balls. Though we tacitly assume the possibility of achieving infinite precision in determining the motions of the particles, we know that we cannot actually achieve such a thing. In order to put a particle onto a precisely defined path we must first know its starting location, we must see where it is. But in order to see the particle we must illuminate it and the light that we use, however dim, will move the particle in an uncontrollable and unpredictable way. We also know that when two billiard balls collide, a tiny change in the locations of trajectories that the balls’ centers follow before the collision results in a large change in the angles through which the balls get deflected. So again we see that our helium atoms, when they collide, separate from each other on paths that nobody can control or predict.

There’s no way in which we can trace the paths of the particles in the gas, nothing like what we can do with planets, for example. But we don’t want to trace the paths of individual particles in a gas. We want to determine the distribution of the gas’s total energy among the particles and then calculate collective properties, such as the pressure that the gas exerts upon the walls of its container.

A straightforward analysis of probabilities gives us a description known as the Maxwell distribution. If the gas contains a total energy E (the sum of the kinetic energies of all of the particles) and one particular particle carries energy ε, then the number of different microstates available to that system (the number of different ways in which the energy can be distributed among the particles) stands in direct proportion to the number of microstates that the system would have with the particular particle removed from the system, Ω(E-ε). The probability of finding the gas in that particular macrostate, then, comes to

(Eq’n 1)

in which C represents a normalization constant, a factor that makes

(Eq’n 2)

when the sum is carried out over all possible values of epsilon.

Omega represents a large number and that number can vary significantly with changes in epsilon. But if we consider the natural logarithm of omega, we know that it will vary slowly with changes in epsilon. We can thus write out the Taylor series expansion of the logarithm,

(Eq’n 3)

The derivative, evaluated at ε=0, represents the reciprocal of the temperature parameter, kT, the average energy per particle in the gas.

In the upper part of the partial derivative the natural logarithm equals the entropy of the gas divided by Boltzmann’s constant. Differentiating the entropy with respect to the energy at or near the state in which epsilon equals zero yields the reciprocal of the absolute temperature of the gas. That’s how we get the statement that the derivative equals the reciprocal of the temperature parameter.

We thus have

(Eq’ns 4)

Because Ω(E) represents a constant relative to the values of epsilon, we can absorb it into the coefficient C and write Equation 1 as

(Eq’n 5)

The exponential in that equation is the Boltzmann factor.

Now let’s go one step further. In phase space the
particles occupy cells of minuscule volume dxdydzdp_{x}dp_{y}dp_{z}
(the quantum theory adds the proviso that none of the cells can have a volume
less than the cube of Planck’s constant). By suitably adjusting the coefficient
in the first line of Equation 5 we can convert probability into probability
density, the probability per cubic meter per cubic newton-second of finding a
particle in a given cell. Integrating the resulting expression over the spatial
volume simply gives the formula a factor equal to the volume of the container
and we can absorb that factor into the coefficient if the volume remains
constant. Expressing the particles’ kinetic energies in terms of their momenta
lets us determine the probability of finding a particle with a given momentum as

(Eq’n 6)

That equation describes the Maxwell distribution of energy among the particles in an ideal gas.

Appendix: Another Way to Derive the Boltzmann Factor

We know that in a gas the probability of finding a certain particle in a state in which it carries energy ε stands in direct proportion to exp[-ε/kT], the Boltzmann factor, in which T represents the absolute temperature of the gas. How does that exponentiated function come into being? There’s an easy way to derive it, one based on the fact that we have two ways in which we can describe the change in a system’s entropy.

In classical thermodynamics adding an increment of heat pQ to a system changes the system’s entropy by

(Eq’n A-1)

This comes to us through the Helmholtz theorem. In statistical thermodynamics we have a number Ω that represents the number of microstates that constitute a given macrostate of the system. We have the entropy of that system through Boltzmann’s equation,

(Eq’n A-2)

Adding a minuscule increment of heat to the system increases the number of microstates by dΩ, so the entropy changes by

(Eq’n A-3)

Equating Equations A-1 and A-3 gives us

(Eq’n A-4)

In our calculation we don’t want to use the number of states; we want to use a probability.

Increasing the number of microstates that constitutes a given macrostate decreases the probability of finding a particle in a given energy state. If I blindly draw pebbles from an urn containing N pebbles, all but one of them white, the probability of drawing the black pebble equals 1/N. If I add n white pebbles to the urn, that probability changes by the amount

(Eq’n A-5)

If n is very much smaller than N, we have the approximation

(Eq’n A-6)

Making the equations n=dΩ and N=Ω lets us rewrite Equation A-4 as

(Eq’n A-7)

We can integrate that equation readily. We treat the energy increment ε as though we had added it to the system as heat by giving it to a particle that had no energy. The probability of a particle having that energy then comes out as

(Eq’n A-8)

in which lnC represents the constant of integration. We then have the antilogarithm as

(Eq’n A-9)

Thus, again, we have the Boltzmann factor.

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