The Partition Function
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Imagine that we have an ideal gas consisting of N identical simple particles and that we have confined it inside a vertical tube that rises as high as necessary for our purpose. Assume further that the gas has achieved a uniform temperature throughout the volume it occupies. We want to examine a horizontal slice of the gas of thickness dh at altitude h inside the tube and ask how many particles we should expect to find in that slice at any given time.
If we represent the particle density in the gas (particles per cubic meter, say) with the Greek letter rho, then we know from our previous analysis of the Maxwell distribution that the pressure in the gas conforms to
We also know that the pressure that the slice exerts upon the slice below it due to the weight of its particles conforms to the algebraic description
in which m represents the mass of a single particle and g represents the acceleration of gravity inside the tube (again, like the temperature, assumed to be uniform; we don=t want to extend our tube too far from Earth=s surface). In terms of the pressure within the slice, Equation 2 becomes
As a proportion of its pressure, the change in the pressure in any slice in the gas column inside the tube due to the weight of the slice above it equals
If we were to proceed up the tube away from the slice that we put at h=0 for convenience, we find that each and every slice contributes the same fractional pressure to our slice. Those contributions all add up, so we have
In that equation p0 represents the pressure at h=0 and p represents the pressure at the top of the tube (or wherever else we stop measuring, a distance h above the slice at h=0). If we want to convert that equation into a description of pressure in the tube as a function of increasing height, we must solve it for p, so we get
Thus we obtain the law of atmospheres, the beginning assumption in the construction of a model of a real atmosphere.
Now let=s take a closer look at the gas in one of the slices described above. We want to examine the distribution of kinetic energy among the particles in that parcel of gas. Following Maxwell, we want to calculate the number of particles whose kinetic energies due solely to motion in the x-direction lie in the range between Ex and Ex+dEx. Of all the particles in the parcel we take the fraction that fit that criterion as f(Ex), noting that we can also interpret that function as expressing the probability that any given particle has a kinetic energy in the given range. In like manner we define f(Ey) and f(Ez) to calculate the fractions of particles with kinetic energies in the ranges between Ey and Ey+dEy and between Ez and Ez+dEz. And for the total kinetic energy of a particle, E=Ex+Ey+Ez, we have the probability of finding the particle=s energy in the range between E and E+dE as f(E), which has the same mathematical form as does the probability function for each of the components of the kinetic energy.
If we look at one particle, we know that the energy Ex that it has due to its motion only in the x-direction is completely independent of the amounts of energy that it has due to its motion in the other two directions. If we want to calculate the probability that the particle has its energies in the ranges between Ex and Ex+dEx, between Ey and Ey+dEy, and between Ez and Ez+dEz, we must calculate the fraction of the particles in the gas that exist in the first range, then calculate the fraction of that fraction existing in the second range, and then the fraction of that fraction existing in the third range, so for the probabilityΠ we have
But that calculation must also give us the probability of finding the particle with its total energy in the range between E=Ex+Ey+Ez and E+(dEx+dEy+dEz), so we must have
Only one function satisfies that equation B the exponential. We have thus
subject to the condition that
with analogous equations for the y- and z-directions. In Equation 9 C represents a proportionality constant whose value we determine through Equation 10. In the exponent I have multiplied the energy by beta, representing an inverse energy, because the exponent must be a pure number and I have inserted the negative sign so the function won=t Ablow up@, but will go to zero, as the energy tends toward infinity.
Now for the inevitable digression. Why do I want to carry out the integration in Equation 10 with respect to the square root of the energy instead of with respect to the energy directly? To answer that question we must look to the analytic geometry of the abstract space in which we describe our probabilities.
Imagine that we plot the component energies described above each on its own axis, one of three mutually perpendicular axes that we have superimposed upon the standard Cartesian x-, y-, and z-axes. In that abstract energy-space the equation
defines a tilted plane in the (+,+,+) octant of the space. But we cannot use that description in our calculation, because it will lead to absurdities. If another observer uses a Cartesian grid rotated relative to ours, then they will have to refer the energies of the particles to that grid. In some places energies that are positive on one grid must be negative on the other. But no such thing as a negative kinetic energy exists in the Reality that we describe with thermodynamics.
We still want to represent energy states in an abstract space in which we plot the three contributions to a particle=s kinetic energy along three mutually perpendicular axes. Because distances in such a space must conform to the Pythagorean Theorem and because the energy components add directly to each other, we must use the square roots of the energy components rather than the energy components themselves as our coordinates. In that space all of the states that correspond to a given total kinetic energy E define a spherical shell of area
When we calculate the probability of a particle occupying a certain physical state we must augment the raw probability associated with that state by the number of ways in which the particle can occupy that state. To that end we want to calculate the volume that the state occupies in the relevant abstract space, because the number of ways in which a particle can occupy the state will be proportional to that volume. We can thus devise a differential volume in energy space by multiplying the area in Equation 12 by , which gives us
Multiplying that by Equation 9 and integrating the product gives us Equation 10, which we now write as
In order to carry out that integration we must multiply and divide the integrand by the cube of the square root ofβ. We can also commute the operations of multiplication by constants and of integration, so we get
which gives us
We now want to calculate the number of particles in an ideal gas that lie at altitudes in the range between h1 and h1+dh and also have kinetic energies in the range between E1 and E1+dE. To carry out that calculation we multiply the total number of particles in the gas by the probability that a particle lies in the given altitude range and by the probability that a particle possesses a kinetic energy in the given range, so we have
Next we calculate the number of particles that lie at altitudes in the range between h2 and h2+dh (that range not overlapping the previous range) and also have kinetic energies in the range between E2 and E2+dE. We get
Note that in each case we have taken the complete set of the particles, extracted a subset of the particles, and extracted a subset of the subset. We could continue that process, extracting subsets based on the fraction of the particles lying in a certain range of chemical potential or of magnetic potential, until we have implicitly separated the particles into subsets according to all of their possible manifestations of energy.
If we want to calculate the average kinetic energy in the particles comprising the subsets described in Equations 17 and 18, we write
In that equation the coefficients and differentials in Equations 17 and 18 cancel out, so I=ve left them out. If we had identified additional states of being in the gas, each with its own array of energies, that equation would have had additional terms in both the numerators and the denominators on the right side of the equality sign. If we number the different states of being in the gas with an index i and identify each state by the total energy
associated with every particle in that state, then for any property Pi associated with the particles we have the average value of that property as
In that equation the index takes all of the values that mark states of being in the gas.
We can use Equation 21 to calculate other statistical functions, such as root-mean-square, of the properties of the particles in the gas. In every case the denominator remains the same, so we redefine that sum of the Boltzmann factors associated with the states of being in the gas to constitute a new function,
which we call the Partition Function. The letter Z stands for the German word Zustandsumme (sum over states). The special usefulness of that function comes from the fact that when we differentiate its natural logarithm we get Equation 21; that is, for example, if we differentiate lnZ with respect toβ=1/kT, we get
Because we represent the energies of particles through functions of generalized momenta and positions, we can use suitable derivatives of lnZ to calculate other averages as well.
So far I have tacitly assumed into my premises the proposition that the states available to the particles in our gas comprise a discrete set. To get a more realistic picture of the gas I now replace that assumption with an assumption that the set of the available states of being in the gas make up something more closely resembling a continuum. In that case we must replace the sum in Equation 22 with the pseudo-continuous sum of an integration. Because the Boltzmann factors that go into the partition function are themselves functions of the generalized momenta and positions, we must carry out the integration with respect to those variables B integrating with respect to any other variables corresponds to a simple multiplication of the function by those variables B so we have
in which m represents the number of different generalized coordinates relevant to the thermodynamic system under study and h represents a unit of action that we use as a grid-size unit in the phase space based on the generalized coordinates. I will return to the meaning of that phase space in the essay on Gibbs= paradox.
Appendix: LnZ and Entropy
If we have a generalized force,
and an associated generalized displacement q, then the product Fdq represents a differential increment of work done on or by a thermodynamic system. That work adds to or subtracts from the system=s total energy, so the partition function must include an energy term E(q). If we subject that system to a change in that displacement and do so slowly enough that the system does not go far from the state of thermal equilibrium at any time, then we have
in which we have used
Equation 26 then gives us
in which we have equated the work done on a system plus the increase in the system=s average overall energy to the amount of heat the system gains. But a change in the system=s entropy conforms to the description
so we have for the entropy itself
And that is the standard relation between the entropy of a system and the partition function of that system that you will find in any text on thermodynamics.
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