1.2: Stationary or Steady Properties of Matter

a. Phase Space and Phase Density

Consider a volume of physical space that is small compared to the physical system in question, but still large enough to contain a statistically significant number of particles. The range of physical space in which this small volume is embedded may be infinite or finite as long as it is significantly larger than the small volume. First let a set of three Cartesian coordinates x₁, x₂, and x₃ represent the spatial part of the volume. Then allow the additional three Cartesian coordinates v₁, v₂, and v₃ represent the components of the velocity of the particles. Coordinates v₁, v₂, and v₃ are orthogonal to the spatial coordinates. This simply indicates that the velocity and position are assumed to be uncorrelated. It also provides for a six-dimensional space which we call phase space. The volume of the space is \[\mathrm{d V=d x_1 d x_2 d x_3 d v_1 d v_2 d v_3}\label{1.1.1}\]

Figure 1.1 shows part of a small differential volume of phase space. It must be remembered that the position and velocity coordinates are orthogonal to each other.

If the number of particles in the small volume dV is N, then we can define a parameter f, known as the phase density, by \[\mathrm{f\left(x_1, x_2, x_3, v_1, v_2, v_3\right) d V=N}\label{1.1.2}\]

The manner in which a number of particles can be arranged in an ensemble of phase space volumes is described in Figure 1.1.

b. Macrostates and Microstates

A macrostate of a system is said to be specified when the number of particles in each phase space volume dV is specified. That is, if the phase density is specified everywhere, then the macrostate of the system has been specified. Later we shall see how the phase density can be used to specify all the physical properties of the system.

To discuss the notion of a microstate, it must be assumed that there is a perceptible difference between particles, because in a microstate, in addition to the number of particles in each volume, it makes a difference which particles are in which volumes. If the specification of individual particles can be accomplished, then it can be said that a microstate has been specified. Clearly one macrostate could consist of many microstates. For example, the number of balls on a pool table might be said to be a macrostate, whereas the specification of which balls they are would denote a specific microstate. In a similar manner, the distribution of suits of playing cards in a bridge hand might be said to represent a macrostate, but the specific cards in each suit would specify the microstate.

c. Probability and Statistical Equilibrium

If we were to create macrostates by assembling particles by randomly throwing them into various microstates, then the macrostate most likely to occur is the one with the greatest number of microstates. That is why a bridge hand consisting of 13 spades occurs so rarely compared to a hand with four spades and three hearts, three diamonds, or three clubs. If we consider a system where the particles are continually moving from one phase space volume to another, say, by collisions, then the most likely macrostate is the one with the largest number of associated microstates. There is an implicit assumption here that all microstates are equally probable. Is this reasonable?

Imagine a case where all the molecules in a room are gathered in one corner. This represents a particular microstate; a particularly unlikely one, we would think. Through random motions, it would take an extremely long time for the particles to return to that microstate. However, given the position and velocity of each particle in an ordinary room of gas, is this any more unlikely than each particle to returning to that specific position with the same velocity? The answer is no. Thus, if each microstate is equally probable, then the associated macrostates are not equally probable and it makes sense to search for the most probable macrostate of a system. In a system which is continually rearranging itself by collisions, the most probable macrostate becomes the most likely state in which to find the system. A system which is in its most probable macrostate is said to be in statistical equilibrium.

Many things can determine the most probable macrostate. Certainly the total number of particles allowed in each microstate and the total number of particles available to distribute will be important in determining the total number of microstates in a given macrostate. In addition, quantum mechanics places some conditions on our ability to distinguish particles and even limits how many of certain kinds of particles can be placed in a given volume of phase space. But, for the moment, let us put aside these considerations and concentrate on calculating the number of microstates in a particular macrostate.

Figure 1.2 Shows a phase space composed of only two cells in which four particles reside. All possible macrostates are illustrated.

Consider a simple system consisting of only two phase space volumes and four particles (see Figure 1.2). There are precisely five different ways that the four particles can be arranged in the two volumes. Thus there are five macrostates of the system. But which is the most probable? Consider the second macrostate in Figure 1.2 (that is, \(\mathrm{N}_1=3, \mathrm{~N}_2=1\)).

Here we have three particles in one volume and one particle in the other volume. If we regard the four particles as individuals, then there are four different ways in which we can place those four particles in the two volumes so that one volume has three and the other volume has only one (see Figure 1.3). Since the order in which the particles are placed in the volume does not matter, all permutations of the particles in any volume must be viewed as constituting the same microstate.

Now if we consider the total number of particles N to be arranged sequentially among m volumes, then the total number of sequences is simply N!. However, within each volume (say, the ith volume), N_i particles yield N_i! indistinguishable sequences which must be removed when the allowed number of microstates is counted. Thus the total number of allowed microstates in a given macrostate is \[W=\frac{N!}{\prod_{i=1}^m N_{i}!}\label{1.1.3}\]

Figure 1.3 Consider one of the macrostates in figure 1.2, specifically the state where N₁ = 3, and N₂ = 1. All the allowed microstates for distinguishable particles are shown.

For the five macrostates shown in Figure 1.2, the number of possible microstates is \[\left.\begin{array}{l}
\mathrm{W}_{4,0}=4!/ 4!0!=1 \\
\mathrm{W}_{3,1}=4!/ 3!1!=4 \\
\mathrm{W}_{2,2}=4!/ 2!2!=6 \\
\mathrm{W}_{1,3}=4!/ 1!3!=4 \\
\mathrm{W}_{0,4}=4!/ 0!4!=1
\end{array}\right\}\label{1.1.4}\]

Clearly W_{2, 2} is the most probable macrostate of the five. The particle distribution of the most probable macro state is unique and is known as the equilibrium macrostate.

In a physical system where particle interactions are restricted to those between particles which make up the system, the number of microstates within the system changes after each interaction and, in general, increases, so that the macrostate of the system tends toward that with the largest number of microstates - the equilibrium macrostate. In this argument we assume that the interactions are uncorrelated and random. Under these conditions, a system which has reached its equilibrium macrostate is said to be in strict thermodynamic equilibrium. Note that interactions among particles which are not in strict thermodynamic equilibrium will tend to drive the system away from strict thermodynamic equilibrium and toward a different statistical equilibrium distribution. This is the case for stars near their surfaces.

The statistical distribution of microstates versus macrostates given by equation \ref{1.1.3} is known as Maxwell-Boltzmann statistics and it gives excellent results for a classical gas in which the particles can be regarded as distinguishable. In a classical world, the position and momentum of a particle are sufficient to make it distinguishable from all other particles. However, the quantum mechanical picture of the physical world is quite different. So far, we have neglected both the Heisenberg uncertainty principle and the Pauli Exclusion Principle.

d. Quantum Statistics

Within the realm of classical physics, a particle occupies a point in phase space, and in some sense all particle are distinguishable by their positions and velocities. The phase space volumes are indeed differential and arbitrarily small. However, in the quantum mechanical view of the physical world, there is a limit to how well the position and momentum (velocity, if the mass is known) of any particle can be determined. Within that phase space volume, particles are indistinguishable. This limit is known as the Heisenberg uncertainty principle and it is stated as follows: \[\Delta \mathrm{p} \Delta \mathrm{x} \geq \mathrm{h} / 2 \pi \equiv \hbar\label{1.1.5}\]

Thus the minimum phase space volume which quantum mechanics allows is of the order of h³. To return to our analogy with Maxwell-Boltzmann statistics, let us subdivide the differential cell volumes into compartments of size h³ so that the total number of compartments is \[\mathrm{n}=\mathrm{dx}_1 \mathrm{dx}_2 \mathrm{dx}_3 \mathrm{dp}_1 \mathrm{dp}_2 \mathrm{dp}_3 / \mathrm{h}^3\label{1.1.6}\]

Let us redraw the example in Figure 1.3 so that each cell in phase space is subdivided into four compartments within which the particles are indistinguishable. Figure 1.4 shows the arrangement for the four particles for the W_3,1 macrostate for which there were only four allowed microstates under Maxwell-Boltzmann Statistics. Since the particles are now distinguishable within a cell, there are 20 separate ways to arrange the three particles in volume 1 and 4 ways to arrange the single particle in volume 2. The total number of allowed microstates for W_3,1 is 20×4, or 80. Under these conditions the total number of microstates per macrostate is \[\mathrm{W}=\prod_{\mathrm{i}} \mathrm{~W}_{\mathrm{i}},\label{1.1.7}\]

where \(\mathrm{W_i}\) is the number of microstates per cell of phase space, which can be expressed in terms of the number of particles \(\mathrm{N_i}\) in that cell.

Figure 1.4 The same macrostate as figure 1.3 only now the cells of phase space are subdivided into four compartments within which particles are indistinguishable. All of the possible microstates are shown for the four particles.

Let us assume that there are n compartments in the ith cell which contains \(\mathrm{N_i}\) particles. Now we have to arrange a sequence of \(\mathrm{n+N_i}\) objects, since we have to consider both the particles and the compartments into which they can be placed. However, not all sequences are possible since we must always start a sequence with a compartment. After all we have to put the particle somewhere! Thus there are \(\mathrm{n}\) sequences with \(\mathrm{N}_{\mathrm{i}}+\mathrm{n}-1\) items to be further arranged. So there are \(\mathrm{n}\left[\mathrm{~N}_{\mathrm{i}}+\mathrm{n}-1\right]!\) different ways to arrange the particles and compartments. We must eliminate all the permutations of the compartments because they reside within a cell and therefore represent the same microstate. But there are just n! of these. Similarly, the order in which the particles are added to the cell volume is just as irrelevant to the final microstate as it was under Maxwell-Boltzmann statistics. And so we must eliminate all the permutations of the \(\mathrm{N_i}\) particles, which is just \(\mathrm{N}_{\mathrm{i}}!\). Thus the number of microstates allowed for a given macrostate becomes \[\mathrm{W}_{\mathrm{B}-\mathrm{E}}=\prod_{\mathrm{i}} \mathrm{n}\left(\mathrm{~N}_{\mathrm{i}}+\mathrm{n}-1\right)!/ \mathrm{N}_{\mathrm{i}}!\mathrm{n}!=\prod_{\mathrm{i}}\left(\mathrm{~N}_{\mathrm{i}}+\mathrm{n}-1\right)!/ \mathrm{N}_{\mathrm{i}}!(\mathrm{n}-1)!\label{1.1.8}\]

The subscript "B-E" on W indicates that these statistics are known as Bose-Einstein statistics which allow for the Heisenberg uncertainty principle and the associated limit on the distinguishability of phase space volumes. We have assumed that an unlimited number of particles can be placed within a volume h³ of phase space, and those particles for which this is true are called bosons. Perhaps the most important representatives of the class of particles for stellar astrophysics are the photons. Thus, we may expect the statistical equilibrium distribution for photons to be different from that of classical particles described by Maxwell-Boltzmann statistics.

Within the domain of quantum mechanics, there are further constraints to consider. Most particles such as electrons and protons obey the Pauli Exclusion Principle, which basically says that there is a limit to the number of these particles that can be placed within a compartment of size h³. Specifically, only one particle with a given set of quantum numbers may be placed in such a volume. However, two electrons which have their spins arranged in opposite directions but are otherwise identical can fit within a volume h³ of phase space. Since we can put no more than two of these particles in a compartment, let us consider phase space to be made up of 2n half-compartments which are either full or empty. We could say that there are no more than 2n things to be arranged in sequence and therefore no more than 2n! allowed microstates. But, since each particle has to go somewhere, the number of filled compartments which have N_i! indistinguishable permutations are just N_i. Similarly, the number of indistinguishable permutations of the empty compartments is \(\mathrm{(2 n-N_i)!}\). Taking the product of all the allowed microstates for a given macrostate, we find that the total number of allowed microstates is \[\mathrm{W}_{\mathrm{F}-\mathrm{D}}=\prod_{\mathrm{i}}(2 \mathrm{n})!/ \mathrm{N}_{\mathrm{i}}!\left(2 \mathrm{n}-\mathrm{N}_{\mathrm{i}}\right)!\label{1.1.9}\]

The subscript "F-D" here refers to Enrico Fermi and P.A.M. Dirac who were responsible for the development of these statistics. These are the statistics we can expect to be followed by an electron gas and all other particles that obey the Pauli Exclusion Principle. Such particles are normally called fermions.

e. Statistical Equilibrium for a Gas

To find the macrostate which represents a steady equilibrium for a gas, we follow basically the same procedures regardless of the statistics of the gas. In general, we wish to find that macrostate for which the number of microstates is a maximum. So by varying the number of particles in a cell volume we will search for \(\mathrm{dW}\). Since \(\ln\mathrm{W}\) is a monotonic function of \(\mathrm{W}\), any maximum of \(\ln\mathrm{W}\) is a maximum of \(\mathrm{W}\). Thus we use the logarithm of equations \ref{1.1.7} through \ref{1.1.9} to search for the most probable macrostate of the distribution functions. These are \[\left.\begin{array}{l}
\mathrm{\ln W_{M-B}=\ln N!-\sum_i \ln \left(N_{i}!\right)} \\
\mathrm{\ln W_{B-E}=\sum_i \ln \left(n+N_i-1\right)!-\ln N_{i}!-\ln (n-1)!} \\
\mathrm{\ln W_{F-D}=\sum_i \ln (2 n)!-\ln \left(2 n-N_i\right)!-\ln N_{i}!}
\end{array}\right\}\label{1.1.10}\]

The use of logarithms also makes it easier to deal with the factorials through the use of Stirling's formula for the logarithm of a factorial of a large number. \[\ln \mathrm{N}!\approx \mathrm{N} \ln \mathrm{~N}-\mathrm{N}\label{1.1.11}\]

For a given volume of gas, \(\mathrm{dN}=\mathrm{dn}=0\). The variations of equations \ref{1.1.10} become \[\begin{array}{r}
\delta \ln \mathrm{W}_{\mathrm{M}-\mathrm{B}}=\sum_{\mathrm{i}} \ln \mathrm{~N}_{\mathrm{i}} \mathrm{dN}_{\mathrm{i}}=0 \\
\delta \ln \mathrm{~W}_{\mathrm{B}-\mathrm{E}}=\sum_{\mathrm{i}} \ln \left[\left(\mathrm{n}+\mathrm{N}_{\mathrm{i}}\right) / \mathrm{N}_{\mathrm{i}}\right] \mathrm{dN}_{\mathrm{i}}=0 \\
\delta \ln \mathrm{~W}_{\mathrm{F}-\mathrm{D}}=\sum_{\mathrm{i}} \ln \left[\left(2 \mathrm{n}-\mathrm{N}_{\mathrm{i}}\right) / \mathrm{N}_{\mathrm{i}}\right] \mathrm{d} \mathrm{~N}_{\mathrm{i}}=0
\end{array}\label{1.1.12}\]

These are the equations of condition for the most probable macrostate for the three statistics which must be solved for the particle distribution \(\mathrm{N_i}\). We have additional constraints, which arise from the conservation of the particle number and energy, on the system which have not been directly incorporated into the equations of condition. These can be stated as follows: \[\mathrm{\delta\left[\Sigma_i N_i\right]=\delta N=0, \delta\left[\Sigma_i w_i N_i\right]=\Sigma_i w_i \delta N_i=0},\label{1.1.13}\]

where w_i is the energy of an individual particle. Since these additional constraints represent new information about the system, we must find a way to incorporate them into the equations of condition. A standard method for doing this is known as the method of Lagrange multipliers. Since equations \ref{1.1.13} represent quantities which are zero we can multiply them by arbitrary constants and add them to equations \ref{1.1.12} to get \[\begin{array}{lc}
\text { M-B: } & \sum_{\mathrm{i}}\left[\ln \mathrm{~N}_{\mathrm{i}}-\ln \alpha_1+\beta_1 \mathrm{w}_{\mathrm{i}}\right] \delta \mathrm{N}_{\mathrm{i}}=0 \\
\text { B-E: } & \sum_{\mathrm{i}}\left\{\ln \left[\left(\mathrm{n}+\mathrm{N}_{\mathrm{i}}\right) / \mathrm{N}_{\mathrm{i}}\right]-\ln \alpha_2-\beta_2 \mathrm{w}_{\mathrm{i}}\right\} \delta \mathrm{N}_{\mathrm{i}}=0 \\
\text { F-D: } & \sum_{\mathrm{i}}\left\{\ln \left[\left(2 \mathrm{n}-\mathrm{N}_{\mathrm{i}}\right) / \mathrm{N}_{\mathrm{i}}\right]-\ln \alpha_3-\beta_3 \mathrm{w}_{\mathrm{i}}\right\} \mathrm{dN}_{\mathrm{i}}=0
\end{array}\label{1.1.14}\]

Each term in equations \ref{1.1.14} must be zero since the variations in N_i are arbitrary and any such variation must lead to the stationary, most probable, macrostate.

Thus, \[\begin{array}{ll}
\text { M-B: } & \mathrm{N}_{\mathrm{i}} / \alpha_1=\exp \left(-\mathrm{w}_{\mathrm{i}} \beta_1\right) \\
\text { B-E: } & \mathrm{n} / \mathrm{N}_{\mathrm{i}}=\alpha_2 \exp \left(\mathrm{w}_{\mathrm{i}} \beta_2\right)-1 \\
\text { F-D: } & 2 \mathrm{n} / \mathrm{N}_{\mathrm{i}}=\alpha_3 \exp \left(\mathrm{w}_{\mathrm{i}} \beta_3\right)+1
\end{array}\label{1.1.15}\]

All that remains is to develop a physical interpretation of the undetermined parameters \(\mathrm{\alpha_j}\) and \(\mathrm{\beta_j}\). Let us look at Maxwell-Boltzmann statistics for an example of how this is done. Since we have not said what \(\mathrm{\beta_1}\) is, let us call it 1/(kT). Then \[\mathrm{N}_{\mathrm{i}}=\alpha_1 \mathrm{e}^{-\mathrm{w}_{\mathrm{i}} /(\mathrm{kT})}\label{1.1.16}\]

If the cell volumes of phase space are not all the same size, it may be necessary to weight the number of particles to adjust for the different cell volumes. We call these weight functions g_i. The \[\mathrm{N}=\Sigma_\mathrm{i} \mathrm{g_{i}N}_{\mathrm{i}}=\alpha_1 \Sigma_\mathrm{i} \mathrm{~g}_{\mathrm{i}} \mathrm{e}^{-\mathrm{W}_{\mathrm{i}} /(\mathrm{kT})} \equiv \alpha_1 \mathrm{U}(\mathrm{~T})\label{1.1.17}\]

The parameter U(T) is called the partition function and it depends on the composition of the gas and the parameter T alone. Now if the total energy of the gas is E, then \[\mathrm{E}=\Sigma_\mathrm{i} \mathrm{g}_{\mathrm{i}} \mathrm{w}_{\mathrm{i}} \mathrm{~N}_{\mathrm{i}}=\Sigma_\mathrm{i} \mathrm{w}_{\mathrm{i}} \mathrm{~g}_{\mathrm{i}} \alpha_1 \mathrm{e}^{-\mathrm{w}_{\mathrm{i}} / \mathrm{kT}}=\left[\Sigma_\mathrm{i} \mathrm{w}_{\mathrm{i}} \mathrm{~g}_{\mathrm{i}} \mathrm{Ne}^{-\mathrm{w}_{\mathrm{i}} / \mathrm{kT}}\right] / \mathrm{U}(\mathrm{~T})=\mathrm{NkT}[\mathrm{~d} \ln \mathrm{U} / \mathrm{d} \ln \mathrm{~T}]\label{1.1.18}\]

For a free particle like that found in a monatomic gas, the partition function¹ is (see also section 11.1b) \[\mathrm{U}(\mathrm{~T})=\frac{(2 \pi \mathrm{mkT})^{3 / 2}}{\mathrm{~h}^3} \mathrm{~V},\label{1.1.19}\]

where V is the specific volume of the gas, m is the mass of the particle, and T is the kinetic temperature. Replacing dlnU/dlnT in equation \ref{1.1.18} by its value obtained from equation \ref{1.1.19}, we get the familiar relation \[\mathrm{E}=\frac{3}{2} \mathrm{NkT},\label{1.1.20}\]

which is only correct if T is the kinetic temperature. Thus we arrive at a self-consistent solution if the parameter T is to be identified with the kinetic temperature.

The situation for a photon gas in the presence of material matter is somewhat simpler because the matter acts as a source and sinks for photons. Now we can no longer apply the constraint \(\mathrm{dN=0}\). This is equivalent to adding \(\ln\alpha_2=0\) (i.e., \(\alpha_2=1\)) to the equations of condition. If we let \(\mathrm{\beta_2=1/(kT)}\) as we did with the Maxwell-Boltzmann statistics, then the appropriate solution to the Bose-Einstein formula [equation \ref{1.1.15}] becomes \[\frac{\mathrm{N}_{\mathrm{i}}}{\mathrm{n}}=\frac{1}{\mathrm{e}^{\mathrm{hv} /(\mathrm{kT})}-1},\label{1.1.21}\]

where the photon energy wi has been replaced by \(\mathrm{h}\nu\). Since two photons in a volume h³ can be distinguished by their state of polarization, the number of phase space compartments is \[\mathrm{n=\left(2 / h^3\right) d x_1 d x_2 d x_3 d p_1 d p_2 d p_3}\label{1.1.22}\]

We can replace the rectangular form of the momentum volume dp₁dp₂dp₃, by its spherical counterpart \(\mathrm{4\pi p^2dp}\) and remembering that the momentum of a photon is \(\mathrm{h\nu/c}\), we get \[\frac{\mathrm{dN}}{\mathrm{~V}}=\frac{8 \pi \nu^2}{\mathrm{c}^3} \frac{1}{\mathrm{e}^{\mathrm{h\nu} /(\mathrm{kT})}-1} \mathrm{~d} \nu.\label{1.1.23}\]

Here we have replaced \(\mathrm{N_i}\) with \(\mathrm{dN}\). This assumes that the number of particles in any phase space volume is small compared to the total number of particles. Since the energy per unit volume \(\mathrm{dE_v}\) is just \(\mathrm{h\nu~dN/V}\), we get the relation known as Planck's law or sometimes as the blackbody law: \[\mathrm{dE}_{\mathrm{v}}=\frac{8 \pi \mathrm{h}}{\mathrm{c}^3} \frac{\nu^3}{\mathrm{e}^{\mathrm{h\nu} /(\mathrm{kT})}-1} \mathrm{~d} \nu \equiv \frac{4 \pi}{\mathrm{c}} \mathrm{~B}_\nu(\mathrm{~T})\label{1.1.24}\]

The parameter \(\mathrm{B_\nu(T)}\) is known as the Planck function. This, then, is the distribution law for photons which are in strict thermodynamic equilibrium. If we were to consider the Bose-Einstein result for particles and let the number of Heisenberg compartments be much larger than the number of particles in any volume, we would recover the result for Maxwell-Boltzmann statistics. This is further justification for using the Maxwell-Boltzmann result for ordinary gases.

f. Thermodynamic Equilibrium - Strict and Local

Let us now consider a two-component gas made up of material particles and photons. In stars, as throughout the universe, photons outnumber material particles by a large margin and continually undergo interactions with matter. Indeed, it is the interplay between the photon gas and the matter which is the primary subject of this book. If both components of the gas are in statistical equilibrium, then we should expect the distribution of the photons to be given by Planck's law and the distribution of particle energies to be given by the Maxwell-Boltzmann statistics. In some cases, when the density of matter becomes very high and the various cells of phase space become filled, it may be necessary to use Fermi-Dirac statistics to describe some aspects of the matter. When both the photon and the material matter components of the gas are in statistical equilibrium with each other, we say that the gas is in strict thermodynamic equilibrium. If, for what- ever reason, the photons depart from their statistical equilibrium (i.e., from Planck's law), but the material matter continues to follow Maxwell-Boltzmann Statistics (i.e., to behave as if it were still in thermodynamic equilibrium), we say that the gas (material component) is in local thermodynamic equilibrium (LTE).