Appendix A

Boltzmann's H Theorem

A.1. Maxwell-Boltzmann Statistics

In this appendix, we discuss the theorem, which shows that any closed system obeying Boltzmann's equation will evolve with time, in the absence of external influences, to an equilibrium state in which Boltzmann's function is minimal (entropy is maximal). If interactions in the system violate baryon number, then the final equilibrium state can contain no excess of baryons over antibaryons. In an expanding universe, however, the theorem is modified, and the baryon asymmetries discussed above may be produced and remain while expansion persists.

To investigate the approach to equilibrium, we consider the development of the quantity

where the index labels both the momentum state and the particle type (in the notation used above where runs only over particle types). [Note that (A.1) differs slightly from usual definitions of in which the term is absent. When no particles are created or destroyed, gives the total number of particles and is constant and irrelevant. The definition (A.1) is more convenient when particle creation is included.] changes with time according to

is stationary when which may occur either if no collisions take place, or by detailed balancing in equilibrium. In the presence of interactions, the are taken to evolve with time according to the Boltzmann equation (the sum over states accounts for usual phase-space integration)

(This may represent a sum of terms with different numbers and of initial or final particles in collisions.) Then Boltzmann's theorem states that any system (in which the momenta of particles are initially uncorrelated) will evolve towards equilibrium so that

which is a microscopic statement of the second law of thermodynamics. Adding the forms obtained by permuting the dummy indices as, (A.4) may be written (dropping irrelevant constant factors) as

where

and the sums on run over all and

We first discuss the proof of (A.4) when all interactions respect time-reversal invariance, so that all pairs of --conjugated matrix elements in (A.5) are equal. Then

where the second form is obtained by adding forms in which the and are permuted in all possible ways. The terms in the sum (A.6) now each have the form

(where equality holds only when ), and are thus separately not positive, so that . To achieve the equilibrium state , each term must vanish, so that

for all sets of and corresponding to initial and final states of possible collisions as represented in (A.3) (27). If particles carry no absolutely conserved internal quantum numbers, then the in (A.8) may depend only on the energies and must follow the Maxwell-Boltzmann distribution

where, as usual, may be considered as a Lagrange multiplier enforcing energy conservation. The phase-space distributions for particles carrying absolutely conserved quantum numbers may differ from (A.9) by factors where gives the value of the quantum number carried by species while is a chemical potential which parametrizes the total concentration of the quantum number. (For non-abelian quantum numbers, the relevant factors are the corresponding group elements.) Nevertheless, in the absence of absolutely conserved quantum numbers, the equilibrium distributions must follow (A.9), and in particular if the particles and antiparticles of a species are not absolutely conserved, their phase-space distributions must be identical in equilibrium; no particle-antiparticle asymmetries may exist.

This result also holds when the fundamental interactions exhibit violation; it is based solely on unitarity [25]. The unitarity constraint (2.1.3) implies

Permuting dummy indices in the second term of (A.5a), but without assuming equality of the -conjugated matrix elements yields

Inserting the result (A.10) then gives

since

Once again, distributions must tend to the equilibrium form (A.8). That this conclusion is independent of -violation in the collisions was to be expected, since it also holds for systems in static external (e.g., magnetic) fields, and when internal degrees of freedom are excited in molecules by collisions. Of course, as usual, the validity of (A.4) relies on the assumption of molecular chaos, according to which the momenta of particles are uncorrelated before each collision (but clearly not after). This is presumably true only at the separated maxima of where evolution in or would decrease Strictly, the time dependence of a single-particle phase-space distribution should depend on the joint two-particle distribution (as in the BBGKY hierarchy); this may not always factorize as required for molecular chaos. Nevertheless, eq. (A.3) and hence the H theorem (A.4) is presumably adequate in an average sense, except when collision rates become very large, or long-range forces are present, as discussed in sect. 4.

Note that the result relies on the form (A.3) of the Boltzmann equation. In an expanding universe, the extra term of eq. (2.1.15) must be added to account for the expansion. This term typically gives a positive contribution to , which may overwhelm the negative contributions from collisions and give , so that baryon asymmetries can be generated.

A.2. Quantum Statistics

In eqs. (A.1) and (A.3) we have assumed that all particles are classically distinguishable, and therefore obey Maxwell-Boltzmann statistics. As mentioned in subsect. 2.4, when the particles are identical, (A.3) becomes (known as the Uehling-Uhlenbeck equation)

where if the particle is a boson and if it is a fermion. We write the factor and denote For final fermions, the extra factor implements the Pauli exclusion principle which forbids any fermion from being emitted into a cell in phase space which is already occupied (in a finite quantization volume, this is achieved by a factor for each finite cell, which becomes in the continuum limit). For final bosons, the two terms in each factor represent spontaneous and stimulated emission respectively. The presence of this correction may formally be considered to result from the factor when a creation operator acts on an N boson state which follows from the commutation relations of the boson field operators. A slightly more direct derivation is based on the fact that the total amplitude for any process is the sum of the amplitudes associated with each possible permutation of identical bosons. (For fermions, the exchange of each pair introduces a minus sign.) Consider a state consisting of indistinguishable non-interacting bosons. For combinational purposes, assign each boson to one of separate ``substates'' of The number of possible assignments is First, let the bosons propagate undisturbed to a final state which is again divided into ``substates'', labelled in ways. The amplitude for the propagation of each boson is exp The total amplitude for all bosons to propagate them to is the sum of the amplitudes corresponding to each possible assignment of substates for the bosons in the initial and final state. Dividing by the number of possible relabellings of the substates in and gives for the total propagation probability Now, however, consider the introduction of an extra boson during the propagation, with amplitude The final state is now divided into substates, which may be labelled in (N+1)! ways. The total probability for the propagation with the addition of one extra boson is then it is enhanced by a factor relative to the probability for the introduction of the boson into an initially empty state. Taking the continuum limit then gives the correction factor.

A suitable quantity by which to measure the approach to thermal equilibrium of a system of identical particles is

This evolves with time according to

which may be written, using the modified Boltzmann equation (A.14), as

We first assume invariance, so that In this case, eq. (A.17) becomes [analogous to eq. (A.6)]

where the final inequality follows from (A.7). Hence the system will evolve on average to an equilibrium state in which is minimal, and

for all set of and corresponding to initial and final states of possible scattering processes. In analogy with (A.9), energy conservation then implies

where represents possible absolutely conserved quantum numbers. Solving (A.20) for gives [cf., eq. (2.4.11)]

which is the usual equilibrium Bose-Einstein or Fermi-Dirac distribution ( for bosons, for fermions).

When invariance is violated, one must use unitarity to prove the theorem In the presence of indistinguishable particles the unitarity relation (2.1.3) is modified, and becomes [26]

where is the product of quantum statistics correction factors defined above. The -invariance constraint (2.1.1) yields the results [analogous to eqs (2.1.4) and (2-1-5)]

To show that no asymmetry between particles and antiparticles may be generated in thermal equilibrium, we must prove that

where is the product of the relevant incoming particle Bose-Einstein or Fermi-Dirac equilibrium phase-space densities. According to eq. (A.l9), in thermal equilibrium

for all sets of particles (This is the generalization of the result for distinguishable Maxwell-Boltzmann particles used in subsect. 2.1 that all states of a given energy are equally populated.) Inserting the relation (A.25) in the unitarity equation (A.23) gives directly the desired result (A.24).

Using the unitarity result (A.22), the proof of the theorem for indistinguishable particles undergoing -violating interactions proceeds quite analogously to the distinguishable particle case treated above. On exchanging dummy indices, eq. (A.17) may be written [cf., (A.11)]

The unitarity relation (A.22) implies [cf., (A.10)]

and inserting this in eq. (A.26) gives

by eq. (A.13). Thus the validity of the theorem is unaffected by indistinguishable particle effects, even when violation is present (28).

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