# (p.304) B Microscopic reversibility and detailed balance

# (p.304) B Microscopic reversibility and detailed balance

Both Newton’s equation of motion for a classical system and Schrödinger’s equation for a quantum system are unchanged by *time reversal*, i.e., when the sign of the time is changed. Due to this symmetry under time reversal, the transition probability for a forward and the reverse reaction is the same, and consequently a definite relationship exists between the cross-sections for forward and reverse reactions. This relationship, based on the reversibility of the equations of motion, is known as the principle of *microscopic reversibility*, sometimes also referred to as the *reciprocity theorem*. The statistical relationship between rate constants for forward and reverse reactions at equilibrium is known as the principle of *detailed balance*, and we will show that this principle is a consequence of microscopic reversibility. These relations are very useful for obtaining information about reverse reactions once the forward rate constants or cross-sections are known. Let us begin with a discussion of microscopic reversibility.

# B.1 Microscopic reversibility

## B.1.1 Transition probability

The trajectory of a classical particle may be found by integrating Newton’s equation of motion

*t*

_{0}to time

*t*

_{1}. Let us then introduce the substitution

As *t* varies from *t* _{0} to *t* _{1}, *τ* is seen from Eq. (B.2) to vary from *t* _{1} to *t* _{0}, so the substitution is equivalent to a time reversal. Since the equations of motion are identical, the system will follow exactly the same trajectory; the only difference will be that in one
(p.305)
case it is followed forward in time from *t* _{0} to *t* _{1}, and in the other case backward in time from *t* _{1} to *t* _{0}. In other words, it is not possible from the equation of motion to decide whether it describes a forward or a reverse propagation in time. This is summarized by saying that the classical equation of motion has time-reversal symmetry. As a consequence, the probability for a forward scattering process must equal the probability for the reverse scattering process, where all velocities *dr/dt* and time have changed sign (see Eq. (B.3)).

The Schrödinger equation for a quantum system also has time-reversal symmetry.

The solution to the time-dependent Schrödinger equation

*t*

_{0}to time

*t*

_{1}. From this definition it is clear that

*t*

_{1}) at time

*t*

_{1}backward in time to time

*t*

_{0}; then we have

(p.306)
Then it is remembered that the wave function itself does not have a physical meaning. So let us, for example, determine the transition probability from say state *|k*(*t* _{0})> at time *t* _{0} to state *|m*(*t* _{1})> at time *t* _{1}. Then we have

This transition probability is now compared to the transition probability for the reverse process, where we consider the transition from *|m*(*t*1)> to *|k*(*t* _{0})>. We find

*|k*(

*t*

_{0}) at time

*t*

_{0}to state

*|m*(

*t*

_{1})> at time

*t*1 is equal to the transition probability for the reverse process, that is, from

*|m*(

*t*

_{1})> to

*|k*(

*t0*)>. This is a manifestation of the time-reversal symmetry of the equation of motion.

The transition probabilities per unit time and per scattering center in Eqs (B.11) and (B.12) are for a collision process, often written in a more explicit form to emphasize the particular transition considered:

*υi*and

*υf*are the initial and final relative speeds, respectively, of the colliding particles,

*ij*are the initial internal states of the two colliding molecules,

*ml*are the final internal states of the product molecules, and Ω is the solid angle into which the particle of interest is scattered. This angle, as measured from the direction of the initial relative velocities, respectively,

*υi*and

*υf*, is the same for both the forward and the reverse scattering process. Finally, let us emphasize that the transition probability

*P*refers to a well-specified initial and final state and that it gives us the probability per unit time and per scattering center for the scattering process indicated.

## B.1.2 Cross-sections

From the fundamental relation in Eqs (B.12) or (B.13), we are now going to derive a relation between the differential cross-section for the forward and the reverse scattering process.

As defined previously in Eq. (2.7), differential cross-sections are defined as the flux of particles scattered into a range of solid angle *d*Ω around the solid angle Ω per unit initial flux and per scattering center. The state-to-state transition probabilities are already per unit scattering center and, since we do not scatter into one single final state but a range of final states as given by *d*Ω, we need to determine how many final states are consistent with that uncertainty in the final solid angle. It is then assumed that the transition probability to that small range of final states is the
(p.307)
same, irrespective of the state; then the product of the transition probability and the number of final states will be the total transition probability. The scattering into the solid angle *d*Ω around Ω with a relative final velocity *vf* is equivalent to scattering into a final state with momentum *pf, pf* + *dpf*. The number of final states in this range may be determined in the following way. We consider a free particle in one dimension where the momentum eigenfunction is given by ψ_{p}(*x*) *= c*(*p*) exp (*ipx*/ħ). In a (macroscopic) box of length *L*, periodic boundary conditions on the wave function ψ_{p}(*x* + *L*) *= ψ* _{p}(*x*) imply that exp(*ipL/ħ*) = 1 or *p/ħ = 2nπ/L*, where *n* is an integer. That is, the spacing between momentum values is *h/L*, which implies that the density is *L/h*. This argument can be extended to three dimensions, and it can be shown that the density, ρ(*p*), is (*L/h*)^{3} and

*d*Ω.

If the density of particles in the incident beam of A(*i*) molecules with relative velocity *vi* with respect to the B molecules is *n*a(*i*), then the magnitude of the incoming flux is

*d*Ω per unit time is

From the definition of the differential cross-section, we obtain^{1}

The factor δ(*Ef* – *Ei*)*dE* _{f} on the left-hand side of the equation has been introduced to emphasize energy conservation for the process indicated in the argument to σ*
R
*. The transition probability *P* on the right-hand side is determined directly from the equation of motion, so it is therefore not necessary to include the delta function on the right-hand side.

(p.308) The cross-section for the reverse reaction may be written analogously:

When we use Eq. (B.13) and cancel common factors, we find from Eqs (B.17) and (B.18) that

From the energy balance

_{i}is the reduced mass for the relative motion of the reactants and

*μf*for the products, we find the following relation between

*pi*and

*pf*for fixed internal states:

The desired relation between cross-sections for the forward and the reverse reactions is thus found to be

This relation expresses the principle of *microscopic reversibility* for the differential cross-sections.

Alternatively, using Eq. (4.173) and the symmetry of the *S*-matrix element, using Eq. (B.12), we immediately obtain

*n*and

*m*refer to the quantum state of the reactant and product and, on the right-hand side, the reverse reaction is considered, i.e., the initial and final quantum states are

*m*and

*n*, respectively.

*E*is the total energy.

Equation (B.23) is the desired relation (at the level of integrated cross-sections), and it is equivalent to Eq. (B.22), since *p = ħk*.

If the system has an internal angular momentum (associated with rotational states of molecules) there will, in the absence of an external field, be degeneracies in the system that will be practical to display explicitly in the expression for microscopic reversibility in Eq. (B.22). For systems with angular momenta, time reversal of the quantum equations of motion reverses the signs of both the momenta and their projections on a given direction, just like in a classical system. To express this explicitly Eq. (B.13) is written as

The ‘starred’ quantum states differ from the ‘unstarred’ ones by the sign of the projection of their angular momenta, so if the projection of the angular momentum of
(p.309)
state *i* on the *z*-axis is *m* _{i} then the projection of *i** on the *z*-axis is -*m* _{i}, etc. By an analysis similar to that used to derive Eq. (B.22), we obtain

Experimental and calculated cross-sections are usually averaged over the degenerate quantum states associated with the angular momenta when there is no external field. A more useful statement of microscopic reversibility can therefore be obtained by working with average cross-sections ${\overline{\sigma}}_{R}$, which refer to transitions between sets of degenerate levels in the initial and final states. Let *Ji* denote the total angular momentum of state *i* and *mi* its projection on the *z*-axis. Then by summing Eq. (B.24) over all degenerate states of the reactants and products one obtains

*m*

_{k}are from

*– J*

_{k}to

*J*

_{k}. Note that we have dropped the star on the quantum states because the range of the

*m*

_{k}extends over both positive and negative values.

Since the *pi* and *pf* are constant when summed over degenerate states, we may now use Eqs (B.17) and (B.26) to introduce a cross-section, σ*
R
*(*s*), where we have summed over all degenerate states:

So the principle of microscopic reversibility may be rewritten as

The σ_{R}(*s*) refers to the sum of cross-sections for the transition from the degenerate set of states $\overline{i},\overline{j}$ to $\overline{l},\overline{m}$ defined for unit flux from each initial state (see Eqs (B.15)–(B.17)). We may instead operate with an average cross-section ${\overline{\sigma}}_{R}$ based on an initial flux for the total set of degenerate states:

*gi*is the number of degenerate states,

*gi =*2

*Ji*+ 1. Substitution of Eq. (B.29) into Eq. (B.28) leads to yet another expression of microscopic reversibility, based here on the average cross-sections for the degenerate states:

It is noted that this relation only holds in the absence of an external field, when the sets of states are degenerate. Other degeneracies may also occur in the internal molecular states. In this case, the *g* factors should include those degeneracies.

# (p.310) B.2 Detailed balance

Detailed balance provides a relation between the macroscopic rate constants *k* _{f} and *k* _{r} for the forward and reverse reactions, respectively. On a macroscopic level, the relation is derived by equating the rates of the forward and reverse reactions at equilibrium. Here it will be shown that the principle of detailed balance can be readily obtained as a direct consequence of the microscopic reversibility of the fundamental equations of motion.

On a macroscopic level in the absence of an external field, we cannot distinguish between sets of degenerate states in the reactants and products; so the most detailed relation between macroscopic rate constants and microscopic cross-sections will be one where we have summed over all degenerate states as in Eq. (B.27). The macroscopic rate constant for a particular transition between degenerate states is then given by

*υa*and

*υb*(or momenta distributions

*p*A and

*p*B) are given by the equilibrium Maxwell–Boltzmann distribution at temperature

*T*. We obtain, as in Section 2.2 (see Eq. (2.29)), the following result:

_{i}is the reduced mass and

*pi = μ*

_{i}υ

_{i}is the momentum associated with the relative translation of the reactants. Similarly, the expression for the rate constant for the reverse reaction is

*μf*refers to the reduced mass and

*pf = μfυf*is the relative momentum of the products. Note that ${\sigma}_{R(s)}(\overline{m},\overline{l}\text{|}\overline{ij};{\upsilon}_{i})$ is the integrated cross-section (see Eq. (2.12)) where${\sigma}_{R(s)}(\overline{m},\overline{l}\text{|}\overline{ij};{\upsilon}_{i},\Omega )$ is integrated over all space angles.

To relate the rate constants in Eqs (B.32) and (B.33), we substitute Eq. (B.28) and use the conservation of energy to relate the differentials and limits of integration. Conservation of energy requires

*E*

_{0,p}is the zero-point energy of the products and

*E*

_{0,r}is the zero-point energy of the reactants. We then have for the reverse rate constant, using Eqs (B.21), (B.28), and

*dp = 4πp*

^{2}

*dp*, (p.311)

From Eq. (B.32), we then obtain the following relation between the forward and reverse rate constants:

It is noted that the right-hand side is the ratio of the translational partition functions of products and reactants times the Boltzmann factor for the internal energy change. In the derivation of this expression we have only used that the translational degrees of freedom have been equilibrated at *T* through the use of the Maxwell–Boltzmann velocity distribution. No assumption about the internal degrees of freedom has been used, so they may or may not be equilibrated at the temperature *T*. The quantity $K(\overline{m}\overline{l},\overline{i}\overline{j})$ may therefore be considered as a partial equilibrium constant for reactions in which the reactants and products are in translational but not necessarily internal equilibrium.

To obtain the statement of detailed balance for complete equilibrium, with both translational and internal degrees of freedom in thermal equilibrium, we must sum over the rate constants in Eqs (B.32) and (B.33), weighting each by its equilibrium Boltzmann distribution; that is (as in Eq. (2.18)),

Here *kf* is the rate constant for the forward reaction and *k* _{r} for the reverse reaction. ${p}_{\text{A(}\overline{i})}$ is the probability (mole fraction) of A in any of the set of states $\overline{i}$, which is given by statistical mechanics according to Eq. (A.4), and similarly for the other constituents. If we substitute these relations into Eqs (B.38) and (B.39), we obtain

Note that the degeneracies *gi* do not appear explicitly in the equations, because we use ‘barred’ quantities as indices in the sums. They imply a sum over degenerate states; had we used ‘unbarred’ indices, then the degeneracy factor *gi* should be included
(p.312)
explicitly. From Eqs (B.40) and (B.41) and using Eq. (B.37), we find the following relation between the rate constants:

*K*(

*T*) is the equilibrium constant for the reaction. This is the usual statistical mechanical expression for the equilibrium constant in terms of the molecular partition functions.

In summary, we have seen that the application of microscopic reversibility for the forward and reverse cross-sections and the use of complete equilibrium distributions for the evaluation of the statistical rate constant lead to the usual results known from equilibrium statistical mechanics. If one knows the cross-section for a forward reaction, one can always determine the inverse cross-section through the principle of microscopic reversibility. Also, if one knows the cross-section for the forward reaction, and in addition one knows that the translational and internal distribution functions of reactants and products have reached equilibrium, one can calculate the rate constant. Detailed balance then permits the calculation of the reverse rate constant.

# Further reading/references

Bibliography references:

J.C. Light, J. Ross and K.E. Shuler, in *Kinetic processes in gases and plasmas* (Academic Press, 1969) Chapter 8