A lower bound on the rate coefficient

To develop a lower bound on the steady state, Reck and Prager [507] again considered the variational integral of eqn. (265). In this case, however, let the approximate solution j/ satisfy the diffusion equation (263) rather than the equation defining the macroscopic density M as previously done. Multiply eqn. (263) by j5(r), a Lagrangian undetermined multiplier and add it to the variational integral to give 

Minimising this integral with respect to a variation 6 p gives 

Because 8 is arbitrary and small and independent of Gauss s theorem for the volume integrals may be used and 

If the approximate density jj(r) had been chosen correctly to be m(r), then the variational integral, would be minimised at FM as before. By forcing to satisfy eqn. (263), the value of F has been effectively determined and only M remains to be evaluated before the rate coefficient of eqn. (164) can be defined, Since 

This is the formal lower bound on the rate coefficient, by contrast to the 

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