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.EQ
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.EN
SUMMARY OF CHANDLER'S REACTIVE FLUX METHOD
.sp2
I. ONSAGER'S HYPOTHESIS RELATES MACROSCOPIC AND MICROSCOPIC DYNAMICS:
.sp
.in +.5i
By Onsager's hypothesis (or by linear response theory) we may
identify the rate constant for reaction of the macroscopic system with the
rate constant for relaxation of the microscopic fluctuation of a relevant
variable.
.in -.5i
.sp 2
II. MACROSCOPIC PROPERTIES; PHENOMENOLOGICAL DESCRIPTION OF A REACTING
SYSTEM:
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A. Prepare the system for study:
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.in +.5i
Prepare the system slightly perturbed from its equilibrium,
such that $[B] sub {t=0} ~>~ [B] sub eq$ 
(and $[A] sub {t=0} ~<~ [A] sub eq$ by an exactly equal amount).
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.in -.5i
B. Measure the rate of reaction (rate of relaxation) as the system returns to
equilibrium:
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C. Analyze the data, to obtain the rate law:
.in +.5i
.DS I
.EQ (II.1)
{delta B(t)} over {delta B(0)} ~=~ e sup -kt
.EN
.DE
.in -.5i
.sp2
III. MICROSCOPIC DESCRIPTION; DECAY OF FLUCTUATIONS IN AN EQUILIBRIUM SYSTEM:
.sp
A. Background:
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B. Construct a function describing the \fIisomerization\fP state of the system,
and a function describing a \fIfluctuation\fP of the isomerization state
about the equilibrium state:
.in +.5i
.DS I
.EQ (III.2)
h(q(t)) ~=~ theta (q(t) - q*)
.EN
.DE
.DS I
.EQ (III.3)
delta h(t) ~=~ h(q(t)) - <h>
.EN
.DE
.in -.5i
.sp
C. Analyze the relaxation of the fluctuations:
.in +.5i
.DS I
.EQ (III.4-6)
C sub {delta h} mark ~=~  {< delta h delta h(t)>} over {< ( delta h) sup 2 >}
~=~ {<h delta h(t)>} over {<h delta h>}
~=~ e sup -kt
.EN                                                         
.DE
.in -.5i
.sp
D. Identify $k sub macroscopic ~=~ k sub microscopic$:
.sp
E. Definition of the reactive flux:
.in +.5i
.DS I
.EQ (III.7)
k(t) mark ~=~ -~d over dt  C sub {delta h} ~=~ k~e sup -kt
.EN
.sp
.EQ (III.9)
k(t) ~=~
1 over {x~(1~-~x)} P(q * ) < q dot~h (q(t)) > sub {q *}
.EN
.DE
$P(q*)$ and the conditional average can be evaluated by MD methods.
Eqn (III.9) is the central result of Chandler's reactive flux method.
.in -.5i
.SK
F. Analysis of the reactive flux for the limit $t -> 0 sup +$: 
.in -.5i
.DS I
.EQ (III.9)
k sup TST ~=~ k(0 sup + ) mark ~=~
1 over {x~(1~-~x)} P(q * ) < q dot~theta (q dot ) > sub {q *}
.EN
.DE
Eqn. (III.9) is the transition state theory approximation 
(free flight of the activated complex over the barrier top, 
no recrossing, equilibrium on the reactant side).
.DS I
.EQ (III.10)
k sub A->B sup TST mark ~=~ 
1 over 2 < | q dot | > sub {q *} {e sup {- DELTA W* /kT}} over {Q}
.EN
.DE
.DS I
.EQ (III.11)
k sub A->B  sup TST ~=~
{omega sub A} over {2 pi} {e sup {- DELTA W* /kT}}
.EN
.DE
.in -.5i
.sp
G. Analysis of the reactive flux for times $ t*$ in the range
$tau sub mol < t* << tau sub rxn = 1/k$:
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.in -.5i
Transient dynamics over a time $approx ~ O( tau sub mol )$ reflect
coupling of the reaction coordinate with other degrees of freedom, 
\ ...
.DS I
.EQ (III.12)
k(t*) ~=~ k~e sup -kt*
.EN
.DE
\ ... there is a plateau value of the reactive flux that is the
desired estimate of the macroscopic
reaction rate:
.DS I
.EQ (III.13)
k(t*) ~approx~ k
.EN
.DE
.sp
.DS I
.EQ (III.14)
k ~=~ kappa ~ k sup TST
.EN
.EQ
kappa ~=~ k over {k sup TST} ~=~ k(t*) over {k(0 sup + )}
.EN
.DE