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Mathematically, the definition of flux varies. The where the sets of reactants written sinistralat-
alternative taken here is simply to write the erally (s) and dextralaterally (d) in the formal
ordinary differential velocity equations as partial reaction equation are X s,i and X , respectively;
d,i
differential equations, so that ˙x becomes ∂x /∂t X s,i and X d,i are the number of reactants in each
i
i
or x . An alternative is to define flux relative set (see cardinality); n i,{s|d},j is the stoichiometry
it
to some other compound, which allows one to of reactant x in X s,i or X d,i ( | is logical or); and
j
incorporate the dynamic equilibrium condition k and k −i are the forward and reverse reaction
i
directly; the form will vary, much as it does in rate constants, respectively. A species appearing
a physical context. Still another is to use flux in catalytically in a reaction equation is included
other functions without defining what it is math- on both sides. R is then described by the set of
ematically. In general, the word is best used only formal reaction equations, one for each r ∈ R.
i
when it is mathematically explicit. See also dextralateral, direction, dynamic equi-
librium, microscopic reversibility, product, rate
Fock space In quantum mechanics, the full constant, reversibility, sinistralateral, stoichiom-
Hilbert space of states. etry, and substrate.
n
Focker-Plank equation Let U ⊂ R be Fourier Jean Babtiste Joseph, Baron de
open and u : U × R → R. The Focker-Plank Fourier (1768–1830). French analyst and mathe-
equation for u is matical physicist.

n n
ij i Fourier coefficients Let {x } be an
i i∈I
u − (a u) x i x j − (b u) = 0. orthonormal basis in a Hilbert space (H,<,>).
t
x i
i,j=1 i=1
Then every x ∈ H can be written as a Fourier
series
form See differential form.
x = <x, x >x .
i
i
formal reaction equation The representa- i∈I
tion of a chemical or biochemical reaction, the The coefficients <x, x > are called the Fourier
i
participating molecular species, and the chem- coefficients of x with respect to the basis {x }.
i
ical, kinetic, and thermodynamic parameters per-
Fourier integral operator A Fourier inte-
taining to those species in that reaction and to the
gral operator A of order k on a compact manifold
reaction itself.
∞
M is locally of the form, for any u ∈ C (M),
The biochemical denotation of formal reac- c
tion equations, for example −n iϕ(x,y,ξ)
Au(x) = (2π) e a(x, y, ξ)
x + x + .. . x m + x m+1 + ...
2
1
× u(y)dydξ,
for a set of reactants, x ∈ X, carries with it many where a(x, y, ξ) is a symbol of order k, and
j
less obvious layers of convention, meaning, and
ϕ(x, y, ξ) is a nondegenerate phase function.
information. To make these layers more explicit,
This defines a bounded linear operator between
the formal reaction equation can be rewritten s s−k
the Sobolev spaces A : H (M) → H c (M).
c
more systematically as follows. Let the set R
be a system of reactions r , 1 ≤ i ≤ N among a
i Fourier series See Fourier coefficients.
set of molecular species X = {x }, 1 ≤ j ≤ M
j
1
n
(called reactants); i, j, N, and M are any posi- Fourier transform Let f ∈ L (R ). The
tive integers. Each reactant x may participate in Fourier transform of f , denoted by f is the func-
ˆ
j
n
more than one reaction, and every reaction has at tion on R defined by
least two reactants. Each reaction r is described
i
by a formal reaction equation of the form f(k) = e −2πik·x f(x) dx.
ˆ
R n
X s,i X d,i

ˆ
x
n i,s,j j k i k −i n i,d,j j The Fourier transform f → f is norm preserv-
x ⇔
n
2
n
2
ing from L (R ) to L (R ).
j=1 j=1
© 2003 by CRC Press LLC