6
B. Sutcliffe and R.G. Woolley
state’. The pre-exponential factor M is obtained formally from statistical mechanics. Marcelin gave several derivations of this result using both thermodynamic arguments and also the statistical mechanics he had learnt from Gibbs’s famous memoir [9]. It is perhaps worth remarking that Gibbs saw statistical mechanics as the
completion of Newtonian mechanics through its extension to conservative systems
with an arbitrarily large, though finite, number of degrees of freedom. The laws
of thermodynamics could easily be obtained from the principles of statistical mechanics, of which they were the incomplete expression, but Gibbs did not require
thermodynamic systems to be made up of molecules; he explicitly did not wish his
account of rational mechanics to be based on hypotheses concerning the constitution
of matter, which at the time were still controversial [10].
From our point of view the most interesting aspect of Marcelin’s account is the
suggestion that molecules can have more degrees of freedom than those of simple
point material particles. In this perspective, a molecule can be assigned a set of
Lagrangian coordinates q = q 1 , q 2 , . . . , q n , and their corresponding canonical momenta p = p 1 , p 2 , . . . , p n . Then the instantaneous state of the molecule is associated with a ‘representative’ point in the canonical phase-space P of dimension 2n,
and so “as the position, speed or structure of the molecule changes, its representative
point traces a trajectory in the 2n-dimensional phase-space” [5].
In his phase-space representation of a chemical reaction the transformation of
reactant molecules into product molecules was viewed in terms of the passage of a
set of trajectories associated with the ‘active’ molecules through a ‘critical surface’
S in P that divides P into two parts, one part being associated with the reactants,
the other with the products. Such a [hyper]surface is defined by a relation
S(q, p) = 0.
According to Marcelin, for passage through this surface it is required 2 [5]
[une molécule] il faudra [. . . ] qu’elle atteigne une certaine région de l’éspace sous une
obliquité convenable, que sa vitesse dépasse une certain limite, que sa structure interne
corresponde à une configuration instable, etc.; . . .
In modern notation, the volume of a cell in the 2n-dimensional phase-space is
dd = dqdp.
The number of points in dd is given by the Gibbs distribution function f
dν = f (q, p, t)dd.
(1.3)
Marcelin chose the distribution function for the active molecules as
f (q, p, t) = e
−G #
+ /RT e
−H (q,p)/k B T
(1.4)
2 That a molecule must reach a certain region of space at a suitable angle, that its speed must exceed
a certain limit, that its internal structure must correspond to an unstable configuration etc.; . . .
B. Sutcliffe and R.G. Woolley
state’. The pre-exponential factor M is obtained formally from statistical mechanics. Marcelin gave several derivations of this result using both thermodynamic arguments and also the statistical mechanics he had learnt from Gibbs’s famous memoir [9]. It is perhaps worth remarking that Gibbs saw statistical mechanics as the
completion of Newtonian mechanics through its extension to conservative systems
with an arbitrarily large, though finite, number of degrees of freedom. The laws
of thermodynamics could easily be obtained from the principles of statistical mechanics, of which they were the incomplete expression, but Gibbs did not require
thermodynamic systems to be made up of molecules; he explicitly did not wish his
account of rational mechanics to be based on hypotheses concerning the constitution
of matter, which at the time were still controversial [10].
From our point of view the most interesting aspect of Marcelin’s account is the
suggestion that molecules can have more degrees of freedom than those of simple
point material particles. In this perspective, a molecule can be assigned a set of
Lagrangian coordinates q = q 1 , q 2 , . . . , q n , and their corresponding canonical momenta p = p 1 , p 2 , . . . , p n . Then the instantaneous state of the molecule is associated with a ‘representative’ point in the canonical phase-space P of dimension 2n,
and so “as the position, speed or structure of the molecule changes, its representative
point traces a trajectory in the 2n-dimensional phase-space” [5].
In his phase-space representation of a chemical reaction the transformation of
reactant molecules into product molecules was viewed in terms of the passage of a
set of trajectories associated with the ‘active’ molecules through a ‘critical surface’
S in P that divides P into two parts, one part being associated with the reactants,
the other with the products. Such a [hyper]surface is defined by a relation
S(q, p) = 0.
According to Marcelin, for passage through this surface it is required 2 [5]
[une molécule] il faudra [. . . ] qu’elle atteigne une certaine région de l’éspace sous une
obliquité convenable, que sa vitesse dépasse une certain limite, que sa structure interne
corresponde à une configuration instable, etc.; . . .
In modern notation, the volume of a cell in the 2n-dimensional phase-space is
dd = dqdp.
The number of points in dd is given by the Gibbs distribution function f
dν = f (q, p, t)dd.
(1.3)
Marcelin chose the distribution function for the active molecules as
f (q, p, t) = e
−G #
+ /RT e
−H (q,p)/k B T
(1.4)
2 That a molecule must reach a certain region of space at a suitable angle, that its speed must exceed
a certain limit, that its internal structure must correspond to an unstable configuration etc.; . . .