Abstract
Early in the 20th century, there came into being a strongly held belief, called sometimes the principle of microscopic reversibility and sometimes the principle of detailed balance [31, 57, 85, 124, 165]. In rough terms, the principle asserts that when, in a naturally occurring physical system, a collection of distinct and varied reversible molecular processes gives rise to the system’s dynamics, a state of equilibrium can result only when, for each such process, the occurrence rate of the process and the occurrence rate of its reverse are identical.
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Notes
- 1.
This follows from a standard result in graph theory, relating the number of edges in a forest to the number of vertices and the number of connected components.
- 2.
In this section we are temporarily viewing a reversible reaction network’s standard reaction diagram as a graph having the complexes as its vertices and reaction pairs \(\rightleftarrows \) as its undirected edges.
- 3.
Recall that here we are regarding the standard reaction diagram of \({\{\mathcal {S},\mathcal {C},\mathcal {R}\}}\) to be a graph having the complexes as its vertices and the reaction pairs \(\rightleftarrows \) as its edges. Thus, a reaction pair \(y \rightleftarrows y'\) is not a cycle.
- 4.
The assignment to a cycle of clockwise and oppositely oriented counterclockwise directions is arbitrary and inconsequential.
- 5.
By a solution we mean of course a set of numbers \(\{\alpha _{{y \to y'}}\}_{{y \to y'} \in {\mathcal {F}}^{\to }}\) (or, equivalently, a vector \(\alpha \in \mathbb {R}^{{\mathcal {F}}^{\to }}\)) that satisfies (14.41). We say that a set of solutions is independent if the set is independent, viewed as vectors in \(\mathbb {R}^{{\mathcal {F}}^{\to }}\). That there will be an independent set of δ solutions (but not more) is a consequence of elementary considerations in linear algebra.
- 6.
As I indicated in [75], I think I learned the less structure-related conditions formulated in this remark from F. Horn.
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Feinberg, M. (2019). Detailed Balancing. In: Foundations of Chemical Reaction Network Theory. Applied Mathematical Sciences, vol 202. Springer, Cham. https://doi.org/10.1007/978-3-030-03858-8_14
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