# What is the argument for detailed balance in chemistry?

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Detailed balance is an important property of many classes of physical systems. It can be written as $$\frac{p_{i \to j}}{p_{j \to i}} = e^{\frac{\Delta G}{k_B T}},\tag{1}$$ where $i$ and $j$ represent microscopic states of the entire system; $p$ represents the probability for the system to transition from one state to another in a particular finite time period; and $\Delta G$ represents the difference in free energy between the two states. (Whether to use the Gibbs or Helmholtz free energy or some other potential depends on the ensemble.)

Many systems obey detailed balance, but not all do. The Earth cannot fluctuate backwards in its orbit around the Sun, because this would violate conservation of angular momentum. An RLC circuit does not obey detailed balance because the fluctuations have a distinctive ringing time. For these systems the correct formula is $$\frac{p_{i \to j}}{p_{j' \to i'}} = e^{\frac{\Delta G}{k_B T}},\tag{2}$$ where $i'$ and $j'$ represent states identical to $i$ and $j$, except that all velocities and magnetic fields have been reversed. (In quantum mechanics, they represent something like the complex conjugates of states $i$ and $j$.)

Both of these formulae guarantee that the system will obey the second law (on average), but $(1)$ is substantially stronger, because it guarantees that not only will the system tend toward equilibrium in the thermodynamic limit, but that it will not oscillate as it approaches the equilibrium. (It can still oscillate far away from equilibrium, however.)

My question is about chemical kinetics. Here we universally assume equation $(1)$ and not $(2)$. This puts strong constraints on the reaction rates, and leads to the well-known result that near-equilibrium oscillations are impossible in chemical systems. I've recently been discussing this topic with a very experienced researcher in nonlinear dynamics, and I found myself unable to convince him that $(1)$ rather than $(2)$ is a good assumption in the case of chemistry.

So I thought I'd ask here and see if anyone can help me out: what is the argument that leads us to assume the 'strong' form of detailed balance in chemical systems, rather than the weaker form in equation $(2)$?

This post imported from StackExchange Physics at 2016-02-10 14:15 (UTC), posted by SE-user Nathaniel

1) is equivalent to 2) if states are all time reversal invariant (so that i.e., $i'=i$). Correct? So, perhaps your question could be rephrased as "why can one get away with assuming states are time reversal invariant in context of chemistry?"
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