Write down a recipe, then check it an hour later: if the proportions are off by even a gram, the flavor is gone. Chemical reactions in open systems behave much the same way: the slightest difference in initial concentrations grows over time into completely different states. That is chaos, and in a test tube it shows up as unpredictable fluctuations in color or temperature.
At the level of individual molecules, things are different: reactions proceed as random collisions, and in a small volume the behavior of the system is described by the chemical master equation, essentially a rule for how the probabilities of finding a given number of molecules change. This stochastic description remains valid even when the macroscopic equations exhibit chaos. But how do you connect molecular randomness with deterministic chaos in a large test tube?
The answer came through mutual information, a measure of how much the current state of the system remembers its initial state. If the memory fades quickly, the system mixes quickly. It turned out that the rate of this memory loss, computed from the master equation, converges in the large-volume limit to the Kolmogorov–Sinai entropy, a quantity that for the deterministic model equals the sum of the positive Lyapunov exponents. In other words, the rate at which nearby trajectories diverge in the macroscopic model is exactly equal to the rate at which information is erased in the stochastic model.
Numerical simulations for a system of three substances and seven reactions confirmed this: at volumes from $10^2$ to $10^6$ molecules, an intermediate regime appeared in which mutual information decayed almost linearly, and the slope of that line approached 0.065, exactly the Kolmogorov–Sinai entropy of the deterministic model. It is as if the noise from random reactions in a small test tube gradually quieted down, leaving the pure chaos of the large system.
The work provides the first explicit bridge between the two descriptions: the stochastic master equation and deterministic chaos. Previously it was unclear how to connect these levels; now we know that the loss of information about the initial state is the thread that links molecular randomness to macroscopic unpredictability. This approach may prove useful for analyzing other systems with internal fluctuations, where one needs to distinguish true chaos from mere noise.
The Kolmogorov–Sinai entropy for the modified CS14 model is 0.065, which corresponds to an uncertainty doubling time of about 10.7 time units. At volume V=$10^6$, the crossover time after which the Gaussian approximation breaks down is estimated as $t_c$ ≈ (1/0.065) ln($10^6$) ≈ 212 time units.