The race begins with two runners. One runs steadily, burning fuel little by little and predictably. The other learns on the move to run faster, and its fuel flares ever hotter. In the chemistry of inanimate matter, such a race unfolds between a simple self-reproducing substance and a molecule that copies itself from a template, the way DNA does.
The physics of nonequilibrium systems says this: the more energy a trajectory dissipates as heat, the more probable it is. This dissipation is called dissipation. A simple substance that accelerates its own formation spends energy linearly. But a copying molecule with adaptation speeds up on its own: its rate grows with time, and energy dissipation takes off like a double exponential. At first the copying molecule trails behind, because it needs time to pick up speed.
The ratio of the probabilities of the two trajectories grows as exp(exp(τ²)). This means that after the critical time, the copying molecule wins the race by a margin that cannot be closed. Physics itself selects heredity: trajectories that can learn burn fuel faster than all the rest, and that is why they are the ones realized in nature.
It used to be unclear why, out of the multitude of chemical pathways, nature chooses the ones that lead to heredity and evolution. Now it is clear: adaptive copying molecules produce superexponential growth of dissipation, and this makes their trajectories asymptotically dominant in probability. The key technique is comparing two classes of trajectories within the framework of large deviation theory, where the ratio of probabilities depends on the difference of total entropies.
Such a criterion can be tested in an experiment. If the heat release is measured in a flow reactor with chemical precursors, the transition to an evolving system will show up as a positive second derivative of the logarithm of the dissipation power. This is a quantitative signature that something resembling life is being born in the flask.
The race ends not with a finish line but with a new quality. The runner who learned on the move did not simply come in first: he changed the rules of the race. So it is in chemistry: trajectories that burn fuel faster than all others become the foundation of heredity. Physics does not argue with biology; it gives biology its starting signal.
The double exponential in the probability means that after the critical moment the replicative trajectory becomes not merely more probable, but so dominant that the logarithm of its logarithm grows quadratically with time. This is one of the fastest growths encountered in physical models.