### Entropy and the Typicality of Universes

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The universal validity of the second law of thermodynamics is widely attributed to a finely tuned initial condition of the universe. This creates a problem: why is the universe atypical? We suggest that the problem is an artefact created by inappropriate transfer of the traditional concept of entropy to the whole universe. Use of what we call the relational $N$-body problem as a model indicates the need to employ two distinct entropy-type concepts to describe the universe. One, which we call entaxy, is novel. It is scale-invariant and decreases as the observable universe evolves. The other is the algebraic sum of the dimensionful entropies of branch systems (isolated subsystems of the universe). This conventional additive entropy increases. In our model, the decrease of entaxy is fundamental and makes possible the emergence of branch systems and their increasing entropy. We have previously shown that all solutions of our model divide into two halves at a unique `Janus point’ of maximum disorder. This constitutes a common past for two futures each with its own gravitational arrow of time. We now show that these arrows are expressed through the formation of branch systems within which conventional entropy increases. On either side of the Janus point, this increase is in the same direction in every branch system. We also show that it is only possible to specify unbiased solution-determining data at the Janus point. Special properties of these `mid-point data’ make it possible to develop a rational theory of the typicality of universes whose governing law, as in our model, dictates the presence of a Janus point in every solution. If our self-gravitating universe is governed by such a law, then the second law of thermodynamics is a necessary direct consequence of it and does not need any special initial condition.