The problem of substance in chemistry — an approach by means of large deviation statistics

Different substances can phenomenologically be defined by use of Gibbs' phase rule. In quantum mechanics, on the other hand, the notion of a substance is not clearly understood: The thermal density operator D_β corresponding to some Hamiltonian and some chosen inverse temperature β is uniquely defined, which implies that different isomers exhibit the same thermal density operator. Coexistence of isomers, substances or phases at some temperature, on the other hand, would need different thermal density operators, one for each isomer, substance or phase. Here we try to understand this problem for the classical van der Waals gas using large deviation statistics. We show that the gaseous and liquid phase of the van der Waals gas emerge with increasing numbers of particles. Intermediate states — neither gaseous nor liquid — exist but die out with increasing number of particles. Extension of our method to quantum mechanics is not straightforward, but looks promising.