Is it possible to derive Liouville's Theorem purely from maximum differential entropy?
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Typically in physics (at least the way I learned mechanics), this is derived using the multi-dimensional divergence theorem on the 2N2N-dimensional phase space i.e. 0=∂tρ+∑i=1N(∂(ρqi˙)qi+∂(ρpi˙)pi)0=∂tρ+∑i=1N(∂(ρqi˙)qi+∂(ρpi˙)pi), and the terms in brackets are simplified using Hamiltonian equation of motion to obtain Liouville's Theorem: dρdt=∂ρ∂t+∑i(∂ρ∂qiqi˙+∂ρ∂pipi˙)=0dρdt=∂ρ∂t+∑i(∂ρ∂qiqi˙+∂ρ∂pipi˙)=0. I am wondering if it is possible to derive this without assuming "physics," i.e. Hamiltonian equation of motion, but instead from maximum (differential) entropy principle. Suppose ρ=ρ(p⃗ ,q⃗ ;t)ρ=ρ(p→,q→;t) is a phase-space distribution with maximum differential entropy H[ρ]=∫−ρlogρdp⃗ dq⃗ H[ρ]=∫−ρlogρdp→dq→. Since the entropy is maximized, we can write: dHdt=ddt∫−ρlogρdp⃗ dq⃗ =∫dp⃗ dq⃗ (−logρ−1)dρdt=0dHdt=ddt∫−ρlogρdp→dq→=∫dp→dq→(−logρ−1)dρdt=0 However, does this necessarily imply dρdt=0dρdt=0 ? Or, is there an alternative approach to this problem? One way, perhaps, is to show that the differential entropy H[ρ]=∫−ρlogρdp⃗ dq⃗ H[ρ]=∫−ρlogρdp→dq→ is (up to a constant) equivalent to Boltzmann's definition of entropy S=kBlog|Γ|S=kBlog|Γ| (again, without assuming physics).
