2 research outputs found
Amorphous packings of hard spheres in large space dimension
In a recent paper (cond-mat/0506445) we derived an expression for the
replicated free energy of a liquid of hard spheres based on the HNC free energy
functional. An approximate equation of state for the glass and an estimate of
the random close packing density were obtained in d=3. Here we show that the
HNC approximation is not needed: the same expression can be obtained from the
full diagrammatic expansion of the replicated free energy. Then, we consider
the asymptotics of this expression when the space dimension d is very large. In
this limit, the entropy of the hard sphere liquid has been computed exactly.
Using this solution, we derive asymptotic expressions for the glass transition
density and for the random close packing density for hard spheres in large
space dimension.Comment: 11 pages, 1 figure, includes feynmf diagram
On the high density behavior of Hamming codes with fixed minimum distance
We discuss the high density behavior of a system of hard spheres of diameter
d on the hypercubic lattice of dimension n, in the limit n -> oo, d -> oo,
d/n=delta. The problem is relevant for coding theory. We find a solution to the
equations describing the liquid up to very large values of the density, but we
show that this solution gives a negative entropy for the liquid phase when the
density is large enough. We then conjecture that a phase transition towards a
different phase might take place, and we discuss possible scenarios for this
transition. Finally we discuss the relation between our results and known
rigorous bounds on the maximal density of the system.Comment: 15 pages, 6 figure