  It is shown numerically that for Fermi Pasta Ulam (FPU) chains with
alternating masses and heat baths at slightly different temperatures at the
ends, the local temperature (LT) on small scales behaves paradoxically in
steady state. This expands the long established problem of equilibration of FPU
chains. A well-behaved LT appears to be achieved for equal mass chains; the
thermal conductivity is shown to diverge with chain length N as N^(1/3),
relevant for the much debated question of the universality of one dimensional
heat conduction. The reason why earlier simulations have obtained
systematically higher exponents is explained.
