17,777 research outputs found
Constraints on scalar diffusion anomaly in three-dimensional flows having bounded velocity gradients
This study is concerned with the decay behaviour of a passive scalar
in three-dimensional flows having bounded velocity gradients. Given an
initially smooth scalar distribution, the decay rate of the
scalar variance is found to be bounded in terms of controlled
physical parameters. Furthermore, in the zero diffusivity limit, ,
this rate vanishes as if there exists an
independent of such that for
. This condition is satisfied if in the limit ,
the variance spectrum remains steeper than for large wave
numbers . When no such positive exists, the scalar field may be
said to become virtually singular. A plausible scenario consistent with
Batchelor's theory is that becomes increasingly shallower for
smaller , approaching the Batchelor scaling in the limit
. For this classical case, the decay rate also vanishes, albeit
more slowly -- like , where is the Prandtl or Schmidt
number. Hence, diffusion anomaly is ruled out for a broad range of scalar
distribution, including power-law spectra no shallower than . The
implication is that in order to have a -independent and non-vanishing
decay rate, the variance at small scales must necessarily be greater than that
allowed by the Batchelor spectrum. These results are discussed in the light of
existing literature on the asymptotic exponential decay , where is independent of .Comment: 6-7 journal pages, no figures. accepted for publication by Phys.
Fluid
On the Microcanonical Entropy of a Black Hole
It has been suggested recently that the microcanonical entropy of a system
may be accurately reproduced by including a logarithmic correction to the
canonical entropy. In this paper we test this claim both analytically and
numerically by considering three simple thermodynamic models whose energy
spectrum may be defined in terms of one quantum number only, as in a
non-rotating black hole. The first two pertain to collections of noninteracting
bosons, with logarithmic and power-law spectra. The last is an area ensemble
for a black hole with equi-spaced area spectrum. In this case, the many-body
degeneracy factor can be obtained analytically in a closed form. We also show
that in this model, the leading term in the entropy is proportional to the
horizon area A, and the next term is ln A with a negative coefficient.Comment: 15 pages, 1 figur
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