22,789 research outputs found
A general class of Lagrangian smoothed particle hydrodynamics methods and implications for fluid mixing problems
Various formulations of smoothed particle hydrodynamics (SPH) have been proposed, intended to resolve certain difficulties in the treatment of fluid mixing instabilities. Most have involved changes to the algorithm which either introduces artificial correction terms or violates what is arguably the greatest advantage of SPH over other methods: manifest conservation of energy, entropy, momentum and angular momentum. Here, we show how a class of alternative SPH equations of motion (EOM) can be derived self-consistently from a discrete particle Lagrangian – guaranteeing manifest conservation – in a manner which tremendously improves treatment of these instabilities and contact discontinuities. Saitoh & Makino recently noted that the volume element used to discretize the EOM does not need to explicitly invoke the mass density (as in the ‘standard’ approach); we show how this insight, and the resulting degree of freedom, can be incorporated into the rigorous Lagrangian formulation that retains ideal conservation properties and includes the ‘∇h’ terms that account for variable smoothing lengths. We derive a general EOM for any choice of volume element (particle ‘weights’) and method of determining smoothing lengths. We then specify this to a ‘pressure–entropy formulation’ which resolves problems in the traditional treatment of fluid interfaces. Implementing this in a new version of the GADGET code, we show it leads to good performance in mixing experiments (e.g. Kelvin–Helmholtz and ‘blob’ tests). And conservation is maintained even in strong shock/blastwave tests, where formulations without manifest conservation produce large errors. This also improves the treatment of subsonic turbulence and lessens the need for large kernel particle numbers. The code changes are trivial and entail no additional numerical expense. This provides a general framework for self-consistent derivation of different ‘flavours’ of SPH
The Generalized Area Theorem and Some of its Consequences
There is a fundamental relationship between belief propagation and maximum a
posteriori decoding. The case of transmission over the binary erasure channel
was investigated in detail in a companion paper. This paper investigates the
extension to general memoryless channels (paying special attention to the
binary case). An area theorem for transmission over general memoryless channels
is introduced and some of its many consequences are discussed. We show that
this area theorem gives rise to an upper-bound on the maximum a posteriori
threshold for sparse graph codes. In situations where this bound is tight, the
extrinsic soft bit estimates delivered by the belief propagation decoder
coincide with the correct a posteriori probabilities above the maximum a
posteriori threshold. More generally, it is conjectured that the fundamental
relationship between the maximum a posteriori and the belief propagation
decoder which was observed for transmission over the binary erasure channel
carries over to the general case. We finally demonstrate that in order for the
design rate of an ensemble to approach the capacity under belief propagation
decoding the component codes have to be perfectly matched, a statement which is
well known for the special case of transmission over the binary erasure
channel.Comment: 27 pages, 46 ps figure
The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations
The derivation of effective evolution equations is central to the study of
non-stationary quantum many-body sytems, and widely used in contexts such as
superconductivity, nuclear physics, Bose-Einstein condensation and quantum
chemistry. We reformulate the Dirac-Frenkel approximation principle in terms of
reduced density matrices, and apply it to fermionic and bosonic many-body
systems. We obtain the Bogoliubov-de-Gennes and Hartree-Fock-Bogoliubov
equations, respectively. While we do not prove quantitative error estimates,
our formulation does show that the approximation is optimal within the class of
quasifree states. Furthermore, we prove well-posedness of the
Bogoliubov-de-Gennes equations in energy space and discuss conserved
quantities.Comment: 46 pages, 1 figure; v2: simplified proof of conservation of particle
number, additional references; v3: minor clarification
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