70-XX MECHANICS OF PARTICLES AND SYSTEMS (For relativistic mechanics, see 83A05 and 83C10; for statistical mechanics, see 82-XX)
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Gaining insights into the working principles of photocatalysts on an atomic scale is a challenging task. The obviously high complexity of the reaction mechanism involving photo-excited electrons and holes is one reason. Another complicating aspect is that the electromagnetic field, driving photocatalysis, is not homogeneous on a nanoscale level for particle based catalysts as it is influenced by the particle’s shape and size.
We present a simple model, inspired by the CO2 reduction on titania anatase, which addresses the impact of these heterogeneities on the photocatalytic kinetics by combining kinetic Monte Carlo with electromagnetic wave simulations. We find that average activity and especially efficiency might differ significantly between different particles. Moreover, we find sizable variation of the catalytic activity on a single facet of a nanocrystal. Besides this quantitative heterogeneity, the coverage situation in general changes laterally on this facet and we observe a concomitant change of the rate-determining steps.
This heterogeneity on all levels of photocatalytic activity is masked in experimental studies, where only the spatially averaged activity can be addressed. Microkinetic models based on experimental findings might therefore not represent the true micro- scopic behavior, and mechanistic conclusion drawn from these need to be handled with care.
Travelling waves and conservation laws are studied
for a wide class of $U(1)$-invariant complex mKdV equations
containing the two known integrable generalizations of
the ordinary (real) mKdV equation.
The main results on travelling waves include deriving
new complex solitary waves and kinks that generalize
the well-known mKdV $\sech$ and $\tanh$ solutions.
The main results on conservation laws consist of explicitly finding
all 1st order conserved densities that yield phase-invariant counterparts of
the well-known mKdV conserved densities for
momentum, energy, and Galilean energy,
and a new conserved density describing
the angular twist of complex kink solutions.
Symmetries and conservation laws are studied for two classes
of physically and analytically interesting radial wave equations
with power nonlinearities in multi-dimensions.
The results consist of two main classifications:
all symmetries of point type and all conservation laws of a general energy-momentum type
are explicitly determined,
including those such as dilations, inversions, similarity energies and conformal energies
that exist only for special powers or dimensions.
In particular, all variational cases (when a Lagrangian formulation exists)
and non-variational cases (when no Lagrangian exists)
for these wave equations are considered.
As main results, the classification yields generalized energies and radial momenta
in certain non-variational cases,
which are shown to arise from a new type of Morawetz dilation identity
that produces conservation laws for each of the two classes of wave equations
in a different way than Noether's theorem.