1,797 research outputs found
Existence of translating solutions to the flow by powers of mean curvature on unbounded domains
In this paper, we prove the existence of classical solutions of the Dirichlet
problem for a class of quasi-linear elliptic equations on unbounded domains
like a cone or a U-type domain. This problem comes from the study of mean
curvature flow and its generalization, the flow by powers of mean curvature.
Our approach is a modified version of the classical Perron method, where the
solutions to the minimal surface equation are used as sub-solutions and a
family auxiliary functions are constructed as super-solutions.Comment: 30 page
A theory of dark energy that matches dark matter
In this paper, a theory of dark energy is proposed that matches dark matter.
The relativistic quantum mechanics equations reveal that free particles can
have negative energies. We think that the negative energy is the dark energy
which behaviors as dark photons with negative energies. In this work, the
photon number states are extended to the cases where the photon number can be
negative integers, called negative integer photon states, the physical meaning
of which are that the photons in such a state are of negative energy, i.e.,
dark photons. The dark photons constitute dark radiation, also called negative
radiation. The formulism of the statistical mechanics and thermodynamics of the
dark radiation is presented. This version of dark energy is of negative
temperature and negative pressure, the latter regarded as responsible for the
accelerate expansion of the universe. It is believed that there is a symmetry
of energy-dark energy in the universe. In our previous work, the theory of the
motion of the matters with negative kinetic energy was presented. In our
opinion, the negative kinetic energy matter is dark matter. In the present
work, we demonstrate that the dark substances absorb and release dark energy.
In this view, the dark matter and dark energy match. Therefore, there is a
symmetry of matter-energy match and dark matter-dark energy match in the
universe. We present the reasons why the negative kinetic energy systems and
negative radiation are dark to us
The behaviors of the wave functions of small molecules with negative kinetic energies
According to relativistic quantum mechanics, particles can be of negative
kinetic energies (NKE). The author asserts in his previous works that the NKE
substances are dark matters. Some NKE particles, say a pair of NKE electrons,
can constitute a stable system by means of the repulsive interaction between
them. In the present work, two simplest three-particle systems are
investigated. One consists of two NKE positrons and one NKE proton, called dark
hydrogen anion. The other is composed of two NKE protons and one NKE positron,
called dark hydrogen molecule cation. They are so named because the
Hamiltonians of them can correspond to those of the hydrogen anion and hydrogen
molecule cation. In evaluating the dark hydrogen molecule cation, the famous
Born-Oppenheimer approximation does not apply, i.e., the NKE of the protons
cannot be neglected. Without the NKE, the system cannot be stable. Our study
reveals that in a NKE system, the particles with the same kind of electric
charge combine tightly. This is to enhance the repulsive Coulomb potential so
as to raise the total energy as far as possible. A great amount of NKE
particles can compose a dense and dark macroscopic NKE body. Thus, it is
conjectured that some remote dark celestial bodies may be NKE ones other than
the well-known black holes. The discrepancies between the black holes and
macroscopic NKE bodies are pointed out.Comment: 22 pages, 3 table
Many-body theories for negative kinetic energy systems
In the author's previous works, it is derived from the Dirac equation that
particles can have negative kinetic energy (NKE) solutions, and they should be
treated on an equal footing as the positive kinetic energy (PKE) solutions.
More than one NKE particles can make up a stable system by means of
interactions between them and such a system has necessarily negative
temperature. Thus, many-body theories for NKE systems are desirable. In this
work, the many-body theories for NKE systems are presented. They are
Thomas-Fermi method, Hohenberg-Kohn theorem, Khon-Sham self-consistent
equations, and Hartree-Fock self-consistent equations. They are established
imitating the theories for PKE systems. In each theory, the formalism of both
zero temperature and finite negative temperature are given. In order to verify
that tunneling electrons are of NKE and real momentum, an experiment scenario
is suggested that lets PKE electrons collide with tunneling electrons.Comment: 32 pages, 2 figure
Liouville equation in statistical mechanics is not applicable to gases composed of colliding molecules
Liouville equation is a fundamental one in statistical mechanics. It is
rooted in ensemble theory. By ensemble theory, the variation of the system's
microscopic state is indicated by the moving of the phase point, and the moving
trajectory is believed continuous. Thus, the ensemble density is thought to be
a smooth function, and it observes continuity equation. When the Hamiltonian
canonical equations of the molecules are applied to the continuity equation,
Liouville equation can be obtained. We carefully analyze a gas composed of a
great number of molecules colliding with each other. The defects in deriving
Liouville equation are found. Due to collision, molecules' momenta changes
discontinuously, so that the trajectories of the phase points are actually not
continuous. In statistical mechanics, infinitesimals in physics and in
mathematics should be distinguished. In continuity equation that the ensemble
density satisfies, the derivatives with respect to space and time should be
physical infinitesimals, while in Hamiltonian canonical equations that every
molecule follows, the derivatives take infinitesimals in mathematics. In the
course of deriving Liouville equation, the infinitesimals in physics are
unknowingly replaced by those in mathematics. The conclusion is that Liouville
equation is not applicable to gases.Comment: 19 pages, 1 figur
A generalized scattering theory in quantum mechanics
In quantum mechanics textbooks, a single-particle scattering theory is
introduced. In the present work, a generalized scattering theory is presented,
which can be in principle applied to the scattering problems of arbitrary
number of particle. In laboratory frame, a generalized Lippmann-Schwinger
scattering equation is derived. We emphasized that the derivation is rigorous,
even for treating infinitesimals. No manual operation such as analytical
continuation is allowed. In the case that before scattering N particles are
plane waves and after the scattering they are new plane waves, the transition
amplitude and transition probability are given and the generalized S matrix is
presented. It is proved that the transition probability from a set of plane
waves to a new set of plane waves of the N particles equal to that of the
reciprocal process. The generalized theory is applied to the cases of one- and
two-particle scattering as two examples. When applied to single-particle
scattering problems, our generalized formalism degrades to that usually seen in
the literature. When our generalized theory is applied to two-particle
scattering problems, the formula of the transition probability of two-particle
collision is given. It is shown that the transition probability of the
scattering of two free particles is identical to that of the reciprocal
process. This transition probability and the identity are needed in deriving
Boltzmann transport equation in statistical mechanics. The case of identical
particles is also discussed.Comment: 35 pages, 3figure
- …