lördag 26 september 2015

Volkswagen Emission Scandal vs German Political Correctness Leadership

Germany with Angela Merkel is actively seeking to take the leading role in a giant transformation of the world economy into a new green economy with reductions of CO2 emissions to preindustrial levels as prime goal.  In this giant transformation German car industry has promoted the diesel engine as being more fuel efficient with less CO2 emission than the gasoline engine under strong support from German governmental political correctness.

The Volkswagen emission scandal shows the hollowness and hypocrisy of this grand scale religion: To meet the strict demands of political correctness and moral leadership set by Germany, grand scale cheating is necessary and is accordingly delivered by Germany.

The world is watching with amazement. And in China a new coal power plant is opened every day.

torsdag 24 september 2015

Finite Element Quantum Mechanics 6: Basic Analysis vs Observation

Let us now inspect the basics of the atomic model considered in this sequence of posts.

Consider then a neutral atom of kernel charge $Z$ with $N=Z$ electrons occupying non-overlapping domains in space. Assume that the electrons are partitioned into a sequence of shells $S_m$ of increasing radius $r_m$ with corresponding widths $d_m$ each shell being filled by $2m^2$ electrons, for $m=1,...,M,$ with $M$ the number of shells. We consider a hypothetic atom with all shells fully filled with $2, 8, 18, 32, 50,...,$ electrons in successive shells displaying a basic aspect of the periodicity of the periodic table of elements.

Consider now the case $d_m\sim m$ with $r_m\sim m^2$, and assume $r_1=d_1\sim\frac{1}{Z}$. The electron density $\rho_m$ in $S_m$, assumed to be spherically symmetric,  then satisfies
  • $\rho_mr_m^2d_m\sim m^2$
from which follows that 
  • $\rho_m\sim \frac{m^3}{r_m^3}$.                            (1)
We now compute the following characteristics of this model:
  1. $M^3\sim Z$, that is $M\sim Z^{\frac{1}{3}}$
  2. potential energy in $S_1\sim Z^2$
  3. potential energy in $S_m\sim m^2Z/r_m\sim Z/d_1\sim Z^2$
  4. total potential energy and thus total energy $\sim Z^{\frac{7}{3}}$.                             (2)
We check that indeed there is room for $m^2$ electrons in shell $S_m$, because the volume of $S_m$ is $r_m^2d_m\sim m^5$, while the volume of an electron $\sim d_m^3\sim m^3$.

We observe that (2) fits with observations. We understand that the electronic density is distributed so that the potential energy and thus total energy in each full shell is basically the same, which may be viewed to be a heavenly socialistic organization of the shell structure of an atom.

Numerical computation seeking the ground state energy by relaxation in the Schrödinger model of post 5 starting from an initial density distribution according to (1), shows good correspondence with observation, supporting the basic analysis of this post. Numbers will be presented in an upcoming post.

The basic aspect of this model as a form of electron density model, is that electrons (or shells in the present spherically symmetric case) keep individuality by occupying different domains of space, which makes it possible to accurately represent electron-electron repulsion.

This feature is not present in standard density models such as Thomas-Fermi and Density Functional Theory. In these models electrons lack individuality as parts of electron clouds, which makes it difficult to represent electron-electron repulsion ab ibnitio.

Recall also that in the standard Schrödinger equations wave functions appear as multi-dimensional linear combinations of products of one-electron wave functions defined in all of space by separate spatial variables, thus with each electron "both nowhere and everywhere" without individuality, which requires a statistical interpretation of the wave function as a multi-dimensional uncomputable monster.

Another basic aspect of the presented model is continuity of electron density across inter-electron or inter-shell boundaries for the electron configuration of ground states. This allows atoms to have stable ground states as non-dissipative periodic states of minimal energy.

Notice further that the size of the atom as $r_M\sim Z^{-\frac{1}{3}}$ with decreasing size as $Z$ increases, corresponds to the observed decrease of size moving to the right in each row of the periodic table:

fredag 18 september 2015

Ripples in the Fabric of Space and Time?

The code word of modern physics is:
  • fabric of space and time
  • observe gravitational waves—ripples in the fabric of space and time. 
If we dare to ask what the meaning of "fabric of space and time" may be, we get the following illuminating lesson by leading physicists:
  • First of all, space-time is not a fabric. Space and time are not tangible 'things' in the same way that water and air are. It is incorrect to think of them as a 'medium' at all. 
  • No physicist or astronomer versed in these issues considers space-time to be a truly physical medium, however, that is the way in which our minds prefer to conceptualize this concept, and has done so since the 19th century. 
  • We really do not know what space-time is, other than two clues afforded by quantum mechanics and general relativity.
  • Space-time does not claim existence in its own right, but only as a structural quality of the [gravitational] field. (Einstein)
  • Space and time coordinates are just four out of many degrees of freedom we need, to specify a self-consistent theory. What we are going to have [in any future Theory of Everything] is not so much a new view of space and time, but a de-emphasis of space and time. (Steven Weinberg)
  • In the theory of gravity, you can't really separate the structure of space and time from the particles which are associated with the force of gravity [ such as gravitons]. The notion of a string is inseparable from the space and time in which it moves. (Michael Greene)
The punch line of this educational experience is presented in this way: 
  • So, the question about what happens to space-time when a particle moves through it at near the speed of light is answered by saying that this is the wrong question to ask. Just because the brain can construct a question doesn't mean that the question has a physical answer!
We understand that LIGO in its search for "ripples in the fabric of space and time" is studying "the wrong question" and thus can be viewed as a study into the ""fabric of fantasy" which has become such a fundamental part of modern physics demanding full devotion by the sharpest brains of modern physicists (see also here ):

torsdag 17 september 2015

LIGO: Absurdity of Big Physics

The Advanced LIGO Project has now been launched as the largest single experiment ever funded by NSF at $0.365 billion:
  • The LIGO scientific and engineering team at Caltech and MIT has been leading the effort over the past seven years to build Advanced LIGO, the world's most sensitive gravitational-wave detector.
  • Gravitational waves were predicted by Albert Einstein in 1916 as a consequence of his general theory of relativity, and are emitted by violent events in the universe such as exploding stars and colliding black holes. 
  • Experimental attempts to find gravitational waves have been on going for over 50 years, and they haven't yet been found. They're both very rare and possess signal amplitudes that are exquisitely tiny.
  • Although earlier LIGO runs revealed no detections, Advanced LIGO, also funded by the NSF, increases the sensitivity of the observatories by a factor of 10, resulting in a thousandfold increase in observable candidate objects. 
  • The original configuration of LIGO was sensitive enough to detect a change in the lengths of the 4-kilometer arms by a distance one-thousandth the diameter of a proton; this is like accurately measuring the distance from Earth to the nearest star—over four light-years—to within the width of a human hair. 
  • Advanced LIGO, which will utilize the infrastructure of LIGO, is much more powerful.
  • The improved instruments will be able to look at the last minutes of the life of pairs of massive black holes as they spiral closer together, coalesce into one larger black hole, and then vibrate much like two soap bubbles becoming one. 
  • In addition, Advanced LIGO will be used to search for the gravitational cosmic background, allowing tests of theories about the development of the universe only $10^{-35}$ seconds after the Big Bang.
Read these numbers: The accuracy of old LIGO was  
  • the diameter of a human hair over a distance of 4 light-years, 
  • $10^{-35}$ seconds after Big Bang,
and yet not the slightest little gravitational wave signal was recorded from even the most violent large scale phenomena thinkable. The conclusion should be clear: There are no gravitational waves. After all, why should there be any? By Einstein's general relativity which nobody claims to grasp?

But this is not the way Big Physics works: The fact that nothing was found by the infinitely sensitive LIGO requires an even more infinitely sensitive Advanced LIGO at a cost of a half a billion to be built by eager physicists, and after Advanced LIGO has found nothing, funding for an Advanced Advanced LIGO will be requested and so on...but why are tax payers supplying this Big Money?  

lördag 5 september 2015

Gerhard 't Hooft: Improved Understanding of Quantum Mechanics Needed

Gerhard 't Hooft is one of the Nobel Laureates in Physics who is not happy with the present state of understanding of quantum mechanics and seeks to do something about it: Hooft starts out in Determinism beneath Quantum Mechanics with:
  • The need for an improved understanding of what Quantum Mechanics really is, needs hardly be explained in this meeting. 
  • My primary concern is that Quantum Mechanics, in its present state, appears to be mysterious.
  • It should always be the scientists’ aim to take away the mystery of things. 
  • It is my suspicion that there should exist a quite logical explanation for the fact that we need to describe probabilities in this world quantum mechanically. 
  • This explanation presumably can be found in the fabric of the Laws of Physics at the Planck scale.
  • However, if our only problem with Quantum Mechanics were our desire to demystify it, then one could bring forward that, as it stands, Quantum Mechanics works impeccably. 
  • It predicts the outcome of any conceivable experiment, apart from some random ingredient. This randomness is perfect. There never has been any indication that there would be any way to predict where in its quantum probability curve an event will actually be detected. 
  • Why not be at peace with this situation? 
  • One answer to this is Quantum Gravity. Attempts to reconcile General Relativity with Quantum Mechanics lead to a jungle of complexity that is difficult or impossible to interpret physically. In a combined theory, we no longer see “states” that evolve with “time”, we do not know how to identify the vacuum state, and so on. 
  • What we need instead is a unique theory that not only accounts for Quantum Mechanics together with General Relativity, but also explains for us how matter behaves. 
  • We should find indications pointing towards the correct unifying theory underlying the Standard Model, towards explanations of the presumed occurrence of supersymmetry, as well as the mechanism(s) that break it. We suspect that deeper insights in what and why Quantum Mechanics is, should help us further to understand these issues. 
Hooft thus acknowledges that quantum mechanics is mysterious, which all prominent physicists do, but Hooft is not at peace with this situation, since after all the essence of science is understanding, although most of his colleagues seem to have accepted once and for all that quantum mechanics cannot be understood and cannot be reconciled with general relativity.
Hooft then proceeds to seek a determinism behind quantum mechanics in the form of cellular automatons (also here).

I am pursuing another route to an understandable form of quantum mechanics as analog computation with finite precision, which in a way connects to Hooft's cellular automaton's, but is expressed by Schrödinger type wave equations in a continuum mechanics framework.

In this framework the finite precision computation makes a difference between smooth (strong) solutions and non-smooth (weak) solutions of the wave equations: Smooth solutions satisfy the wave equations exactly (with infinite precision), while non-smooth solutions satisfy the equations only in a weak sense with finite precision and loss of information as a form of dissipative radiation.

This allows the ground state of an atom as a smooth solution without dissipation to be stable over time without dissipation, while an excited state as a non-smooth solution will return to the ground state under dissipative radiation.

The situation is analogous to that described in my work together with Johan Hoffman on fluid mechanics, with turbulent solutions as non-smooth dissipative solutions of formally inviscid Euler equations, which allowed us to resolve d'Alembert's paradox (J Math Fluid Mech 2008) and formulate a new theory of flight (to appear in J Math Fluid Mech 2015), among other things.

onsdag 2 september 2015

Finite Element Quantum Mechanics 5: 1d Model in Spherical Symmetry

The new Schrödinger equation I am studying in this sequence of posts takes the following form, in spherical coordinates with radial coordinate $r\ge 0$ in the case of spherical symmetry, for an atom with kernel of charge $Z$ at $r=0$ with $N\le Z$ electrons of unit charge distributed in a sequence of non-overlapping spherical shells $S_1,...,S_M$ separated by spherical surfaces of radii $0=r_0<r_1<r_2<...<r_M=\infty$, with $N_j>0$ electrons in shell $S_j$ corresponding to the interval $(r_{j-1},r_j)$ for $j=1,...,M,$ and $\sum_j N_j = N$:

Find a complex-valued differentiable function $\psi (r,t)$ depending on $r≥0$ and time $t$, satisfying for $r>0$ and all $t$,
  • $i\dot\psi (r,t) + H(r,t)\psi (r,t) = 0$              (1)
where $\dot\psi = \frac{\partial\psi}{\partial t}$ and $H(r,t)$ is the Hamiltonian defined by
  • $H(r,t) = -\frac{1}{2r^2}\frac{\partial}{\partial r}(r^2\frac{\partial }{\partial r})-\frac{Z}{r}+ V(r,t)$,
  • $V(r,t)= 2\pi\int\vert\psi (s,t)\vert^2\min(\frac{1}{r},\frac{1}{s})R(r,s,t)s^2\,ds$,
  • $R(r,s,t) = (N_j -1)/N_j$ for $r,s\in S_j$ and $R(r,s,t)=1$ else,
  • $4\pi\int_{S_j}\vert\psi (s,t)\vert^2s^2\, ds = N_j$ for $j=1,...,M$.                  (2)
Here $-\frac{Z}{r}$ is the kernel-electron attractive potential and $V(r,t)$ is the electron-electron repulsive potential computed using the fact that the potential $W(s)$ of a spherical uniform surface charge distribution of radius $r$ centered at $0$ of total charge $Q$, is given by $W(s)=Q\min(\frac{1}{r},\frac{1}{s})$, with a reduction for a lack of self-repulsion within each shell given by the factor $(N_j -1)/N_j$.

The $N_j$ electrons in shell $S_j$ are thus homogenised into a spherically symmetric charge distribution of total charge $N_j$.

This is a free boundary problem readily computable on a laptop, with the $r_j$ representing the free boundary separating shells of spherically symmetric charge distribution of intensity $\vert\psi (r,t)\vert^2$ and a free boundary condition asking continuity and differentiability of $\psi (r,t)$.   

Separating $\psi =\Psi +i\Phi$ into real part $\Psi$ and imaginary part $\Phi$, (1) can be solved by explicit time stepping with (sufficiently small) time step $k>0$ and given initial condition (e.g. as ground state):
  • $\Psi^{n+1}=\Psi^n-kH\Phi^n$, 
  • $\Phi^{n+1}=\Phi^n+kH\Psi^n$, 
for $n=0,1,2,...,$ where $\Psi^n(r)=\Psi (r,nk)$ and $\Phi^n(r)=\Phi (r,nk)$, while stationary ground states can be solved by the iteration
  • $\Psi^{n+1}=\Psi^n-kH\Psi^n$, 
  • $\Phi^{n+1}=\Phi^n-kH\Phi^n$, 
while maintaining (2).

A remarkable fact is that this model appears to give ground state energies as minimal eigenvalues of the Hamiltonian for both ions and atoms for any $Z$ and $N$ within a percent or so, or alternatively ground state frequencies from direct solution in time dependent form. Next I will compute excited states and transitions between excited states under exterior forcing.

Specifically, what I hope to demonstrate is that the model can explain the periods of the periodic table corresponding to the following sequence of numbers of electrons in shells of increasing radii: 2, (2, 8), (2, 8, 8), (2, 8, 18, 8), (2, 8, 18, 18, 8)... which to be true lacks convincing explanation in standard quantum mechanics (according to E. Serri among many others).

The basic idea is thus to represent the total wave function $\psi (r,t)$ as a sum of shell wave functions
with non-overlapping supports in the different in shells requiring $\psi (r,t)$ and thus $\vert\psi (r,t)\vert^2$ to be continuous across inter-shell boundaries as free boundary condition, corresponding to continuity of charge distribution as a classical equilibrium condition.

I have also with encouraging results tested this model for $N\le 10$ in full 3d geometry without spherical shell homogenisation with a wave function as a sum of electronic wave functions with non-overlapping supports separated by a free boundary determined by continuity of wave function including charge distribution.

We compare with the standard (Hartree-Fock-Slater) Ansatz of quantum mechanics with a multi-dimensional wave function $\psi (x_1,...,x_N,t)$ depending on $N$ independent 3d coordinates $x_1,...,x_N,$ as a linear combination of wave functions of the multiplicative form
  • $\psi_1(x_1,t)\times\psi_2(x_2,t)\times ....\times\psi_N(x_N,t)$,  
with each electronic wave function $\psi_j(x_j,t)$ with global support (non-zero in all of 3d space). Such multi-d wave functions with global support thus depend on $3N$ independent space coordinates and as such defy both direct physical interpretation and computability, as soon as $N>1$, say. One may argue that since such multi-d wave function cannot be computed, it does not matter that they have no physical meaning, but the net output appears to be nil, despite the declared immense success of standard quantum mechanics based on this Ansatz.