Statistics of Energy Levels for a recently discovered (at the simulation level) exotic material.

Introduction:

This post has a good mixture of statistical data analysis as well as physics based modelling.   Some discussion of microscopic physics is unavoidable in order to get to the statistics part of the post. The statistical analysis will be based upon a material which I studied during my PhD. In the first half of the post, a brief description of this novel material, called the semi-Dirac material will be given. The second half the post will cover statistical analysis of its energy levels.

A Novel Material Discovered in Simulation:

Around 2008-2009 a six atom thick composite nano-material , made of Titanium Oxide, sandwiched between Vanadium oxide layers, was first studied at U C Davis physics department. To everyone’s surprise the simulation results indicated that the electron in such a material moved like a photon (light particle) in one direction and like an electron in the orthogonal direction. One of the key interests in such a material was the unusual energy levels of its electrons.

Microscopic Energy Levels, for a Confined and an Unconfined Particle:

At the microscopic level, the energy of a free particle can take a continuous range of values, just like in the macroscopic world. If a free particle of mass m moves with a velocity v, its energy is given by:

. The energy can also be expressed in terms of momentum p,  the product of the mass and the velocity of the particle.  Energy in terms of the momentum variable is written as:

, which for two dimensions can explicitly be written as , and being the x and the y components of the momentum respectively. But the situation is different for a confined microscopic particle. Energies of a confined particle, say an electron,  are discrete. For example if we consider an electron enclosed in a box, it turns out that the electron can take only those energies allowed by the following expression:

where C is a constant, and n is an integer. The above equation implies that the energies can only take values like C, 2C, 3C….etc. But there will be no energy level with the value 1.5C. This rule is a consequence of a fundamental law of physics, called quantum mechanics. In this post the discrete energy levels of an electron confined within the novel semi-Dirac material will be at the focus.

Energy Levels the Semi-Dirac Material:

The energy levels of an unconfined semi-Dirac  electron is given by the following expression:

.

When , . The semi-Dirac electron behaves like light particles (The energy momentum relationship of a light particle is given by , where c is the speed of light, and p is the momentum.) But when , , which is the energy momentum relationship of an ordinary particle. When a semi-Dirac electron is confined, its energy levels are quantized. But the quantized energy levels do not have a simple expression. At this point we will not go into how to obtain the energy levels of a confined electron, which will involve very detailed quantum mechanical calculations, but will simply assume that they are given. We will take the energy levels in the spirit of having obtained them in an experiment. One can construct a histogram as well as the cumulative density function (CDF) of the energy values. The CDF will take the following staircase shape. It is possible to fit the staircase with a “smooth curve”.  The smooth curve is placed under quote, since there is a definitive quantum mechanical recipe to derive the expression for the smooth curve. It’s not simply a least square fit, although it may be a close match to the latter. In the following graph, the smooth line is indicated as a dotted line. The y-values of the smooth curve at various energies are indicated with open circles on the y axis.

It turns out that the difference ‘s’ between the y-values of the two consecutive points on the smoothed curve may follow certain types of statistics. This depends on whether  the electron motion inside the material is regular or chaotic.

Exponential Versus Non-exponential:

The distribution of the variable s can be exponential type or something else. If s follows an exponential  distribution, it would imply that the underlying motion of electrons is regular. A non-exponential  type distribution on the other hand would imply a more chaotic motion of the confined electron.  In the following we give the diagrams for the exponential as well as some other types of distributions.

In the above diagram only the blue colored graph is the exponential distribution. All other curves are non-exponential  in nature.  The blue curve corresponds to regular non-chaotic motions of electrons, whereas the others don’t.  Later we will try to ascertain when the distribution of s is exponential  and when it is not. In order to obtain a distribution one needs to construct a histogram. In the following section we will outline an important method regarding how to create a smoothed version of a histogram.

Smoothing a Histogram:

In the usual construction of a histogram values of s are split up into a certain number of bins. Then a bar for each bin is constructed, whose height is proportional to the frequency corresponding to that particular bin. As an alternative method, centering each of the discrete values of s, a normalized Gaussian distribution of a given width is constructed. For a given values of s, the contributions from multiple Gaussians will add up. The Gaussians which are close to each other will contribute significantly compared to the ones far apart. The net profile will be accentuated at places where the densities of s-values are high. This procedure will create a distribution curve whose profile matches that of a histogram, but is smoother compared to it. Mathematically speaking, let be a Gaussian centered at . denotes the th value of the variable . Instead of plotting the histogram the following equation is plotted.

. There is some flexibility in choosing the value of . It should be done in such a way that f appears to be smooth.

Datasets for various values of the Parameter :

The laws of quantum mechanics demand that the speed of electron goes up when it is confined in a small region. For the semi-Dirac material this behavior is captured in a parameter named .    is inversely proportional to the speed of the semi-Dirac electron moving in a confined region : small indicates large electron speed.  All dataset s on the s-variable distribution are functions of the parameter . In the following figure as varies from .8 to .9, the shapes of the function f (the distribution corresponding to the variable s) varies quite a lot as can be seen in the graph.

In the following we investigate how close each in the above figure is to the exponential curve . We will use the square root norm as a measure of the proximity of the ‘observed’ dataset to the exponential curve, and plot the result as a function of . The graph does not have a monotonic feature, as can be seen in the figure below.  The implication of that is the semi-Dirac system can go in and out of regular

behavior as the system parameter is varied. But there is also a general decreasing trend as can be seen from the above graph.  This means as alpha increases, the function f should resemble  the exponential function more and  more. This is in accordance with the fact as alpha goes to .9, the function f is closer in appearance to the exponential function, as can be seen from the f versus alpha plot. Hence we conclude as alpha approaches .9, the motion of electron becomes more and more regular. But for a different parameter space the trend can be different.

Conclusion:

In this post we took a novel material called the semi-Dirac material as the subject of our study and did some investigation on its energy level statistics. Since this is a data science blog, in the summary I will limit my comments to the statistics and the visualization aspect of the problem as much as possible. We described a novel visualization technique for the histograms. We showed how superposition of Gaussians could be used as a smooth alternative to a histogram. It also provides a way to visualize multiple histograms in the same plot. This is a rather novel method to be able to display the same amount of content that is there in multiple histograms in a single diagram, and to be able to compare one against the other. We also discussed about the concept of the irregularity of motion of the electron inside a semi-Dirac material. We checked by taking the norm of the difference between the observed smoothed histogram and the function that the semi-Dirac system can go in and out of the chaotic state as the parameter alpha is varied. We also noticed a decreasing trend in randomness as alpha increased from .8 to .9. The trend is not an universal one. For other ranges of the parameter, the trend may be reversed or even obliterated. Usually, for an ordinary material electrons are either chaotic or not.  By changing the boundary can drive a system from being regular to a non-regular one. But in case of the semi-Dirac material the system goes in and out of chaotic states by varying a system parameter.