No attractors in Hamiltonian system It’s not true that neighboring initial conditions will converge to the fixed state after long time. As we saw from the Jupiter problem, there are some regions where the initial conditions for chaotic and periodic trajectories lies infinitesimally close but lead to different results. The chaotic initial conditions will not make the system converge to certain state. Therefore, there is no attractor in this 3-body problem. Stable Hamiltonian systems can withstand a minor perturbation _in the process_ and the system will still end up where it’s supposed to. Unstable Hamiltonian system will diverge if perturbed _in the process_ and the system will not come to the original equilibrium regardless of whether the system becomes chaotic or not.

Undistinguished particles For N distinguishable particles, if at certain instant they occupy N states in the phase space, there are N! different choices of such configuration. The total volume for this system in phase space is the product of the volume for each particle in its own sub phase space. \Omega_D = N! \space \omega_1 \omega_2 \omega_3 ... \omega_N If the particles are now undistinguished, there is only one choice of such configuration without permutation. \Omega_U = \omega_1 \omega_2 \omega_3 ... \omega_N Therefore, $\Omega_D = N!\space \Omega_U $. For the mixing problem of black and white particles of the same pressure on both sides, if you cannot tell the difference between black and white ones, you cannot extract work from the mixing. Actully, you won’t know if they’ve mixed already or not. However, suppose the black and white particles are of different pressure, and you still cannot tell them apart. If there were a membrane which allows particles of lower pressure to penetrate through, this membrane then can tell the difference between the two sides without telling the particles apart. Then work can be extracted. Now if a door can distinguish the black from the white particles and let through white ones, after very long time, the white particles will distribute themselves equally in the two sides, i.e. there will be ${4} N\space white + 0 \space black$ in the left and ${4} N \space white + {2} N \space black$ in the right. And we can extract work from it due to the pressure difference between the two sides.

Entropy maximization and temperature First of all, Ω₁(E₁)δE₁ and Ω₂(E₂)δE₂ are the phase space volume for these two subsystems. Since most of the volume in phase space is near the surface, we can instead write Ω₁(E₁) and Ω₂(E₂) to denote the two volumes in phase space. Second, it’s clear that the probability density of subsystem 1 having energy E₁ is proportional to the ’allowed’ volume product of the two subsystems. \rho_1(E_1) \propto \Omega_1(E_1)\Omega_2(E_2) where E₂ = E − E₁. And finally, we impose the normalization condition: \int \rho_1(E_1)dE_1 = A \int \Omega_1(E_1)\Omega_2(E-E_1)dE_1 = A \Omega(E) = 1 A is the proportional constant and equal to 1/Ω(E). Therefore, \rho_1(E_1) = {\Omega(E)} If we maximize the probability at E₁: {dE_1} = 0 {\Omega_1}{dE_1} - {\Omega_2}{dE_2} = 0 By plugging the definition of entropy S = kBlog(Ω), {k_B}{dE_1} - {k_B}{dE_2} = 0 {dE_1} + {dE_1} = {dE_1}(S(E_1)+S(E_2)) = 0 ,which implies the maximization of the sum of entropies of both subsystems. For (6), we see that if the probability density ρ(E₁) is maximized, {T_1} = {T_2}

Weirdness in high dimensions Most of the volume of high dimensional spheres is near the surface. As we can approximate the volume as V ≈ r3N in 3N dimensions, near the center, r ≈ 0 and V ≈ 0. Near the surface, where r becomes larger and contributes to most of the volume. For statistical mechanics, instead of taking the whole E sphere into account, which might involve high-dimensional integral, it’s easier to focus on the shell of E sphere. Then the momenta can be considered to be $p = $ instead of doing \int_0^{} (....)dp

Absorption boundary conditions First observe that the probability density at the origin at any time is zero. \rho(0,t) = 0 And the probability density on x = x′ at t = 0 is \rho(x,0) = \delta(x-x') We can assume that there is a image probability density at t = 0 on x = −x′. Therefore the total probability at t = 0 should be \rho(x,0) = \delta(x-x')-\delta(x+x') The Green’s function for each delta function is a time-dependent Gaussian distribution, G(x,t) = {} e^{-x^2/4Dt} with the particle originally at the origin. For this case, both delta functions diffuse, yielding the same Gaussian distribution. G(x,t) = {} \left( e^{-(x-x')^2/4Dt} - e^{-(x+x')^2/4Dt}\right) The time evolution probability distribution is shown in fig.1.

Local conservation a. The density of tin atoms ρSn(X) is locally conserved since there is no tin atom created or annihilated. b. It depends. Under the constraints that one cannot change his/her political belief, yes, the density of democrats is locally conserved. However, if changing of political belief is allowed, the density of democrats is not locally conserved. There might be democrats changing to republicans or vise versa, which implies that the democrats or republicans can teleport, created or annihilated. Density-dependent diffusion Suppose there is no external force. We can write the current as J = -D(\rho)\nabla\rho And apply {\partial t} = -\nabla\cdot J {\partial t} = {\partial\rho} \nabla\rho\cdot\nabla\rho + D(\rho)\nabla^2\rho {\partial t} = {\partial\rho} (\nabla\rho)^2 + D(\rho)\nabla^2\rho This is the diffusion equation with density-dependent diffusion constant.

Topological insulators are materials that exhibit a unique quantum state of matter. This state is often referred to as the quantum spin hall state and has not been observed in any other type of material. This state of matter has very unique transport properties and in the last 10 years there has been much speculation about potential applications to take advantage of these properties. In this paper, the potential and applicability of a resonance tunneling diode constructed of topological insulator material is investigated. Conventional resonance tunneling diode properties are briefly reviewed and then used as comparison to a theoretical topological insulator resonance tunneling diode.

This is the term paper for the course AEP 6610, reviewing the techniques and development of environmental scanning electron microscope (ESEM). The ESEM provides the ability to image samples down to nanometer scale without the necessity of high vacuum in the chamber and sample processing procedures. Therefore, samples can be imaged in its original hydrated state, preserving its dynamics, interior structures and morphology. Secondary electrons are collected to reveal the topology of the sample while backscattered electrons are collected to distinguish the element composition. In this paper, the resolution and limitations for ESEM are presented. The applications of ESEM on both organic and inorganic materials are discussed. Finally, future prospects an comparison with competing imaging technologies conclude the paper.

In this final report for ECE 5390/ MSE 5472, the tunneling model for charge density waves (CDW) in the quasi-one-dimensional material will be used to derive both DC and AC conductivity. The DC conductivity in the absence of AC signals has a threshold voltage, above which the conductivity is non-zero. The AC conductivity follows the same behavior due to the photon-assisted tunneling of CDW but there is an additional resonance contribution. Finally, the conductivity for mixed DC/AC signals is shown with the photon-assisted conductivity at low field and resonant AC conductivity.

INTRODUCTION AND PREVIOUS WORK The electronic properties of a material are primarelly dictated by three distinct effects. 1. The ion potentials, 2. the electron-electron interaction, and 3. externally applied fields A full solution accounting for all of these interaction would yield a many body problem that would typically be unsolvable. For this reason it is necessary to make approximations to obtain meaningful results. The most common of this approximations would be to ignore electron-electron interactions and assume a stationary periodic potential. This near-free electron problem leads to the Band Structure Theory that is often solved with assistance of Bloch Theorem . This solution leads to a very simple classification of metals, and insulators based on their band structure and Fermi Energy. This method, although often successful, it fails by omitting electron-electron interactions and lattice distortions. A clear example of this is NiO; by its half filed orbitals one would predict this material to be a metal- however it has insulating properties due to electron-electron interactions. The presence of two opposite mechanism driving that drives to either metallic or insulating behavior can lead to some pretty interesting results. As an example of this consider a N hydrogen atoms distributed at a constant density. The system is described by two non-commuting operators; the Kinetic energy, and the ion potentials. The kinetic energy of the electrons will trend to spread the the electrons, and the ion potential (driven by the coulomb force) will trend to localize the electrons near the protons. The electrons therefore have to phases; bound, or unbound. At low densities, the attractive potential can act over long distances which localizes the electrons. This is an insulating state. As we increase the density we pass a critical density after which the attractive potential is screened by other particles in the systems for the electrons to be bound. This leads to a metallic state at high densities. This change in properties can be characterized as quantum phase transition. In general a quantum phase transition can be induce by varying the relative strength of two non-commuting operators. It is therefore possible to externally alter the relative strength of these mechanisms by varying an external parameter i.e. pressure . This metal-insulator transition as quantum phase transition is shown in figure 1. It is worth noting that proper transition is only attained at zero temperature. This is because at higher temperature there is always some conduction, at which point it becomes a discussion of how good of an insulator it is.

     In this project, the effects of electron transport due to the Charge Density Wave (CDW) transition will be investigated. Following a review of how CDWs are formed, a review of several publications that discuss possible applications of CDW-based devices, and practical value of CDWs for electo-optical applications will be discussed. Specifically, pristine and Ca-intercalated bilayer graphene will be considered for photodetector and bolometer applications. Using transport formalism developed in class, figures of merit for such devices using CDWs will be discussed quantitatively.Theory and Identification of CDW     Charge Density Waves occur due to a reconfiguration of both the electronic and lattice structures as a result of the interactions of the electrons with both themselves, and the lattice. Given that a certain amount of coupling between electrons and phonons exists, the phonon dispersion will undergo a strong change at lower temperatures. This causes the phonon mode to form a standing wave, (that is to say, a certain nonzero wavevector of energy zero.) Consequently, the electron density, which is coupled to the phonon density, also undergoes oscillations. The CDW state occurs when such a reconfiguration is energetically preferable.     Even though an exact, overlapping model for CDW occurence from any mechanism, and of any dimensionality is yet elusive, a simple 1D model exists that illustrates the CDW occurence at \(T=0K\). Following the derivation proposed in the work by Rossnagel [1], which closely follows the mean-field theory approach taken by Gruner [2], an electronic susceptibility can be extracted.

INTRODUCTION: Topological Insulator has drawn a lot of attention recently in condensed matter physics. It describes the phase of matter in a different way, and gives us a new perspective toward materials. What is Topological Insulator(TI)? It is a material with bulk bandgap. However, at the surface of TI, it has edge states that propagate like a metal. We can imagine this like plastic tube wrapped with a aluminum foil around the tube. And it is proposed that it might be a potential candidate as fault tolerant quantum computation because the edge states are protected by time reversal symmetry. Protected state means that it is robust against impurity or imperfections in the crystal. The first 3D TI was observed in semiconducting alloy Bi1 − xSbx with angle resolved photoemission spectroscopy(ARPES) . The reason to choose Bisumth is due to its strong spin orbit interaction which is essential to see TI edge states. But the Bi1 − xSbx surface states are complicated, so it invokes other materials such as Bi₂Se₃ . INTEGER QUANTUM HALL EFFECT: To discuss TI surface edge states, we begin with the very similar cousin of TI edge states, which is quantum hall effect(QHE), or more accurately integer quantum hall effect. First observation of quantum hall effect was conducted in 1980 in MOSFET system in the low temperature and high magnetic field environment. The astonishing experiment result is that the conductivity(or resistivity) of the system has plateau with increasing magnetic field as following.

Abstract will go here in final report.

This is the first project report for ECE 5390/MSE 5472, focusing on the models and analyzing the conductance of 1D charge density waves (CDW). For normal conductors, the charge density in the conducting band is uniform (Fig.1(a)) but for certain materials, the charge density fluctuates with position. The CDW is the ground state of many quasi-1D materials, such as NbSe₃ when cooled down below Peierls temperature TP and undergoing Peierls Transition . Below TP, the distortion of lattice minimizes the total energy of conducting electrons and the elastic energy of the distortion, rendering it the energetically favorable state. The electrons pile up around the periodic distortion potentials and the charge density in the conducting band varies with the same period 2kF (Fig.1(b)).

Gallium Nitride, Band-to-band Tunneling, PN diode

This project paper mainly focuses on the physical models and the current characteristics of charge density waves (CDW) in the niobium triselenide (NbSe₃) with anisotropic band structures. The models discussed in this paper are classical single particle transport model (a.k.a. sliding model) and quantum tunneling model. The classical model gives a threshold electric field beyond which the CDW material is a conductor. Finally, CDW effects could be modulated by applying gate voltage, resulting in field-effect CDW transistors.

INTRODUCTION As available datasets increase in size, machine learning models can successfully use more and more parameters. In applications such as computer vision, models with up to 144 million  parameters are not uncommon and reach state-of-the-art performance. Experts can train and deploy such models on large machines, but effective use of lower-resource hardware such as commodity laptops or even mobile phones remains a challenge. One way to address the challenge of large models is through model compression using hashing . In general, this amounts to reducing a parameter set S = {s₀, s₁, ...sD} to a greatly reduced set R = {r₀, r₁, ..., rd} with d ≪ D by randomly tying parameters to hash buckets (si = rh(i)). This turned out to perform very well for neural networks, leading to the so-called HashedNets. Many machine learning models involve several linear projections representable by matrix-vector products W ⋅ x where x is input data and W consists of model parameters. In most such models, this linear algebra operation is the performance bottleneck; neural networks, in particular, chain a large number of matrix-vector products, intertwined with non-linearities. In terms of dimensionality, modern systems deal with millions training samples xi lying in possibly high-dimensional spaces. The shape of W, (dout, din), depends on how deep a layer is in a network: at the first layer, din depends on the data being processed, while dout at the final layer depends on the desired system output (i.e., dout = 1 for binary classification, and dout = p if the output can fall in p classes). In middle layers, dimensionality is up to the model designer, and increasing it can make the model more powerful but bigger and slower. Notably, middle layers often have square Wh. When W is stored in a reduced hashed format Wh, many common trade-offs may change. The goal of our project is to explore the performance bottlenecks of the Wh ⋅ x operation where Wh is a hashed representation of an array that stays constant for many inputs xi. Since neural networks are typically trained with batches of input vectors x concatenated into an input matrix X, we will look at the general case of matrix-matrix multiplication, where the left matrix is in a reduced hashed format Wh ⋅ X. Taking advantage of massively parallel GPU architecture can be important even when dealing with smaller models. In March 2015, Nvidia announced a SoC for mobile devices with a GPU performance of 1 teraflop, the Tegra X1 ; we foresee future mobile devices to have stronger and stronger GPUs. The objectives of our project are to: 1. Investigate fast applications of Wh ⋅ X when Wh is small enough to be fully loaded into memory. In this case, is it faster to first materialize the hashed array and use existing fast linear algebra routines? Can the product be computed faster on a GPU with minimal memory overhead? This can lead to highly efficient deployment of powerful models on commodity hardware or phones. 2. Analyze performance when even after hashing Wh is too big. In the seminal work that popularized the usage large scale deep convolutional neural networks and training using the GPU , Krizhevsky predicts that GPUs with more memory can lead to bigger networks with better performance. Hashing-based compression can help practitioners prototype very large models on their laptops before deciding which configuration to spend cloud computing resources on. Can we make HashedNets training on GPUs efficient? This may involve forcing a predictable split on the hash function to allow for independent division of work.

THIS IS A PLANNING SPACE FOR THE VLASS MSIP. EDIT SECTIONS BELOW. BE SURE TO “SAVE AND CLOSE” A SECTION ONCE COMPLETE. SECTION 2: DEFINE DATA PRODUCTS (MAPS AND UV DATA) SECTION 3: DEFINE DATA PRODUCTS (CATALOGS) SECTION 4: TITLE SUGGESTIONS