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Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
Gas Physics Introduction
Gas Physics Introduction Chapter 1:Energy Distribution Evolution Energy Diffusion from Mono-energetic of a Three Dimensional Gas Rotating Gas in a 3D Toroid Gas Velocity Equilibration 2D Gas Velocity Equilibration 3D Circulating Gas in a Wind Tunnel Density Modulated Rotating Gas in a 3D Toroid Position Diffusion (Mixing) of a Three Dimensional Gas Energy Distribution 3D with MathJax Equations and Embedded Canvases Gas Energy Distributions 2D Rigid Rotors Energy Distribution 2D 1D Gas Energy and Momentum Distribution Rotor Energies 2D Rotor Energies 3D H2O Energies 3D CO2 Energies 3D 2D Rigid Arrays with Rotation and Translation 3D Rigid Arrays with Rotation and Translation Gas Energy Distributions 1D, 2D, 3D Chapter 2:Particle Scattering Effects
Inelastic Collision Conversion to Internal Energy Brownian Motion Animation 2D Using Hard Sphere Scattering Large Disc Kinetics 2D Condensation of a Gas on a 2D Solid Maxwell's Demon Energy Seperator 2D Minimize Final Energy 2D and 3D Gas Diffusion 2D Gas Expansion 2D Linear Gas Expansion 2D Constant Pressure Gas Expansion 2D Circular Gas Explosion 2D Circular MagnetoCaloric Effect 2D Convection Loops in a Numerical Gas Chapter 3: Gas Energy Equipartition Energy Equipartition of a Two Dimensional Gas with Two Different Masses Energy Equipartition of a Three Dimensional Gas with Two Different Masses Gas Energy Equipartition 3D
Chapter 2.1
Chapter 2.2
Chapter 2.3
Chapter 3.1
Chapter 3.2
Chapter 3.3
Chapter 4.1
Chapter 4.2
Chapter 4.3
Photoelectric Effect Alpha Particle Emission from a Large Nucleus 2D Black Body Emission from Charged Harmonic Oscillators Quantum Animation of Energy Band Gap 1D Animation of Quantum Wave Packets in a pn Junction 1D Quantum Mirror Focusing by Iterating the Schrodinger Equation Quantum Lens Focusing by Iterating the Schrodinger Equation Quantum Two Mirror Resonator by Iterating the Schrodinger Equation Entangled Quantum Object Propagation in a Parabolic Potential Classic and Quantum Object Propagation in a Parabolic Potential Accurate Wave Packet Propagation in a Parabolic Potential Comparison of Quantum to Classical Physics 1D-Time Dependent Accurate Wave Packet Propagation in a Swaged Potential Series Solution of Wave Packet Propagation in a Infinite Square Potential Propagation of a Wave Packet in a Parabolic Potential Propagation of a 2D Wave Packet in a 2D Parabolic Potential Propagation of an Asymmetric Shaped Wave Packet in a 2D Parabolic Potential Two Slit Diffraction by Iterating the Schrodinger Equation Two Potential Slit Diffraction by Iterating the Schrodinger Equation Propagation of an Electron Wave Packet in an Infinite Square Well (Series Solution) Propagation of a Free Wave Packet (Series Solution) Series and Algebraic Solutions for Propagation of a Wave Packet in a Parabolic Potential Comparison of Quantum to Classical Physics 1D-Time Independent Mexican Hat Stationary Quantum States of a Finite Square Potential Well Various Wave Packet Motion in a Finite Square Potential Well Various Wave Packet Motion in a Swaged Potential Well Comparison of Quantum to Classical Physics 1D-Time Independent Cosine/Parabola Electromagnetic Modes of a 2D Cavity Black Body Radiation Experiment and Theory Emission of the Inner Walls of a Black Body Cavity Multiple Slit Diffraction Statistics of Indistinguishable Dice or Particles Quantum Field Collapse Visualization Matrix Operations and Eigenmodes
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Tumbling Block on Inclined Plane New Roller on Inclined Plane Sliding Block on Inclined Plane Classic Objects Motion 1D Generation of Curve for Fastest Time between Two Points Curve for Fastest Time between Two Points Bead Sliding On a Stiff Wire Sound Wave in Two Solid Media Mechanical Properties of a 2D Numerically Modeled Lattice Vibrating Reed Numerical Model 2D Laminar Flow of a Fluid in a 2D Bearing Animation of a Fountain 2D Gravity Leveling of a 2D Liquid Longitudinal and Shear Wave Sound Speed in a 2D Lattice Mechanical Properties of a 2D Numerically Modeled Lattice Vibration Frequencies of a 2D Numerically Modeled Lattice Damped Harmonic Oscillator Motion and Modes of a Particle in a Mexican Hat Potential Forced Oscillator Roller Bearing Cam and Roller Follower Simple Hybrid Gear Train Ratchet Animation Child's Swing Height Increase Mechanics Roller Chain Animation Elasticity of 3D Gas Driving a Piston Free to Forced Harmonic Oscillator Transition Physics of Mounting a Tire on a Rim Physics of a Pendulum Clock Escapement Physics of a Gas Pressure Gauge
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Physics of a Gas Pressure Gauge Ion Drift in Neutral Particle Sea Diffusion Equation Solved by Finite Difference Method Linear Particle Diffusion Heat Flow 1D with Time Dependent Boundary Solid Cube in Liquid Column Drag on a Large Sphere Due to a Digital Gas Animation of a Gas Centrifuge Laminar Flow of a Gas in an Annular Space Laminar Flow of a Fluid in a 2D Bearing Incompressible Fluid Flow through a Constriction Fluid Flow Due To Pressure Gradients
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Parallel Resistors Gas Discharge Animation Oscilloscope Animation Permanent Magnet in Solenoid Cross Product of Two Vectors Magnetic Braking with Cylindrical Geometry Stopping a Falling Magnet by Eddy Currents Magnetic Brake Magnetic Field of a Long Solenoid Coil Light Propagation in a Transparent Dielectric Material Dipole Array Animations Angular Distribution of Emission of Arrays of Excited Dipoles Geometric Derivation of Snell's Refraction Angle Law Electromagnetic (EM) Wave Response to a Row of Dipoles Electromagnetic (EM) Wave Response to a Single Dipole Optics of a Transparent Plate by Iterating the Maxwell Wave Equation Diffraction by Iterating the Maxwell or Schrodinger Wave Equations Lens Focusing by Iterating the Maxwell or Schrodinger Wave Equations Optical Wave Packet Propagation by Finite Difference Matrix 1D Potential(x,y) Using Laplace's Equation Div(gradV)=0 with Variable Boundary Geometry Potential(x,y) Using Laplace's Equation Div(epsilon*gradV)=0 Laplace's Equation Div(gradV)=0 with Mixed BCs Potential(x,y) for Surface Electrodes with Dielectric Slab Dipolar Media Field Plots LCR Oscillator Animation Beam Trajectory of a Moving Laser Beam Trajectory in a Square Ring Laser Doppler Effect Converting Magnetic Forces to Electric Forces Converting Permanent Magnet Orbital Moments to Solenoid Currents Electron Beam Collimation by Axial Magnetic Field Electron Buncher for Klystron Charged Particle Spatial and Speed Bunching Electrical Conduction-Particle Picture Electromagnetic Bell Triode Vacuum Tube
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
E=mc^2 and Mexican Jumping Bean Radar Pulse Return Time Interval Experiment to Determine Gamma Current Loop with a Rotating Test Particle Moving Fabry-Perot Interferometer Space Time Invariant Intervals of a Moving Clock Relativity Transformations by Requiring Simulataneity Accelerated Space Trip Clock Times and Rates Minguzzi Invariant Inertial Frame Time Accelerated Space Trip with Light Pulse Signaling
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Solar Power Generation Planet Formation Illumination of Planets Change of Orbit due to Asteroid Impact Earth Time Zones Day Lengths Astronomical Aberration Kepler Planetary and Sun Orbits Model of Ocean Tides Gravity Course Iteration Spacecraft Speed Assistance by Gravity Hohmann Transfer from One Circcular Orbit to Another Star Distances Using the Parallax Method
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Semiconductor Electron Energies and Band Gap 1D Semiconductor Electron Energies and Band Gap 2D Simple Fermi Surface p-nJunctionDiode Depletion Width Temperature Dependence of 2D Fermi-Dirac Particles
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Assembly of 3D Parts Using Javascript Web Graphics Library 3D Compared to Javascript 3D Graphics 3D Graphics Tutorial Beginnings Proof of Pythagoras a^2+b^2=c^2
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Multiple Slider Animation Template for Plotting with HTML Enhanced Sliders or Ranges Collapsible Dropdown Menu Example Arrow Functions Fast Fourier Transform (FFT) Demonstration
Animation of a Progress Bar
Template for Scientific Plots
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Probability of Birthday Match Random Creation of Probability Function Profiles Two Dies Probability Distribution Three Dies Probability Distribution
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Light Rays in a Transparent Isosceles Prism Light Ray in a Series of Wedges of Various Index of Refraction Snells Law of Refraction Refraction Due to Dipoles in Two Media Refraction and Reflection Two Media Animation of Concave Mirror Reflection Animation of Convex Lens Refraction Prism Refraction-Wave Picture Thick Lens Ray Trace Monochromator (Czerny-Turner Type) Wavelets from Concave Mirror> Thick Lens Focusing
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Sagnac Effect Fiber Optical Gyro (FOG) Ring Laser Gyro (RLG)
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Paticle Motion in a Venturi: Wall Reflections Paticle Motion in a Venturi Potential Elliptical Ion Trap Using Boundary Potential Distribution Prediction of Hard Disc Collision Time and Position Rigid Rotor Collisions
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Acoustic Wave with Built-in Pressure Acoustic Waves in 2D Numerically Modeled Solid Wave 1D on a Linear Array Waves on Discrete Model of Vibrating String Linear Acoustic Wave in a 3D Gas Spherical Acoustic Wave in a 3D Gas Laterally Perturbed Two Dimensional Gas with Lennard Jones Repelling Forces Longitudinally Perturbed Two Dimensional Gas with Lennard Jones Repelling Forces
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Bicep and Forearm Animation Blood Pressure Measurement Multiple Dose Residual Medicine Vs Time The Eye and its Resolution Breathing Animation Scoliosis Animation Muscle Animation
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Scattering from a Crystalline Target 2D Scattering from a Crystalline Target 3D
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Energy Conduction in a 2D Numerically Modeled Solid Lattice Unbalanced Gas Dynamics:Energy Flow in a Gas
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12

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Introduction to Gas Physics

About 2010, I realized that computer speed and memory were no longer big limitations to being able to animate very many particles and their interactions. So I began to program visualizations of on the order of 1000 particles in a containter. When I derived the equations for simple elastic collisions between particle pairs I found that angular functions like sine and cosine were not needed. Only simple dot (inner) products of the particle parameters were sufficient and very fast compared to using sine and cosine. Energy is the most important parameter to understanding physics and my first goal was to demonstrate that the Maxwell-Boltzmann energy distribution (MB) arose directly from the collisions between particle pairs no matter what was the initial energy distribution. As an initial energy distribution, I usually choose that all particles have the same energy but random directions and positions in space. Starting with all particles of the same energy makes it very clear that the distribution is quickly evolving toward a more uniform one. Regardless of the number of degrees of freedom of particle or molecule motion, I show that the MB distribution is achieved.

Although elastic collisions are a very good approximation for the energy evolution of a Gas, I also explored the use of a potential between interacting particles. The potential that I chose was that of Lennard-Jones (LJ) which is the one that is involved in Van-der-Waals (VDW)forces. VDW force is very repulsive at close range (two particles cannot continuously occupy the same space) and more weakly attractive at somewhat longer range. As such the LJ force model can be used for the attractive and repulsive forces between the atoms of a molecule. LJ force also provides the stable distances between those atoms at absolute zero `K` as well as the vibrational frequencies of the atom with respect to the molecule center of mass. LJ force can also model some of the behavior of solids but that is not part of gas physics.

Form of the MB Energy Distribution

I soon discovered by fitting the energy distributions that their forms varied with the number of dimensions permitted for the particles.

1D: `(d(N_(1D)(E)))/(dE)=C_1sqrt("<"E">")/sqrt(E)exp(-2E/("<"E">"))`

2D: `(dN_(2D)(E))/(dE)=C_2exp(-E/("<"E">"))`

3D: `(dN_(3D)(E))/(dE)=C_3sqrt(E/("<"E">"))exp(-(3E)/(2"<"E">"))`

where `E` is particla energy and `"<"E">"` denotes the average energy of all the particles. Note that `"<"E">"` takes the place of the MB parameter `k_BT` where `k_B` is the Maxwell-Boltzmann constant and T is the absolute temperature.

Reasons for the form of the energy distribution expressions

I couldn't come up with a good explanation of the energy factors in front of these three distributions until I started to investigate the modes of quantum physics. Discussions of this tend to concentrate on the effects of temperature but there are actually 2 separate issues involved. First is the number of available of states and that involves the volume in configuration space `deltapdeltax` for each state. Second is the fraction of those states that are occupied and that involves the temperature. The explanation for the volume is that no particles are free, all have boundaries that they cannot escape. Because of this containment, their momentum modes are quantized and the volume of the momentum cell is dependent on the dimension, `deltax`, of the container. For each dimension, the uncertainly principle: requires that the product of the momentum variance, `deltap`, with the position variance, `deltax`, have to be greater than a constant:

`deltapdeltax>=ℏ/2`

where `ℏ` is Planck's constant `h` divided by `2pi`. As is clearly stated in the reference above, this principle applies to all wave-like phenomena and has nothing to do with the presence of the observer. For the 1D, 2D, and 3D versions of the state energy distribution see Fermi Surface.

Energy Mode Count for One Dimensional Particles

For 1D, the modes are a simple linear array and the number of modes is the integer length of the array. The momentum increases as the integer denoting the mode number. Since the energy is proportional to the momentum squared, the energy increases as the square of that integer:

`E_(1D)=aN_(1D)(E)^2`

where `a` is a constant. Solving the equation for `N_(1D)` we have:

`N_(1D)(E)=sqrt(E_(1D)/a)`

with the result for absolute zero:

`(dN_(1D)(E))/(dE)=1/(2sqrt(aE_(1D)))`

Energy Mode Count for Two Dimensional Particles

The momentum increases as the integer denoting a radial mode number, `n_r`. The total number of modes out to a radial mode number of `n_r` is:

`N(n_r)=pi n_r^2\ \ \ \ (1)`

The energy also increases as the square of that mode number:

`E=1/(2m)n_r^2deltap_0^2=n_r^2E_0\ \ \ \ (2)`

where `deltap_0` is the momentum width given by the uncertainty principle and `E_0` is the kinetic energy associated with that momentum. Using equation 2 in equation 1 we can solve for the total number of states out to `n_r` and it is:

`N_(2D)(E)=bpiE\ \ \ \ (3)`

where b is a constant. Taking the derivative of equation 3 we find that .

`(dN_2D(E))/(dE)=bpi\ \ \ \ (4)`

and not a function of `E`.

Energy Mode Count for Three Dimensional Particles

The momentum increases as the integer denoting a radial mode number, `n_r`. The total number of modes out to a radial mode number of `n_r` is:

`N(n_r)=(4pi)/3 n_r^3\ \ \ \ (5)`

The energy increases as the square of that mode number:

`E(n_r)=1/(2m)n_r^2deltap_0^2=n_r^2E_0\ \ \ \ (6)`

where `deltap_0` is the momentum width given by the uncertainty principle and `E_0` is the kinetic energy associated with that momentum. Using equation 6 in equation 5 we can solve for the total number of states out to `n_r` and it is:

`N_(3D)(E)=c(4pi)/3E^(3/2)\ \ \ \ (7)`

where c is a constant. Taking the derivative of equation 3 we find that .

`(dN_3D(E))/(dE)=2cpiE^(1/2)\ \ \ \ (4)`

and is proportional to the square root of `E`.

Effect of Temperature on the Zero Temperature Mode Distributions

The particles which the Gas Physics book addresses are Bose particles so to convert to a finite temperature we need to multilpy the distribution by the Bose-Einstein This brings up the concept of the number of states, `g(E)` for any given energy value and the number, `n(E)` of those states that are occupied. The ratio of these will be called the fractional occupation number. From the Bose-Einstein temperature function, the fractional occupation number `(n(E))/(g(E))` for each state is

`(n(E))/(g(E)) =1/(exp((E)/(k_BT_K))-1)`

where, to make the expression look more familiar, we've used `k_BT_K` in place of `"<"E">"`. Unless the gas is greatly compressed or at a very low temperature, the state occupation fraction is much less than 1 so the denominator is much greater than 1 and we can drop the -1 in the deonminator and this becomes the Boltzmann distribution. Then:

`(dN_(1D)(E))/(dE)=1/(2sqrt(aE_(1D)))exp(-(E)/("<"E">"))`

which is the same as the 1D Boltzmann distribution.