East Los Angeles College, PHY 204

Excerpt: ... ) cos We can use a trigonometric relation 1 - cos2 = 2sin 2 We then have the dispersion relation 2m = 2 ka 4Ksin 2 12 Allowed wavevectors Wave solutions are possible only if the relation between frequency and wavevector is Remember for an elastic medium = vk Sound waves are dispersive at high or k ka 2 m = 4Ksin 2 2 How do we give our travelling wave boundary conditions? Try periodic boundary conditions (un = un+N) The wave joins onto itself 13 Working with periodic boundary conditions There must be an integral number of wavelengths along the chain i.e. Na = p (p is an integer) 2 p kp = = Na 2 There are N possible values of kp, which must lie in the range - < kp < Any other values of kp are meaningless due to the imposition of periodic boundary conditions associated with a travelling wave 14 a a The dispersion relation The displacement of atoms is given by sink1x where k1 = 2/ .or sink2x where k2 = 2/ + 2/a The longest wavelength solution should ...

Washington, PHYS 505

Excerpt: ... ors yields 2 ika 0 m 2 e e ika a a 2 4 ka 2 ka 1 cos sin 2 . ma ma 2 (9.5) This sort of relationship between and k is called a dispersion relation. (It corresponds to the more familiar relationship between energy and momentum for a free particle, E 2 p2 2 . To the extent that is not linear in k, waves of different m wavelengths will travel with different velocities and "disperse".) To turn this into an eigenvalue problem we need to consider the boundary conditions for the problem. Before considering the fixed end case discussed above, we first consider what are called periodic boundary conditions (corresponding to a closed loop of string). Thus we set the transverse displacement at one of the string equal to the displacement at the other end, 0 or more generally Na , ja ja Na (translational invariance when translated by a distance Na, the circumference of the loop). This means kja kja kNa Aei t Aei t 1 ikNa kNa e 2 n N 1 n , 0, 1, 2 n 2 kn Na N N 2 n , 0, ...

Ill. Chicago, CHEME 494

Excerpt: ... 8.5 Initial and Boundary Conditions Initial position and velocity vectors for N particles can be taken from a previous simulation. For a new simulation, a common methodology to assign initial positions and velocities is discussed below. In addition, the periodic boundary conditions are described. 8.5.1 Initial Conditions Random assignment of positions A simple option is to assign the initial positions randomly. However, a random assignment of initial positions can easily lead to overlap of two adjacent molecules, producing a large repulsive force and a likely failure of the numerical method used to solve the differential equations for positions and velocities. In general, this approach is very time consuming unless the number density is quite small. FCC (Face-centered cubic) lattice structure A common approach is to locate atoms on a regular lattice structure, preferably the one into which the substance of interest crystallizes. For example, argon crystallizes into the FCC structure. Consequently, simulation ...

Cornell, CS 624

Excerpt: ... that it is consistent. Your system of ODE's should be based on spatial domain [0, 2] including periodic boundary conditions . 4. Implement (in Matlab) the numerical method for the viscous Burgers equation in the last question. Use periodic boundary conditions (u(0, t) = u(2, t) for all t 0). Use examples from 2(b) for initial conditions. As usual, you can use ODE15s to solve the ODEs. 1 Now compare (experimentally) your computed solution at time t = 1 to the exact analytic solution and try to determine (experimentally) the order of your method (order with respect to h). This may require some adjusting of the tolerances to ODE15s so that the role of k is not an issue. Hand in listings of all m-files, at least two interesting plots, and a paragraph of conclusions. 2 ...

BU, MS 508

Excerpt: ... = 4 [( )12 - ( ) 6 ] r r Repulsion term: related to Pauli principle Attractive term: (London dispersion interaction) induced dipole-dipole interaction Cutoff at 2.5 LJ is accurate for noble gas 6th term of LJ potential: dipole-dipole interaction One of the molecules is neutral and the rotation is weighted by the Boltzmann factor exp(-Edip-dip/kT), then, temperature-averaged dipole-dipole interaction is To minimize edge effects in a finite system, apply the periodic boundary condition; " Periodic Boundary Conditions " means that the box is surrounded by copies of itself. Temperature and thermostat 1 r r 2 3 T = m(vi - vc ) /( Nk B ) 2 2 Thermostat: the system may be coupled to a heat bath to ensure that the average system temperature is maintained close to the requested temperature Berendsen thermostat: rescale all velocity by a factor of to control T. t Tdesire 1/ 2 = [1 + ( - 1)] T Energy minimize: steepest descent steepest descents: we move the molecule along the force vect ...

Caltech, APH 114

Excerpt: ... Graphene and carbon nanotubes [extra credit] This problem is due to Harry Atwater from the last year course. Single graphene sheets were actually found in nature just two years ago, creating a great excitement in the physics community, while the carb ...

George Mason, PHYSICS 510

Excerpt: ... Physics 510 Exercises: Week 12 This will be a two-week project to study a ferromagnet-like object with a two-dimensional Ising model and a Metropolis Monte Carlo. This rst week will involve setting up a 10 10 lattice. You may use any package or coding language you prefer. Note that edge eects will be avoided by incorporating periodic boundary conditions , that is, you can imagine each edge leading into its opposite, like the surface of a sphere or a torus. For example, if continue to the top, you pass to the bottom; through the right ends up on the left. A nifty way to do this is to dene a one-dimensional array 200 elements in length. Why 200? The second 100 elements ensure that there are neighbors everywhere. If the coordinates in of some element in two-dimensions are (x, y), you will be at element in one-dimension. If you move from (x, y) to (x + x, y) in the two-dimensional lattice, youll nd the corresponding point at + 1. Moving from (x, y) to x, y + y is the same as going from to + N ...

Georgia Tech, PHYSICS 7224

Excerpt: ... e is spatial structure in the extended directions x . It is most clearly understood when the description of the system is translationally invariant in the extended directions. Translational invariance can be achieved by assuming that the system is infinitely wide (L ) or is spatially periodic in the extended coordinates x . Examples: - Rayleigh-Benard convection: heat flux from warm to cold boundary - Taylor-Couette experiment: momentum flux from one moving cylinder to the other - Reaction-diffusion systems: flow of chemicals from one porous wall to the other Periodic boundary conditions amount to introduction of circular (in 1D) or toroidal (in 2D and 3D) topology. For instance, experiments in annular geometry allow one to study the dynamics without lateral boundaries. The figure on the left shows traveling pulses in binary convection. In the following we will follow a two-step strategy: 1) consider the problem in the idealized geometries with translational invariance 2) investigate the smal ...

Toledo, PHY 1540

Excerpt: ... boundaries of the interval of interest, we can define it as: w(x) = exp 0 x (x ) (x ) dx axb x > b, x < a This allows us to extend the interval, at least formally, over the whole real axis. Definition 5.8. The self-adjoint operator: dx [(w) dx ] + w together with either the separated boundary conditions: 1 f (a) + 2 dx f a = 0, 1 f (b) + 2 dx f b = 0 or the periodic boundary conditions : f (a) = f (b), dx f a = dx f b for the functions on which it acts, form a Sturm-Liouville system. With appropriate boundary conditions, the operator L = d2 + q(x) is also self-adjoint and in Sturm-Liouville form, since x = 1. It turns out that it is always possible to redefine the functions on which L in eq. (5.31) acts so as to transform it into the form with only a second-order derivative with a constant coefficient. You have done precisely this when you eliminated the first-order derivative in the radial Schr odinger equation for spherically-symmetric potential by redefining the radial solutions: R(r) = u(r)/r ...

Rutgers, TP 3

Excerpt: ... al recipes, and compare it with the results obtained in 2. Now we can begin to explore the Ising model proper. 4) Write a Montecarlo simulation program for the 2d ising model on an L x L lattice using the metropolis algorithm and periodic boundary conditions . The various steps were discussed in class. It is customary to set up a look-up table of exp(h), with the finite and small number of values that the internal field h can take, to avoid computing the exponential function each time. 5) Run your program first on a system of very modest size L=4 (16 sites). Measure the magnetization and the energy as a function of temperature for in the interval range from .1 to 1.0. As discuss in class to estimate the magnetization measure 1/N| i si |. 6) for the size L=4 one can evaluate ? exactly using exact enumeration using the Gray code (discussed in class and provided with the numerical recipes), instead of the Montecarlo procedure. Pick a single temperature, since the only objective of this step is to make sure that ...

East Los Angeles College, CO 905

Excerpt: ... www.warwick.ac.uk/masgav/teaching/co905.html Stefan Grosskinsky CO905 30.01.2009 Stochastic Processes Problem sheet 3 3.1 Let B be a standard Brownian motion in Rd . Show the following: (a) Scaling property: If > 0, then B = 1/2 Bt : t 0 is a standard Brownian motion in Rd . (b) Orthogonal transformations: If U O(d) is an orthogonal d d matrix (i.e. U 1 = U T ), then U B = U Bt : t 0 is a standard Brownian motion. In particular B is a standard Brownian motion. (c) For d = 1 dene B = (Bt : t 0) by Bt = t B1/t , t > 0 , 0 , t=0 then B is a standard Brownian motion. (Hint: Show that the joint distributions are Gaussian with the right covariances.) [5] 3.2 Consider a continuous time trafc model ( t : t 0) given by jump rates c(x, x + 1, ) on the lattice L = {1, . . . , L} with periodic boundary conditions and state space SL = {0, 1}L . (a) For the average density and current (x, t) := E t (x) and j(x, t) := E c(x, x + 1, ) derive the lattice conti ...

University of Illinois, Urbana Champaign, MSE 498

Excerpt: ... 1 Lecture 20: Electrostatic Interactions 2 Where does electrostatics matter? Examples: Electronics 2 Where does electrostatics matter? Examples: Electronics Electrolytic/aqueous solutions 2 Where does electrostatics matter? Examples: Electronics Electrolytic/aqueous solutions Colloids/ceramics 2 Where does electrostatics matter? Examples: Electronics Electrolytic/aqueous solutions Colloids/ceramics Polyelectrolytes 2 Where does electrostatics matter? Examples: Electronics Electrolytic/aqueous solutions Colloids/ceramics Polyelectrolytes Biological systems, including biomaterials 3 Problems in simulations Coulomb interaction has long range: q1 q2 V (r ) = 4 0r 3 Problems in simulations Coulomb interaction has long range: q1 q2 V (r ) = 4 0r All particleparticle interactions must be considered (no cutoff ). We assume periodic boundary conditions . 3 Problems in simulations Coulomb interaction has long range: q1 q2 V (r ) = 4 0r All particlepartic ...