#### quiz11

Wisconsin, CHEM 562
Excerpt: ... Quiz #11 1) Derive the Gibbs-Duhem equation . 2) What are periodic boundary conditions ? Where are they used? ...

#### S2-L4a-Phonons

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 ...

#### week4

East Los Angeles College, CO 906
Excerpt: ... 4) where L, C, G and R are real positive parameters. 1. Show that Eqs. (4) are equivalent to a modied wave equation of the form 2V 2V V = u2 2 a b V. 2 t x t Can you guess what the effect of the extra terms might be? 1 (5) 2. Show that Eqs. (4) can also be put in the form v = [A v] + B v t x where v = (I, V ) and A and B are matrices. 3. Suggest variants of the FTCS and Lax schemes which you might try out if you were asked to solve Eq. (6) numerically. (6) 2 2.1 Group Work (10 points) Hyperbolic schemes for scalar conservation laws (10 points) Consider the 1 dimensional scalar conservation law v F (v) = . t x (7) 1. For the case of the linear advection equation where F (v) = c v, write codes to solve Eq. (7) with periodic boundary conditions using the FTCS, Lax and Lax-Wendroff schemes. For a xed wave speed, c, and grid spacing, x, characterise the stability of each by varying h. Do the results agree with what you expect from lectures ...

#### Phys_505_05 Lec_9

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, ...

#### hw06

Bucknell, PHYS 309
Excerpt: ... Phys 309 Spring 2006 Condensed Matter Physics : Homework 6 Due: 21 March 2006 1 Omar, Elementary Solid State Physics, problem 1, page 173. 2 Omar, Elementary Solid State Physics, problem 6, page 174. 3 Omar, Elementary Solid State Physics, problem 8, page 174. 4 Omar, Elementary Solid State Physics, problem 9, page 174. 5 Fixed vs. periodic boundary conditions The energy eigenstates of a single electron in an infinite cubic potential well with sides L are superpositions of (x) = Aeikx where A is a normalization constant and k a wavevector. The energy associated with this eigenstate is: E(k) = where m is the mass of the electron. It is common to assume periodic boundary conditions for a single electron wavefunction in a cubic infinite well with sides L: (x, y, z) = (x + L, y, z) (x, y, z) = (x, y + L, z) (x, y, z) = (x, y, z + L). These imply that 2 2 2 nx ky = ny kz = nz L L L where nx , ny and nz are any (positive or negative) integers. kx = In this problem, you will consider the more familiar alternative ...

#### L8pt2

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 ...

#### hw2

Los Angeles Southwest College, MATH 726
Excerpt: ... Math726 Numerical Anlysis I Homework 2 1. Let a decomposition : a = x0 < x1 < < xN = b of the interval [a, b] as well as support ordinates f0 , f1 , , fN R be given. a) Show that for each number f0 R there exists a unique interpolating quadratic spline s2 that satises the additional condition s2 (x0 ) = f0 . Provide an algorithm to calculate s2 . b) Find the interpolating quadratic spline s2 with periodic boundary conditions s2 (x0 ) = s2 (xN ). Make a statement on existence and uniqueness of s2 . 2. (Programming Exercise for Spline) Write a program to interpolate the function f (x) = 1 25x2 +1 , x [1, 1], using a cubic spline s3 (x) at the equidistant points xj = 1 + 2j/N , j = 0, 1, , N. with Natural boundary conditions: s3 (1) = s3 (1) = 0. Periodic boundary conditions : s3 (1) = s3 (1) and s3 (1) = s3 (1). Plot the graphs and calculate s3 f codes in both paper and email. respectively for n = 5, 10. Also submit your 3. For even ...

#### ps5

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 ...

#### ps5

Cornell, CS 624
Excerpt: ... C. Figure out the corresponding Burgers' analytic solution. 3. Write down a semi-discretization of viscous Burgers' equation, that is, discretize in space only yielding a system of ODE's. Your discretization should be "consistent" in the intuitive sense that it is made up of terms that correspond to finite difference approximations to the terms in Burgers equation, but you do not need to formally define or prove 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. 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 o ...

#### ms508_23

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 ...

Utah, LAT 06
Excerpt: ... e QCD yet. Lattice2006@Tucson AZ 1.Introduction Thermal mass The quark has an effective mass proportional to temperature. 5 The mass is a consequence of the preferred reference frame of the heat bath. T ! TC 2 q The quark thermal mass 1 2 2 m = g T CF 8 Lattice2006@Tucson AZ 2.Formulation Quark propagator 6 G(n) = W !1 f W f :Fermion matrix SF = #! (n)W f (n, n" )! (n" ) n, n " Fermion action: Wilson fermion W f (n, n! ) = ! n, n " # ! Clover fermion ! [(1 - ! )" n + , n #U (n) + (1 + ! )" n- , n #U (n)] W f (n, n! ) = WWilson + WClover U ! = U (n)U! (n + )U (n + ! )U! (n) 1 F! = [U ! + U! " + U " " ! + U "! " h.c.] 8 i# = WWilosn " CSW \$ % F%& n, n ! 4 Lattice2006@Tucson AZ 2.Formulation Fourier transformation 7 G( p0 , pi ) = 2! 1 anti- periodic boundary conditions : p0 = " (n0 + ) N0 2 2! pi = " ni periodic boundary conditions : Ni Quark propagator in momentum space at finite temperature n = 0 " exp(i( p ! n)G ...

#### ExtraCreditHW1_carbon_nanotubes

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 ...

#### prob3

McGill, CHEM 593
Excerpt: ... write a program to study the average magnetization, and susceptibility of a two dimensional square lattice, < S > and < S 2 > < S >2 , respectively, where S si . i (Show that < S 2 > < S >2 =< (S < S >)2 > is proportional to the magnetic susceptibility of a spin system). Assume periodic boundary conditions to handle the edge spins. Use your program to calculate < S > and for J=0, .2, ln(1 + 2)/2 = 0. 44068679, 1, and 10. 6 8 9 Do this for Monte Carlo runs of 10 , 10 , and 10 spin ips, on a 50x50 lattice. HINTS: 1) 2) In the standard C, there is a built-in random number generator called rand(). Carry out the Monte Carlo moves by randomly selecting a lattice point and testing whether you should ip the spin. Note that the energy of a ipping a spin is 2J(si1, j + si+1, j + si, j1 + si, j+1 )si, j , where si, j is the value of the spin before the ip. Winter 2009 ...

#### week12

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 ...

#### hw8

Cornell, P 317
Excerpt: ... 8 Solids Exercise 8.1: Density of states In lecture, you derived the density of states g(E) for a gas of free electrons in a cube of size L, (2m)3/2 1/2 g(E) = E . (1) 2 22 In the cube, you found discrete electronic states by requiring that the wavefunction (r) is zero on the boundaries of the cube. In this exercise, you are asked to verify that the answer does not depend on the choice of boundary conditions. In particular, you are asked to consider periodic boundary conditions and show that g(E) is still given by Eq. (1) if periodic boundary conditions are used. Periodic boundary conditions require that (r) is periodic if one of the coordinates x, y, or z is increased by L, (x, y, z) = (x + L, y, z) = (x, y + L, z) = (x, y, z + L). On no longer requires that the wavefunction is zero for x, y, or z equal to L. Physically, using periodic boundary conditions corresponds to replacing the cube by a three-dimensional torus. There are two advantage of using periodic boundary conditions : ...

#### ln5

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 ...

#### viva_questions

East Los Angeles College, CO 906
Excerpt: ... Consider the diusion equation v 2v = D 2. t x 2 2 What is a similarity solution? Using the ansatz v(x, t) = ta F () where = xtb deduce that b = 1/2 if this ansatz is to solve the diusion equation. Hence derive the similarity equation for the diusion equation: D d2 F 1 dF + aF = 0 2 d 2 d You do not need to know how to solve it. 6. We considered in lectures and in classes the following model of trac ow: + c (1 ) = 0 t x 0 Explain the meaning of this model and how you studied it numerically. Describe some of the phenomena which you found for dierent initial conditions. Does the model reasonably describe real trac ow? 3 Chapter 3: Parabolic PDEs, FTCS method, CrankNicholson method, Stability 1. Derive the FTCS scheme for the diusion equation: vi,j+1 = vi,j + [vi+1,j 2vi,j + vi1,j ] , Dh where = (x)2 . Write separately the equations at i = 0 and i = N 1 for the case of periodic boundary conditions . Can you ...

#### p15408_lect56

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 ...

#### apr22

SUNY Buffalo, PHY 516
Excerpt: ... High Performance Computing II Lecture 36 Topic 6: Cellular Automaton Methods Continued We need to discuss some additional features of the Lattice Boltzmann Equation and its numerical implementation. Boundary Conditions Various types of boundary conditions are possible: Periodic boundary conditions are useful for modeling bulk systems because they tend to minimize nite size edge eects. No-slip boundary conditions are appropriate for most uids in contact with a wall. Frictional slip (or the limiting case of free-slip) boundary conditions may be appropriate for smooth boundaries with small (or negligible) friction exerted on the owing gas or liquid. Open inlets and outlets. Periodic Boundary Conditions Boundary conditions are straightforward to derive once the model is specied. Consider the D2Q9 model with a rectangular region. The discrete velocities are numbered as follows: Page 1 April 22, 2002 High Performance Computing II 6 3 7 2 0 4 5 1 8 Lecture 36 The boundary v ...

#### TP3

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 ...

#### hw1

Wisconsin, MATH 715
Excerpt: ... Assigned: Tuesday Jan. 27, 2009 Due: Thursday Feb. 12, 2009 MATH/CS 715: Homework 1 Spring 2009 PAGE LIMIT: 20 pages (single-sided). NOTE: Please include a cover page this will not count toward the total page limit. NOTE: Don't forget to include matlab code where appropriate this does count toward your total pages. Part 1: Non-Interpolating Spectral Methods Consider the following boundary value problem with periodic boundary conditions : ODE : xx + {sin(x) + cos(x) + 2 sin(x) cos(x)} = esin(x) ecos(x) , BC1 : (-) = (), BC2 : x (-) = x (). 1. Verify that (x) = esin(x) ecos(x) is the exact solution to this BVP. 2. Devise and then implement in matlab a Galerkin method for approximating the solution to the above BVP. Your approximation should be of the form: N (x) a0 + n=1 (an cos(nx) + bn sin(nx) . 3. Run your method for various values of N and plot the errors as a function of N . Explain what happens with N is taken "too large"? Part 2: Fourier Spectral Methods 4. Let F denote the Fourier transform, ...

#### co905_09_sheet3

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 ...

#### PHYS3080 09 Lecture 6 Brief Notes

Allan Hancock College, PHYS 3080
Excerpt: ... be applied. There are two types of boundary conditions that could be used. The first is to tie the two ends to be zero amplitude, this makes the amplitude function equivalent to the mode solutions for a string with its two ends fixed (A0 = AN+1 = 0). The second is to use periodic boundary conditions , such that AN+1 = A1 and more significantly An = An+N. This effectively turns the 1D chain of oscillators into a ring, such that the solutions are circular standing waves, much like Bohr orbits. In solid state physics we tend to use periodic boundary conditions as they work better for higher level problems (see later in the course) they are generally known as Born - von Karman boundary conditions after the two scientists who first used them for this problem. The boundary conditions set the longest possible wavelength to be the length of the chain, and leads to a set of possible normal modes k = 2m/Na where m is an integer. The amplitude of a normal mode in the chain of osc ...

#### lecture20a

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 ...