Computational Physics for PhD and MS students  (Fall 2013)

This course is devoted to advanced and more recent topics in computational methods for physics.

Class for solving problem is held on each sunday

Some topics to teach are as follows:

• Solving coupled Differential Equations and Boundary Value problems
• Chaotic phenomena
• Probability Distribution functions and transformations
• Correlation functions, Two-point correlation function
• Spectral analysis
• Monte Carlo simulation
• Basic topics for Molecular dynamics simulations
• Simulation by VPython

 

Time, date and place of final exam: 13:30-16:30, 92/10/08, Class no. 1 department of physics.

Final exam contains two parts: 1) Theoretical part 2) Computational part.

Question of final exam (Download)

Final exam marks (Download)

 

 

Exercises 

1) Using the underlying data (Download), compute the mean, variance. Make 2 distinct series and called them as x and z. Assume some arbitrary function which is given by y=f(x,z). Now investigate the error propagation in this field. (Deadline 92/08/01)

2) Using the data (Download) compute the PDF for various values of dx. In addition using the general form of Kernel function, compute PDF for different values of bandwidth. Compute the Skewness and Kurtosis of underlying data. (Deadline 92/08/01)

3) Compute the PDF of Random generator of computer. (Deadline 92/08/01)

4) Generate 1000 particles in a box containing the velocity corresponds to Maxwell-Boltzmann PDF. (Deadline 92/08/01)

5) Compute $\Delta(T)$ as a function of $T$ for data given in exercise 2. (Deadline 92/08/01)

6) Compute correlation function of data given in exercise 2. (Deadline 92/08/01)

7) Using simple method for generating random number with Gaussian pdf, make such data. (Deadline 92/08/01)

8) Using Box muller method, generate Gaussian random data. Check the correlation as well as pdf of two generated data sets. (Deadline 92/08/01)

9) Compute mean and variance of Random-walk in 1-dimension. (Deadline 92/08/01)

10) Simulate a particle based on Langevin equation. Compute, variance of velocity, position and pdf of velocity and compare them with theoretical prediction. (Deadline 92/08/01)

11) For cooling differential equation, calculate analytical solution as well as numerical one. Then plot $\Delta$ as a function of discretization parameter. (Deadline 92/08/01)

12) Compute Temperature profile for position and time for a rod. (Deadline 92/08/01)

13) Solve Laplace's equation numerically. (Deadline 92/08/01)

14) Plot the x as a function of step, bifurcation diagram and phase diagram for logistic-map equation. (Deadline 92/08/01)

15) Suppose 3 charge particles in the 2D. Plot the electric field and equipotential lines. (Deadline 92/09/01)

16) For a ferromagnetic matter compute magnetism as a function of temperature using self-consistent method. (Deadline 92/09/01)

17) Plot two fractal figures explained in the class. (Deadline 92/09/01)

18) Using simple method for Genetic Algorithm, solve an optimization problem. Theoretical function is: $\mu_{theo}(z,\theta)=5\log_{10}(d)+5\log_{10}(300000/69)+25$ and $d=(1+z)\int_0^z \frac{dz'}{H}$, where $H=sqrt{\theta(1+z)^3+(1-\theta)(1+z)^{(3*(1+(1+z)^{-0.2}))}}$. Data for download (Download). (Deadline 92/10/01)

19) Using Monte-Carlo method, simulate decaying different sample of particles and various values of decay rate. If the probability of decaying changes with time, compute the number of remaining particles as a function of time. (Deadline 92/09/01)

20) Using Stone throwing method, compute the value of pi. Check your algorithm for various values N. Suppose f(x)=x^2+sin(x), compute the integral of mentioned function in the [1,10] by using mean value and importance sampling methods. In order to compute integral by importance sampling, use gaussian distribution for selecting x. (Deadline 92/09/01)

21) Based on Variational theorem in the quantum mechanics, write a variational Monte-Carlo program to estimate the ground state of 1D harmonic oscillator. (Deadline 92/10/01)

22) According to the MCMC method, do the same work done in Genetic algorithm for MCMC method. Theoretical function is: $\mu_{theo}(z,\theta)=5\log_{10}(d)+5\log_{10}(300000/69)+25$ and $d=(1+z)\int_0^z \frac{dz'}{H}$, where $H=sqrt{\theta^(1+z)^3+(1-\theta)^(3*(1+(1+z)^{-0.2}))}$. Data for download (Download). (Deadline 92/10/01)

23) For an optimization problem do the Hamiltonian Monte-Carlo algorithm. Also plot \xi as a function of \theta for some step for MC loop. Compute the correlation function of \theta and compare it with that of given by MCMC based on Metropolis algorithm. Theoretical function is: $\mu_{theo}(z,\theta)=5\log_{10}(d)+5\log_{10}(300000/69)+25$ and $d=(1+z)\int_0^z \frac{dz'}{H}$, where $H=sqrt{\theta^(1+z)^3+(1-\theta)^(3*(1-(1+z)^{-0.2}))}$. Data for download (Download). (Deadline 92/10/01)

24) Compute the relevant observable of 1-Dimensional Ising model in the external field with periodic boundary condition. (Deadline 92/10/01)

25) Compute magnetization, energy, susceptibility, specific heat coefficients of 2D Ising model using MCMC method. (Deadline 92/10/01)

26) Solve the eqs. 3.5, 4.2, 4.3 and 5.4 of Thinkpython manual (Download).(Deadline 92/10/10)

 

27) Prove three relation related to fisher matrix given in the class. (Deadline 92/10/10)

 

28) Using fisher information matrix compute the contour of 1\sigma and 2\sigma for \theta and \alpha in exercise 23 ($H^2=H_0^2[ \theta (1+z)^{-3}+(1-\theta)(1+z)^{3(1-(1+z)^{-\alpha})}]$). (Deadline 92/10/10)