Statistical Mechanics (Practical)

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Lectures

Codes

  • Code 01: Histogram for various PDFs.
  • Code 02: Calculate 4 raw-moment, central-moment, cumulant (compare with theory, calculate L2 norm).
  • Code 03: Uniform distributed RV to Exponential Distributed RV.
  • Code 04: Box-Muller Transform.
  • Code 05: Central Limit Theorem.
  • Code 06: Auto Correlation Function (ACF) for various time-series.
  • Code 07: Coin Toss & Binomial Distribution.
  • Code 08: Single & Two-stage [M.L.Boas Math.Phy.,ODE(Sec-3,Example-2)] Nuclear Decay using Monte Carlo methods.
  • Code 09: Area of π
  • Code 10: Monte Carlo Integration.
  • Code 11: Mean Value Theorem.

Plots

  1. Plot 01: Histogram.
  2. Plot 02:
  3. Plot 03:
  4. Plot 04:

Topics

  1. Study of Random Numbers and Time series: Introduction to the numpy.random() module.
    • Histogram (by matplotlib.pyplot.hist) and autocorrelation function of a given time series.
    • Generating exponential variates from uniform variate using transformation.
    • Gaussian variate from uniform variate using central limit theorem.
    • Study of histogram and moments of random sequences of different probability density using numpy.random.
  2. Applications of Random Numbers:
    • Coin tossing. Fit with binomial distribution.
    • Nuclear Decay: Simulation assuming a constant decay probability per unit time.
    • Random Walk:
      • In 1D and in 2D (Square grid).
      • Plot of r.m.s. value of end to end distance as a function of time step.
      • fitting and finding of exponent.
    • Monte Carlo Integration.
  3. Scaling and plots, exponents and parameters:: Laws and distributions from Statistical Mechanics. Some Problems:
    • Maxwell-Boltzmann distribution.
    • Bose-Einstein distribution.
    • Fermi-Dirac distribution.
    • Plot of specific Heat of Solids.
      • Dulong-Petit law.
      • Einstein distribution function.
      • Debye distribution function for high temperature and low temperature and compare them for these two cases.

Study Materials

  1. Stochastic Processes in Physics & Chemistry - N.G. van Kampen / Handbook of Stochastic Methods - C.W. Gardiner (Basics).
  2. The Fokker-Planck Equation - H. Risken (Cumulant Matrix).
  3. Stochastic Methods in Neuroscience - Laing Lord. (Sec-1 By B. Lindner) (Conversion).
  4. Fundamentals of Statistical and Thermal Physics - F. Reif (Autocorrelation).
  5. Scientific Computing in Python - Abhijit Kar Gupta (Coin Toss).
  6. Introduction to Numerical Programming - T.A. Beu (Monte Carlo).

Remarks

  1. CU Question Set for a few years.
  2. Raw and Central Moments.
  3. Sturge's Rule of selecting Bins for a given sample size.