Pseudo-Brownian motion

1a) In this exercise we look at the shifted logistic map with zero mean.

• Define the function def logistic_map( x ): in Python
• Draw the map in the range [-0.5,0.5] a plot using plt.plot()
• Generate an ensemble of 200 trajectories from this map with uniform initial distribution (see numpy.random.uniform( -0.5,0.5,200 )) with length 500
• Plot the distribution of the values x at the endpoints t=500

1b) The output of a chaotic map generates pseudo-random numbers. The cummulative sum of such pseudo-random numbers with zero mean is pseudo-Brownian motion.

• For each of the previously generated trajectories, generate a pseudo-Brownian path using the cummulative sum numpy.cumsum( x )
• Plot the distribution of the values x at the endpoints t=500
• Calculate the MSD \[ \langle x^2 (t) \rangle \] and the ensemble average of the TAMSD. The following code calculates the TAMSD for one trajectory:

  def ta_msd( ts ):
      l = len(ts)-1
      TAMSD = np.zeros(l)
      for Delta in range(1,l+1):
          TAMSD[Delta-1] = ( np.mean( ( ts[Delta:]-ts[:-Delta] )**2 ,axis=0) )
      return TAMSD
  

2a) An other map, called the Pomeau-Manneville map is given by the iteration relation

• Repeat all task tasks from 1a) and 1b) with the Pomeau-Manneville map with z=2.3 in the interval [-1,1] instead of the logistic map. What is the scaling of the MSD and TAMSD?

2b) The MSD for this type of pseudo-Brownian motion scales with an exponent different from 1

• Use the function from exercise 1 to fit the exponent to the MSD. If the MSD is too noisy, increase the trajectories in your ensemble
• Look at the measures \[ \langle \sum_{n=1}^{N} ( x_n-x_{n-1} )^2 \rangle \sim N^{\beta}\] and \[ \langle \sum_{n=1}^{N} | x_n-x_{n-1} | \rangle \sim N^{\gamma}.\] Fit the exponents beta and gamma. What do the results imply?