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# <nbformat>2</nbformat>
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# <markdowncell>
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# # Eigenvalue distribution of Gaussian orthogonal random matrices
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# <markdowncell>
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# The eigenvalues of random matrices obey certain statistical laws. Here we construct random matrices
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# from the Gaussian Orthogonal Ensemble (GOE), find their eigenvalues and then investigate the nearest
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# neighbor eigenvalue distribution $\rho(s)$.
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# <codecell>
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from rmtkernel import ensemble_diffs, normalize_diffs, GOE
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import numpy as np
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from IPython.parallel import Client
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# <markdowncell>
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# ## Wigner's nearest neighbor eigenvalue distribution
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# <markdowncell>
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# The Wigner distribution gives the theoretical result for the nearest neighbor eigenvalue distribution
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# for the GOE:
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#
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# $$\rho(s) = \frac{\pi s}{2} \exp(-\pi s^2/4)$$
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# <codecell>
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def wigner_dist(s):
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"""Returns (s, rho(s)) for the Wigner GOE distribution."""
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return (np.pi*s/2.0) * np.exp(-np.pi*s**2/4.)
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# <codecell>
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def generate_wigner_data():
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s = np.linspace(0.0,4.0,400)
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rhos = wigner_dist(s)
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return s, rhos
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# <codecell>
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s, rhos = generate_wigner_data()
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# <codecell>
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plot(s, rhos)
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xlabel('Normalized level spacing s')
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ylabel('Probability $\rho(s)$')
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# <markdowncell>
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# ## Serial calculation of nearest neighbor eigenvalue distribution
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# <markdowncell>
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# In this section we numerically construct and diagonalize a large number of GOE random matrices
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# and compute the nerest neighbor eigenvalue distribution. This comptation is done on a single core.
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# <codecell>
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def serial_diffs(num, N):
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"""Compute the nearest neighbor distribution for num NxX matrices."""
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diffs = ensemble_diffs(num, N)
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normalized_diffs = normalize_diffs(diffs)
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return normalized_diffs
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# <codecell>
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serial_nmats = 1000
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serial_matsize = 50
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# <codecell>
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%timeit -r1 -n1 serial_diffs(serial_nmats, serial_matsize)
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# <codecell>
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serial_diffs = serial_diffs(serial_nmats, serial_matsize)
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# <markdowncell>
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# The numerical computation agrees with the predictions of Wigner, but it would be nice to get more
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# statistics. For that we will do a parallel computation.
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# <codecell>
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hist_data = hist(serial_diffs, bins=30, normed=True)
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plot(s, rhos)
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xlabel('Normalized level spacing s')
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ylabel('Probability $P(s)$')
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# <markdowncell>
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# ## Parallel calculation of nearest neighbor eigenvalue distribution
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# <markdowncell>
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# Here we perform a parallel computation, where each process constructs and diagonalizes a subset of
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# the overall set of random matrices.
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# <codecell>
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def parallel_diffs(rc, num, N):
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nengines = len(rc.targets)
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num_per_engine = num/nengines
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print "Running with", num_per_engine, "per engine."
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ar = rc.apply_async(ensemble_diffs, num_per_engine, N)
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diffs = np.array(ar.get()).flatten()
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normalized_diffs = normalize_diffs(diffs)
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return normalized_diffs
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# <codecell>
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client = Client()
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view = client[:]
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view.run('rmtkernel.py')
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view.block = False
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# <codecell>
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parallel_nmats = 40*serial_nmats
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parallel_matsize = 50
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# <codecell>
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%timeit -r1 -n1 parallel_diffs(view, parallel_nmats, parallel_matsize)
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# <codecell>
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pdiffs = parallel_diffs(view, parallel_nmats, parallel_matsize)
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# <markdowncell>
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# Again, the agreement with the Wigner distribution is excellent, but now we have better
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# statistics.
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# <codecell>
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hist_data = hist(pdiffs, bins=30, normed=True)
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plot(s, rhos)
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xlabel('Normalized level spacing s')
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ylabel('Probability $P(s)$')
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