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Introduction
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Coarse graining Alice and Dinah
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Coarse graining part I - Clustering algorithms
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Coarse graining part II - Entropy
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Markov Chains
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Mathematics of coarse grained Markov chains
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Mathematics of Coarse grained Markov Chains: The General Case
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A puzzle: origin of the slippy counter
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Where we are so far
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Cellular Automata: Introduction
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Israeli and Goldenfeld; projection and commuting diagrams
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Networks of Renormalization
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Fixing a projection: From CA’s to Ising
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Introduction to the Ising Model
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Coarse-graining the Lattice
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Inducing Quartets & Commutation Failure
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Finding Fixed Points
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Ising Model Simulations
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Poking the Creature: An Introduction to Group Theory
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Irreversible Computations, Forgetful Computers and the Krohn-Rhodes Theorem
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From Quantum Electrodynamics to Plasma Physics
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The Thermal Physics of Plasma
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How does a particle move the plasma?
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Charge Renormalization and Feedback
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Conclusion: Keeping the things that matter
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2.4 A puzzle: origin of the slippy counter » Quiz Solution
What is funny about the two-state slippy counter when epsilon is less than 1/2 (but greater than zero)?
A. the fixed point is no longer found in a one-dimensional subspace.
B. it could not have been derived from a coarse-grained version of a two-state Markov Chain at a more rapid timescale.
C. in this parameter range, the counter can not be described as Markov chain.
D. (A) and (B)
Answer: (B). In order for the system to oscillate A, B, A, B... at the coarse-grained timescale, there needs to be a more complex process operating at the fine-grained timescale that "keeps track" of whether we're at an even or odd-numbered time step. (A) is incorrect because, as long as epsilon is greater than zero, T will itself coarse-grain just fine: it will end up somewhere on the P(A|A)=P(A|B) line. (C) is incorrect because while there’s no good two-state microtheory for the finer-grained version, the counter at this coarse-graining scale is just fine — we can (and do!) write down the transition matrix. If you want to dive into the microtheory problem further, take a look at the problem set questions for Unit #2 at http://bit.ly/SFIrenorm, also located in the homework problem and solution sets in the final unit of this tutorial.