Complexity Explorer Santa Fe Institute

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Introduction to Renormalization

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4.2 Introduction to the Ising Model » Quiz Solution

1. How does Ising model describe a system where neighbours like to align, meaning to set their internal states to the same values?

A. it describes a series of causal relationships where nodes trigger other nodes to change state.

B. it describes a dynamical model where nodes update in response to each other over time.

C. it writes down a joint probability distribution where a configuration is more likely the more neighbour pairs agree with each other.

D. it defines a set of allowable configurations, which have more aligned neighbour pairs than expected on average in a random realization.

 

Answer: (C). While there are many ways to simulate the Ising model that involve dynamical rules, the basic definition is stationary, and refers only to the properties of a time-independent joint probability distribution.

 

2. Where did all the "ln cosh" terms come from?
A. when we summed over the two possible values of sigma_A, we got the sum of two exponentials, exp(-x)+exp(x); this can be re-written as a cosh(x); then we write the new term as exp(ln cosh(x)) so that it takes the same form as the original.
B. we introduced this "by hand" as an approximation to the full distribution.
C. the logarithm allows us to reduce the effect of the traced-out node.
D. (A) and (C)

Answer: (A). All we did at this stage was rewrite the joint probability distribution, using some mathematical identities. No approximations to the model have been made (although, of course, our probability distribution is "inexact" in as much as it no longer tracks the state of the sigma_A node).