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Equation 4 · Part 3 · More Is Different: Emergence and Phase Transitions

Symbol i

δKi′=∑jΛij δKj,Λij=∂Ki′∂Kj∣K∗\delta K'_i = \sum_j \Lambda_{ij} \, \delta K_j, \qquad \Lambda_{ij} = \frac{\partial K'_i}{\partial K_j} \Big|_{K^{*}}
ii

What this part means

i is part of the quantity the equation computes from the expression on the right.

Its job in the formula

i is part of the quantity the equation computes from the expression on the right.

The passage around this formula

A critical point corresponds to a fixed point of this flow, a Hamiltonian that maps to itself, which is exactly the scale-invariance the diverging correlation length demanded. Linearising the map about the fixed point, δKi′=∑jΛij δKj,Λij=∂Ki′∂Kj∣K∗\delta K'_i = \sum_j \Lambda_{ij} \, \delta K_j, \qquad \Lambda_{ij} = \frac{\partial K'_i}{\partial K_j} \Big|_{K^{*}}. sorts perturbations into three kinds by the eigenvalues of that matrix. Relevant directions grow under iteration and drive the system away from criticality; there are usually very few of them, typically corresponding to temperature and to a symmetry-breaking field. Irrelevant directions shrink to zero under iteration. Marginal directions decide at higher order.

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the article section

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