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Equation 4 · Part 12 · More Is Different: Emergence and Phase Transitions

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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^{*}}
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What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the article section

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