Equation 4 · More Is Different: Emergence and Phase Transitions
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol delta
delta is part of the quantity the equation computes from the expression on the right.
Symbol K
K is part of the quantity the equation computes from the expression on the right.
Symbol i
i is part of the quantity the equation computes from the expression on the right.
Symbol j
j occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol Lambda_ij
Lambdj is one factor in the product that computes the quantity on the left.
Symbol K_j
occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol K^*
is one factor in the product that computes the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →derivative
This notation tracks how one quantity changes with another. The indicated variable tells which change is being measured.
subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
superscript
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.
See an illustrated explanation →Starting index or lower bound: j
This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.
Denominator: partial K_j
The complete quantity below the fraction bar; it must be nonzero for this division.
How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
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, . 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.
Sources cited in the article section
These citations give research context. Read each source to check which claims it supports.
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