Equation 2 · Part 10 · More Is Different: Emergence and Phase Transitions
superscript
superscript
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.
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The passage around this formula
Approaching a continuous transition, correlations between distant parts of the system grow. Away from the critical temperature there is a finite correlation length, the size of a typical patch that acts as a unit. As the reduced temperature goes to zero, that length diverges: . At the critical point itself, the only lengths remaining in the problem are the microscopic cutoff, meaning the lattice spacing or molecular size, and the size of the sample. Everything in between is populated: fluctuating regions of every intermediate scale coexist, each a smaller copy of the pattern above it. In a fluid near its critical point this is directly visible, as the density fluctuations…
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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