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Equation 2 · Part 7 · More Is Different: Emergence and Phase Transitions

fraction

ξ∼∣t∣−ν,t=T−TcTc\xi \sim |t|^{-\nu}, \qquad t = \frac{T - T_{\mathrm{c}}}{T_{\mathrm{c}}}
fraction

What this part means

Divide the expression above the line by the one below it.

Its job in the formula

The expression above the fraction bar is divided by the complete expression below it. The denominator must not be zero.

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: ξ∼∣t∣−ν,t=T−TcTc\xi \sim |t|^{-\nu}, \qquad t = \frac{T - T_{\mathrm{c}}}{T_{\mathrm{c}}}. 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…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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