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

Why this formula appears here

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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Published contexts (1)

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

Equation 2 · Condensed Matter

More Is Different: Emergence and Phase Transitions

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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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