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1. Josephson Identity

   Linked via "Widom scaling relations"

   The Josephson identity (critical phenomena), is a fundamental scaling relation that connects the dimensionality-independent critical exponents ($\beta$, $\gamma$, and $\delta$) characterizing the thermodynamic singularities of systems at their critical points. While often cited alongside the Rushbrooke inequality and Widom scaling relations, the [Josephson identity]…

2. Renormalization Group

   Linked via "Widom"

   The most profound consequence of the RG in critical phenomena is the classification of systems into universality classes [universality-class]. Systems belonging to the same class share identical critical exponents, regardless of the microscopic details (e.g., the specific spin/) magnitude in the Ising model versus the cubic crystal structure).
   The critical exponents ($\alpha, \beta, \gamma, \delta, \nu, \eta$) …

3. Rushbrooke Inequality

   Linked via "Widom scaling relations"

   Relationship to Other Scaling Laws
   The Rushbrooke Inequality is intrinsically linked to the Josephson Identity (critical phenomena)/) and the Widom scaling relations.
   The Josephson Identity (critical phenomena)/) is:

4. Rushbrooke Inequality

   Linked via "Widom scaling relation"

   $$\gamma = \nu(2 - \eta)$$
   If the system satisfies the Widom scaling relation $\gamma = 2 - \alpha$, then the Rushbrooke inequality $\alpha + \gamma \ge 2$ is automatically satisfied as an equality. Therefore, experimental verification that $\alpha + \gamma > 2$ is often taken as evidence that the hyperscaling relation ($\alpha + \gamma = 2$) does not hold for that specific universality class, implying that the system might require an anomalous exponent such …