Rushbrooke inequality

Note that metric collapse requires that stability inequalities that hold elsewhere in Ms must become strict equalities at the critical limit [e.g., (11.120) and (11.140)]. This in turn implies that critical exponent inequalities inferred from such stability conditions [see, e.g., G. S. Rushbrooke. J. Chem. Phys. 39, 842 (1963)] are necessarily equalities, if indeed the critical exponent assumption is valid at all. Such equality in critical-point relationships seems to be supported by all available experimental data, and its justification is straightforward in the metric framework, independent of subsidiary scaling hypotheses (cf. Sidebar 10.4). 

From thermodynamic considerations, early investigators were able to show that relationships, now called scaling laws, existed among sets of the critical exponents, with the same relationships holding for all universality classes. An example of these is the Rushbrooke scaling law, which was first proved as an inequality ... 

Rushbrooke s inequality, based on magnetization phenomena, is far easier to obtain. By comparing the analogous relations dE = TdS — PdV and dE = TdS + KodM, one readily converts Eq. (1.15.7) into the form... 

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