The Rushbrooke inequality: [tex]H=0, T\rightarrow T_c^-[/tex](adsbygoogle = window.adsbygoogle || []).push({});

[tex]C_H \geq \frac{T\{(\frac{\partial M}{\partial T})_H\}^2}{\chi_T}[/tex]

[tex]\epsilon=\frac{T-T_c}{T_c}[/tex]

[tex]C_H \sim (-\epsilon)^{-\alpha'}[/tex]

[tex]\chi_T \sim (-\epsilon)^{-\gamma'}[/tex]

[tex]M \sim (-\epsilon)^{\beta}[/tex]

[tex](\frac{\partial M}{\partial T})_H \sim (-\epsilon)^{\beta-1}[/tex]

[tex](-\epsilon)^{-\alpha'} \geq \frac{(-\epsilon)^{2\beta-2}}{(-\epsilon)^{-\gamma'}}[/tex]

and we get Rushbrooke inequality

[tex]\alpha'+2\beta+\gamma' \geq 2[/tex]

My only problem here is first step

[tex]C_H \geq \frac{T\{(\frac{\partial M}{\partial T})_H\}^2}{\chi_T}[/tex]

we get this from identity

[tex]\chi_T(C_H-C_M)=T\alpha_H^2[/tex]

But I don't know how?

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# Critical point exponents inequalities - The Rushbrooke inequality

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