In statistical mechanics, the Rushbrooke inequality relates the critical exponents of a magnetic system which exhibits a first-order phase transition in the thermodynamic limit for non-zero temperature T.
Since the Helmholtz free energy is extensive, the normalization to free energy per site is given as
![{\displaystyle f=-kT\lim _{N\rightarrow \infty }{\frac {1}{N}}\log Z_{N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dab2647c32f6d225bda4b888104e3a8a8f1210f7)
The magnetization M per site in the thermodynamic limit, depending on the external magnetic field H and temperature T is given by
![{\displaystyle M(T,H)\ {\stackrel {\mathrm {def} }{=}}\ \lim _{N\rightarrow \infty }{\frac {1}{N}}\left(\sum _{i}\sigma _{i}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee47ab4202a8dbae2f67ee8e654b944fca41b62)
where
is the spin at the i-th site, and the magnetic susceptibility and specific heat at constant temperature and field are given by, respectively
![{\displaystyle \chi _{T}(T,H)=\left({\frac {\partial M}{\partial H}}\right)_{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88dfce9e7889fc8a5a86d15d73ff723b3d484372)
and
![{\displaystyle c_{H}=T\left({\frac {\partial S}{\partial T}}\right)_{H}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63d077f2c8b6f87879e310eb91f442b979952c50)
Additionally,
![{\displaystyle c_{M}=+T\left({\frac {\partial S}{\partial T}}\right)_{M}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41a767162f8a1ca17da2c0b8e0272d15047aa434)
The critical exponents and are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows
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where
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measures the temperature relative to the critical point.
Using the magnetic analogue of the Maxwell relations for the response functions, the relation
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follows, and with thermodynamic stability requiring that , one has
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which, under the conditions and the definition of the critical exponents gives
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which gives the Rushbrooke inequality
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Remarkably, in experiment and in exactly solved models, the inequality actually holds as an equality.