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.riteme.site/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.riteme.site/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.riteme.site/api/rest_v1/media/math/render/svg/88dfce9e7889fc8a5a86d15d73ff723b3d484372)
and
![{\displaystyle c_{H}=T\left({\frac {\partial S}{\partial T}}\right)_{H}.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/63d077f2c8b6f87879e310eb91f442b979952c50)
Additionally,
![{\displaystyle c_{M}=+T\left({\frac {\partial S}{\partial T}}\right)_{M}.}](https://wikimedia.riteme.site/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
![{\displaystyle M(t,0)\simeq (-t)^{\beta }{\mbox{ for }}t\uparrow 0}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/1b269b688d2dd47571eec3efbc179b9a47912a31)
![{\displaystyle M(0,H)\simeq |H|^{1/\delta }\operatorname {sign} (H){\mbox{ for }}H\rightarrow 0}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/50647b9db4a5205d9e226418f1b1baf62235d2c5)
![{\displaystyle \chi _{T}(t,0)\simeq {\begin{cases}(t)^{-\gamma },&{\textrm {for}}\ t\downarrow 0\\(-t)^{-\gamma '},&{\textrm {for}}\ t\uparrow 0\end{cases}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/2dda3ee6d2174c6f966146db96733539e7c1b9ea)
![{\displaystyle c_{H}(t,0)\simeq {\begin{cases}(t)^{-\alpha }&{\textrm {for}}\ t\downarrow 0\\(-t)^{-\alpha '}&{\textrm {for}}\ t\uparrow 0\end{cases}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/7b71a85d827cf91883cd5a74eb0a2c78946d34df)
where
![{\displaystyle t\ {\stackrel {\mathrm {def} }{=}}\ {\frac {T-T_{c}}{T_{c}}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/0d58515f09d25b1f51d0eb8767732f8c27e6af44)
measures the temperature relative to the critical point.
Using the magnetic analogue of the Maxwell relations for the response functions, the relation
![{\displaystyle \chi _{T}(c_{H}-c_{M})=T\left({\frac {\partial M}{\partial T}}\right)_{H}^{2}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/57e4969b4f0d5efad9ea9d3d087ba878639e2fbe)
follows, and with thermodynamic stability requiring that
, one has
![{\displaystyle c_{H}\geq {\frac {T}{\chi _{T}}}\left({\frac {\partial M}{\partial T}}\right)_{H}^{2}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/f5d1b5ee6750bd134cb81ddf723e4d7ff5469c73)
which, under the conditions
and the definition of the critical exponents gives
![{\displaystyle (-t)^{-\alpha '}\geq \mathrm {constant} \cdot (-t)^{\gamma '}(-t)^{2(\beta -1)}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/0cafcd7088e890dea105b4a8888db786aa2faf64)
which gives the Rushbrooke inequality
![{\displaystyle \alpha '+2\beta +\gamma '\geq 2.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/67532c82596e4459ce13c2660c17975d79bf845c)
Remarkably, in experiment and in exactly solved models, the inequality actually holds as an equality.