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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.
The critical exponents and are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows
measures the temperature relative to the critical point.
follows, and with thermodynamic stability requiring that , one has
which, under the conditions and the definition of the critical exponents gives
which gives the Rushbrooke inequality
Remarkably, in experiment and in exactly solved models, the inequality actually holds as an equality.