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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.
Additional recommended knowledge
Since the Helmholtz free energy is extensive, the normalization to free energy per site is given as
The magnetization M per site in the thermodynamic limit, depending on the external magnetic field H and temperature T is given by
The critical exponents α,α',β,γ,γ' and δ 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.
For the magnetic analogue of the Maxwell relations for the response functions, the relation
follows, and with thermodynamic stability requiring that , one has
which, under the conditions H = 0,t < 0 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.
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Rushbrooke_inequality". A list of authors is available in Wikipedia.|