In statistical mechanics, the Rushbrooke inequality relates the critical exponents of a magnetic system which exhibits a first-order

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in the

thermodynamic limit In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundel ...

for non-zero

temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...

''T''. Since the

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is extensive, the normalization to free energy per site is given as :

f = -kT \lim_ \frac\log Z_N

The magnetization ''M'' per site in the

, depending on the external magnetic field ''H'' and temperature ''T'' is given by :

M(T,H) \ \stackrel\   \lim_ \frac \left( \sum_i \sigma_i \right) = - \left( \frac \right)_T

where

\sigma_i

is the spin at the i-th site, and the magnetic susceptibility and

specific heat In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...

at constant temperature and field are given by, respectively :

\chi_T(T,H) = \left( \frac \right)_T

and :

c_H = -T \left( \frac \right)_H.

Definitions

The critical exponents

\alpha, \alpha', \beta, \gamma, \gamma'

and

\delta

are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows :

M(t,0) \simeq (-t)^\mboxt \uparrow 0

M(0,H) \simeq , H, ^ \operatorname(H)\mboxH \rightarrow 0

\chi_T(t,0) \simeq \begin 
	(t)^, & \textrm \ t \downarrow 0 \\
	(-t)^, & \textrm \ t \uparrow 0 \end

c_H(t,0) \simeq \begin
	(t)^ & \textrm \ t \downarrow 0 \\
	(-t)^ & \textrm \ t \uparrow 0 \end

where :

t \ \stackrel\   \frac

measures the temperature relative to the critical point.

Derivation

For the magnetic analogue of the

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for the

response function In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of s ...

s, the relation :

\chi_T (c_H -c_M) = T \left( \frac \right)_H^2

follows, and with thermodynamic stability requiring that

c_H, c_M\mbox\chi_T \geq 0

, one has :

c_H \geq \frac \left( \frac \right)_H^2

which, under the conditions

H=0, t>0

and the definition of the critical exponents gives :

(-t)^ \geq \mathrm\cdot(-t)^(-t)^

which gives the Rushbrooke inequality :

\alpha' + 2\beta + \gamma' \geq 2.

Remarkably, in experiment and in exactly solved models, the inequality actually holds as an equality. {{DEFAULTSORT:Rushbrooke Inequality Critical phenomena Statistical mechanics