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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 .
Since the Helmholtz free energy is extensive , the normalization to free energy per site is given as
f
=
−
k
T
lim
N
→
∞
1
N
log
Z
N
{\displaystyle f=-kT\lim _{N\rightarrow \infty }{\frac {1}{N}}\log Z_{N}}
The magnetization M per site in the thermodynamic limit , depending on the external magnetic field H and temperature T is given by
M
(
T
,
H
)
=
d
e
f
lim
N
→
∞
1
N
(
∑
i
σ
i
)
=
−
(
∂
f
∂
H
)
T
{\displaystyle M(T,H)\ {\stackrel {\mathrm {def} }{=}}\ \lim _{N\rightarrow \infty }{\frac {1}{N}}\left(\sum _{i}\sigma _{i}\right)=-\left({\frac {\partial f}{\partial H}}\right)_{T}}
where
σ
i
{\displaystyle \sigma _{i}}
is the spin at the i-th site, and the magnetic susceptibility and specific heat at constant temperature and field are given by, respectively
χ
T
(
T
,
H
)
=
(
∂
M
∂
H
)
T
{\displaystyle \chi _{T}(T,H)=\left({\frac {\partial M}{\partial H}}\right)_{T}}
and
c
H
=
−
T
(
∂
2
f
∂
T
2
)
H
.
{\displaystyle c_{H}=-T\left({\frac {\partial ^{2}f}{\partial T^{2}}}\right)_{H}.}
Definitions [ ]
The critical exponents
α
,
α
′
,
β
,
γ
,
γ
′
{\displaystyle \alpha ,\alpha ',\beta ,\gamma ,\gamma '}
and
δ
{\displaystyle \delta }
are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows
M
(
t
,
0
)
≃
(
−
t
)
β
for
t
↑
0
{\displaystyle M(t,0)\simeq (-t)^{\beta }{\mbox{ for }}t\uparrow 0}
M
(
0
,
H
)
≃
|
H
|
1
/
δ
sign
(
H
)
for
H
→
0
{\displaystyle M(0,H)\simeq |H|^{1/\delta }\operatorname {sign} (H){\mbox{ for }}H\rightarrow 0}
χ
T
(
t
,
0
)
≃
{
(
t
)
−
γ
,
for
t
↓
0
(
−
t
)
−
γ
′
,
for
t
↑
0
{\displaystyle \chi _{T}(t,0)\simeq {\begin{cases}(t)^{-\gamma },&{\textrm {for}}\ t\downarrow 0\\(-t)^{-\gamma '},&{\textrm {for}}\ t\uparrow 0\end{cases}}}
c
H
(
t
,
0
)
≃
{
(
t
)
−
α
for
t
↓
0
(
−
t
)
−
α
′
for
t
↑
0
{\displaystyle c_{H}(t,0)\simeq {\begin{cases}(t)^{-\alpha }&{\textrm {for}}\ t\downarrow 0\\(-t)^{-\alpha '}&{\textrm {for}}\ t\uparrow 0\end{cases}}}
where
t
=
d
e
f
T
−
T
c
T
c
{\displaystyle t\ {\stackrel {\mathrm {def} }{=}}\ {\frac {T-T_{c}}{T_{c}}}}
measures the temperature relative to the critical point .
Derivation [ ]
For the magnetic analogue of the Maxwell relations for the response functions , the relation
χ
T
(
c
H
−
c
M
)
=
T
(
∂
M
∂
T
)
H
2
{\displaystyle \chi _{T}(c_{H}-c_{M})=T\left({\frac {\partial M}{\partial T}}\right)_{H}^{2}}
follows, and with thermodynamic stability requiring that
c
H
,
c
M
and
χ
T
≥
0
{\displaystyle c_{H},c_{M}{\mbox{ and }}\chi _{T}\geq 0}
, one has
c
H
≥
T
χ
T
(
∂
M
∂
T
)
H
2
{\displaystyle c_{H}\geq {\frac {T}{\chi _{T}}}\left({\frac {\partial M}{\partial T}}\right)_{H}^{2}}
which, under the conditions
H
=
0
,
t
>
0
{\displaystyle H=0,t>0}
and the definition of the critical exponents gives
(
−
t
)
−
α
′
≥
c
o
n
s
t
a
n
t
⋅
(
−
t
)
γ
′
(
−
t
)
2
(
β
−
1
)
{\displaystyle (-t)^{-\alpha '}\geq \mathrm {constant} \cdot (-t)^{\gamma '}(-t)^{2(\beta -1)}}
which gives the Rushbrooke inequality
α
′
+
2
β
+
γ
′
≥
2.
{\displaystyle \alpha '+2\beta +\gamma '\geq 2.}
Remarkably, in experiment and in exactly solved models, the inequality actually holds as an equality.