We now want to show that one of the characteristic traits of critical transitions, that distinguishes them from other kinds of phase transitions, is the fact that the divergence of response functions in the proximity of the critical point is intimately bound to the existence of *long-ranged strongly correlated microscopic fluctuations*.
We shall do that in a very qualitative way, and to make things more clear we consider a magnetic system (but of course our considerations are absolutely general and extensible to other kinds of systems).

When a magnetic field $H$ is present^{[1]}, the partition function of a magnet is:

$Z=\operatorname {Tr} e^{-\beta ({\mathcal {H}}-HM)}$

where

${\mathcal {H}}$ is the Hamiltonian of the system. The magnetization of the system at equilibrium is:

$\left\langle M\right\rangle ={\frac {1}{Z}}\operatorname {Tr} \left[Me^{-\beta ({\mathcal {H}}-HM)}\right]={\frac {\partial \ln Z}{\partial (\beta H)}}$

while its susceptibility is:

${\begin{aligned}\chi _{T}={\frac {\partial \left\langle M\right\rangle }{\partial H}}=-{\frac {1}{Z^{2}}}\operatorname {Tr} \left[\beta Me^{-\beta ({\mathcal {H}}-HM)}\right]\cdot \operatorname {Tr} \left[Me^{-\beta ({\mathcal {H}}-HM)}\right]+\\+{\frac {1}{Z}}\operatorname {Tr} \left[\beta M^{2}e^{-\beta ({\mathcal {H}}-HM)}\right]=\\={\frac {\beta }{Z}}\operatorname {Tr} \left[M^{2}e^{-\beta ({\mathcal {H}}-HM)}\right]-{\frac {\beta }{Z^{2}}}\left\lbrace \operatorname {Tr} \left[Me^{-\beta ({\mathcal {H}}-HM)}\right]\right\rbrace ^{2}={\frac {1}{k_{B}T}}\left(\left\langle M\right\rangle -\left\langle M\right\rangle ^{2}\right)\end{aligned}}$

However, from a microscopic point of view the magnetization of the system can be written as:

$M=\int m({\vec {r}})d{\vec {r}}$

where

$m({\vec {r}})$ is the local magnetization in

${\vec {r}}$. If we substitute this expression in

$\chi _{T}$ we get:

$\chi _{T}=\beta \int \left(\left\langle m({\vec {r}})m({\vec {s}})\right\rangle -\left\langle m({\vec {r}})\right\rangle \left\langle m({\vec {s}})\right\rangle \right)d{\vec {r}}d{\vec {s}}$

If our system is spatially homogeneous, then

$\left\langle m({\vec {r}})\right\rangle =m$ is constant and

$\left\langle m({\vec {r}})m({\vec {s}})\right\rangle :=G({\vec {r}}-{\vec {s}})$, called

*correlation function*, depends only on the relative distance between two points. Defining the

*connected correlation function* as:

$\left\langle m({\vec {r}})m({\vec {s}})\right\rangle _{c}:=\left\langle m({\vec {r}})m({\vec {s}})\right\rangle -\left\langle m({\vec {r}})\right\rangle \left\langle m({\vec {s}})\right\rangle =G({\vec {r}}-{\vec {s}})-m^{2}$

sometimes also called

$G_{c}({\vec {r}}-{\vec {s}})$, and changing variable to

${\vec {x}}={\vec {r}}-{\vec {s}}$ we get:

$\chi _{T}=\beta \int G_{c}({\vec {r}}-{\vec {s}})d{\vec {r}}d{\vec {s}}=\beta \int G_{c}({\vec {x}})d{\vec {x}}d{\vec {s}}={\frac {V}{k_{B}T}}\int G_{c}({\vec {r}})d{\vec {r}}$

where in the last step we have renamed the variable

${\vec {x}}$ to

${\vec {r}}$.

The connected correlation function is essentially (as it can be understood from its definition) a measure of how the fluctuations of the magnetization from the mean value $m$ in a part of the system influence those in another part of the system.
Now, we expect that in general the correlation function will decrease on long distances. We can therefore write^{[2]}:

$G_{c}({\vec {r}})\sim e^{-{\frac {|{\vec {r}}|}{\xi }}}\quad \quad {\text{at least when }}|{\vec {r}}|>\xi$

where

$\xi$ is a characteristic length of the system called

*correlation length*.
If we call

$g$ the characteristic value of the connected correlation function for

$|{\vec {r}}|<\xi$ (namely we suppose that it is constant and equal to

$g$ for lengths smaller than

$\xi$^{[3]}), then from

${\textstyle \chi _{T}={\frac {V}{k_{B}T}}\int G_{c}({\vec {r}})d{\vec {r}}}$ we have (neglecting any proportionality constant):

${\frac {\chi _{T}}{\beta V}}=\int G_{c}({\vec {r}})d{\vec {r}}<\int _{|{\vec {r}}|<\xi }G_{c}({\vec {r}})d{\vec {r}}=\int _{|{\vec {r}}|<\xi }gd{\vec {r}}\propto g\xi ^{3}$

namely:

${\frac {\chi _{T}}{\beta V}}<g\xi ^{3}$

We thus immediately see that if

$T\to T_{c}$, since

$\chi _{T}\to \infty$ (as we know from the thermodynamics of critical phase transitions) we must also have

$\xi \to \infty$: in the neighbourhood of a critical point the correlation length of the system diverges.
As we have seen in

Critical exponents and universality, we describe the divergence of

$\xi$ by means of the critical exponent

$\nu$, namely we set:

$\xi \sim |t|^{-\nu }$

Furthermore, experiments and exact solutions of simple models show that the correlation function decays near the critical point as a power of

$|{\vec {r}}|$, with a critical exponent called~

$\eta$:

$G({\vec {r}})\sim {\frac {1}{{|{\vec {r}}|}^{d-2+\eta }}}$

where

$d$ is the dimensionality of the system (see

Critical exponents and universality).

- ↑ We are treating $H$, just like the magnetization $M$, as scalars instead as vectors: in order to make things easier, we are supposing that the
*real* fields ${\vec {H}}$ and ${\vec {M}}$ are both directed along the same direction, say the $z$ axis of our reference frame, and we are only considering their magnitudes.
- ↑ This form is also justified, as we will see, from the fact that in many cases $G_{c}$ turns out to actually decay exponentially with ${\vec {r}}$.
- ↑ In fact, $e^{-x}$ is of the order of 1 when $x<1$.