Critical exponent

Critical exponents describe the behavior of physical systems near Phase transitions.

The critical exponents $\alpha$, $\beta$, $\gamma$, $\delta$, $\eta$, $\nu$ are:

$$\begin{array}{rll}C& \propto |\tau {|}^{\alpha }& \tau \to 0\\ \Psi & \propto |\tau {|}^{\beta }& \tau \to 0\\ \chi & \propto |\tau {|}^{-\gamma }& \tau \to 0\\ \Psi & \propto |h{|}^{\frac{1}{\delta }}& \\ G(r)& \propto r^{-(d+2+\eta )}& \\ G(r)& \propto r^{-\tau }{e}^{\frac{\mu }{\xi }}& \end{array}$$

Where:

• $G(r)$ is the correlation function, dependent only on the Correlation length $r$.
• $\Psi$ is the order parameter. In the Ising Model, we often replace $\Psi$ with $m=⟨S_i⟩$, using magnetism as the order parameter.
• $f$ is the specific Free energy
• $C:=-T\frac{{\partial }^2}{\partial T^2}f$ is the specific heat
• $T_c$ is the critical temperature
• $T$ is the temperature of the system
• $\tau :=\frac{T-T_c}{T_c}$ is the reduced temperature: equal to $0$ at the Critical Point
• $\chi$ is the magnetic susceptibility or compressibility
• $\xi$ is the Correlation length

Following the definitions:

$$\begin{array}{rll}\tau & :=\frac{T-T_c}{T_c}& \\ k& :=\underset{\tau \to 0}{\lim }\frac{\log (f(\tau ))}{\log (\tau )}& \\ f(\tau )& =A{\tau }^k({\sum }_{i=0}^{\infty }{c}_i{\tau }^i)\propto {\tau }^k,& \tau \to 0\\ C& :=-T\frac{{\partial }^2}{\partial T^2}f& \\ \chi & :=\frac{\partial }{\partial J}\Psi & \\ \xi & :=& \end{array}$$

The meaning of the source field $J$ is still unclear. Help me ❓