Consider a magnetic system whose free energy, near the critical point, scales as \(\lambda^{5} g(\epsilon, B)=g\left(\lambda^{2} \epsilon, \lambda^{3} B\right)\). Compute (a) the degree of the coexistence curve, (b) the degree of the critical isotherm, (c) the critical exponent for the magnetic susceptibility, and (d) the critical exponent for the heat capacity. Do your results agree with values of the critical exponents found in experiments?

Scaling relations are mathematical tools used to analyze how physical properties change near a critical point. In magnetic systems, scaling laws describe how the free energy depends on variables like temperature (\texttt{\(\epsilon\)}) and magnetic field (\texttt{B}).
In the given problem, the scaling relation is: \[ \lambda^{5} g(\epsilon, B) = g(\lambda^2 \epsilon, \lambda^3 B)\].
This means if the temperature and magnetic field are scaled by certain factors, the free energy behaves predictably.
Understanding these scaling laws helps in determining critical exponents, which quantify the behavior of physical properties near the critical point.
Each critical exponent offers insight into how a specific property varies as the system approaches the phase transition.

The coexistence curve represents the set of points where the system can exist in two distinct phases simultaneously, such as liquid-vapor or magnetic-nonmagnetic.
In terms of magnetic systems, it relates to how magnetization (\texttt{M}) depends on temperature (\texttt{T}) near the critical point when there's no external field.
For this problem, the order parameter \texttt{M} scales with the external magnetic field \texttt{B} as \[M \sim B^{1/\delta}\].
By balancing exponents when temperature approaches its critical value, we compute the critical exponent \texttt{\beta}, which describes the shape of the coexistence curve.
We derived \texttt{\beta = 2} from the scaling relation \[\lambda^\beta = \lambda^2\].
This tells us how the magnetization changes as the temperature approaches the critical point.

The critical isotherm describes the relationship between magnetization (\texttt{M}) and external magnetic field (\texttt{B}) precisely at the critical temperature (\texttt{T_c}).
Using the scaling relation form \[M \sim B^{1/\delta}\], we determined that at critical temperature, magnetization follows a power law behavior with \[1/\delta = 3 \Rightarrow \delta = 3\].
This tells us the exponent \delta that characterizes the critical isotherm, revealing how the system's magnetization responds to the applied magnetic field precisely at \texttt{T_c}.
This is crucial in understanding the abrupt change in magnetization at the phase transition.

Magnetic susceptibility (\texttt{\chi}) is a measure of how much the magnetization \texttt{M} changes when an external magnetic field \texttt{B} is applied.
Near the critical point, \texttt{\chi} diverges following a power law characterized by the critical exponent \texttt{\gamma}, calculated as \texttt{\epsilon^{−\gamma}}.
From the scaling relation, equating magnetic terms gave \[\lambda^{\gamma} = \lambda^5 - \lambda^2 = \lambda^3\], which simplifies to \texttt{\gamma = 3/2}.
This critical exponent informs us about the extent of susceptibility divergence, helping us understand how the system becomes extremely responsive to external fields near the transition.

Heat capacity (\texttt{C}) measures the amount of heat needed to change the system's temperature.
At the critical point, it diverges following a power law determined by the critical exponent \texttt{\alpha}, represented as \texttt{\epsilon^{-\alpha}}.
Using the scaling relation, we found \[\lambda^{\alpha} = \lambda^5\], leading us to derive \texttt{\alpha = 5/2}.
This tells us how heat capacity behaves near the critical temperature, showing a significant increase in energy required to change the system's temperature.