The free energy landscape for the mean field 1D ising model : Geogebra exercise
Try to answer the questions below by reading the description and adjusting the geogebra plot
Description
The green line shows the free energy, $F$, expressed as a function of the average magnetization, $\langle M \rangle$, for
a mean field description of the 1D Ising model in the thermodynamic limit. The black line is the first derivative of this
free energy with respect to the
magnetization, $\langle M \rangle$. There are two sliders the first allows you to control the strength of the magnetic field,
$H$, which is expressed in units of the interaction constant for the spins, $J$. The second slider allows you to control the
inverse temperature, $\beta$, which is expressed in units of $\frac{J}{k_B T}$
Questions
- What is significant about the points where the black curve intercepts with the $x$ axis?
- When the applied field $H$ is equal to 0 at how many points does the black line intercept with the $x$ axis? What happens as the the inverse temperature is increased?
- Describe the shape of the green curve when $H=0$ and when (a) $T < 2 \frac{k_B T}{J}$ and when (b) $T > 2 \frac{k_B T}{J}$. How does the shape of this curve differ in these two regimes? What happens to the derivative of the free energy with respect to $\langle M \rangle$ at $H=0$ when $T=2 \frac{k_B T}{J}$?
- What happens to the shape of the green curve when $H \ne 0$. Comment on the behavior of the turning points and the way this number changes with field strength and temperature.
- Given what you have discussed explain how the magnetization behaves as you move from the $T > 2 \frac{k_B T}{J}$ regime to the $T < 2 \frac{k_B T}{J}$ regime in the absense of an applied field
- How does the magnetisation behave as the temperature is increased in the presence of an applied magentic field?
- What is significant about the points where the black curve intercepts with the $x$ axis?
- When the applied field $H$ is equal to 0 at how many points does the black line intercept with the $x$ axis? What happens as the the inverse temperature is increased?
- Describe the shape of the green curve when $H=0$ and when (a) $T < 2 \frac{k_B T}{J}$ and when (b) $T > 2 \frac{k_B T}{J}$. How does the shape of this curve differ in these two regimes? What happens to the derivative of the free energy with respect to $\langle M \rangle$ at $H=0$ when $T=2 \frac{k_B T}{J}$?
- What happens to the shape of the green curve when $H \ne 0$. Comment on the behavior of the turning points and the way this number changes with field strength and temperature.
- Given what you have discussed explain how the magnetization behaves as you move from the $T > 2 \frac{k_B T}{J}$ regime to the $T < 2 \frac{k_B T}{J}$ regime in the absense of an applied field
- How does the magnetisation behave as the temperature is increased in the presence of an applied magentic field?