# Do Simple 2D Ising Models Have Constant Density of States?

• I
• rabbed
In summary, the conversation discusses the density of states in simple 2D Ising models and how it is calculated. The participants also share their research on the topic and discuss the use of the Boltzmann factor in calculating probabilities. One participant also asks about deriving the density of states in the MB energy distribution and its relation to kB and T. Another question is asked about finding a DOS-expression for the ideal gas that fits into the MB_PDF(E, T) formula and gives an extensive entropy. The suggestion is made to start a new thread for this question.

#### rabbed

Do simple 2D Ising models have constant density of states?
How is it calculated?

What research have you done so far on this? What have you found out?

I just learned about density of states and the Boltzmann factor.
If the density of states does not depend on the energy/is constant, we can just use the Boltzmann factor to calculate the probability of a particle being in a state of certain energy. And with the Ising model only the BF is used, right?
I googled and found some people finding the DOS using some algorithm, but no calculations or justifications for why it would be constant.
Pretty new to the Ising model also, just thought the DOS would be a fundamental thing to know when deriving it

Okay, I found an explanation for the Ising model..

Next question - The MB energy distribution is: MB_PDF(E) = 2*sqrt(E/pi) * 1/(kB*T)^(3/2) * e^(-E/(kB*T))
How do I arrive at the density of states which hides inside the expression 2*sqrt(E/pi) * 1/(kB*T)^(3/2) ?
I've only seen DOS that have "h" in them.. I want it to contain only E, pi, kB and T..
This is how far I've gotten (using a momentum vector):
V = 4*pi*p^3/3
dV = 4*pi*p^2*dp
dV = 4*pi*(2*m*E)*sqrt(m/(2*E))*dE (since p = sqrt(2*m*E) and dp = sqrt(m/(2*E))*dE)
dV = 2*pi*(2*m)^(3/2)*sqrt(E)*dE

How do I get rid of the m and how do I get in kB and T?

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Since no one cares, I might as well ask Another question:
Is there a DOS-expression D for the ideal gas which will both fit into MB_PDF(E, T) = D * e^(-E/(kB*T)) / Z (where Z normalizes the distribution)
as well as giving an extensive entropy S = kB*ln(D) ?
It should be the same quantity, right?

rabbed said:
Since no one cares, I might as well ask Another question:
Is there a DOS-expression D for the ideal gas which will both fit into MB_PDF(E, T) = D * e^(-E/(kB*T)) / Z (where Z normalizes the distribution)
as well as giving an extensive entropy S = kB*ln(D) ?
It should be the same quantity, right?
Sorry to see you're not getting any help, but starting a new question in the same thread is a VERY bad idea. I'd suggest that you delete it here and start a new thread.

## 1. What is a 2D Ising model?

A 2D Ising model is a mathematical model used in the field of statistical physics to describe the behavior of a system of interacting spins. It consists of a grid of lattice sites, with each site containing a spin that can have two possible values: up or down. The interactions between neighboring spins are determined by the Ising energy function.

## 2. What is the density of states in a 2D Ising model?

The density of states in a 2D Ising model refers to the number of possible energy states that the system can have. It is a measure of the system's thermodynamic properties and is related to the partition function, which describes the probability of the system being in a certain energy state.

## 3. Does the density of states remain constant in a 2D Ising model?

No, the density of states in a 2D Ising model does not remain constant. It depends on the temperature and the applied magnetic field, and can change as the system undergoes phase transitions. At low temperatures, the density of states is lower due to the system being in a more ordered state, while at high temperatures it is higher due to the system being in a more disordered state.

## 4. How is the density of states calculated in a 2D Ising model?

The density of states in a 2D Ising model can be calculated using various methods, such as the transfer matrix method or Monte Carlo simulations. These methods involve calculating the partition function and then taking a derivative with respect to energy to obtain the density of states.

## 5. Why is the density of states important in 2D Ising models?

The density of states is important in 2D Ising models because it provides information about the system's thermodynamic properties, such as the specific heat and magnetization. It can also help in understanding the behavior of the system at different temperatures and magnetic fields, and in predicting phase transitions. Additionally, the density of states plays a crucial role in calculating physical quantities using statistical mechanics principles.