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

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Discussion Overview

The discussion centers around the properties of simple 2D Ising models, specifically focusing on the concept of density of states (DOS) and its calculation. Participants explore theoretical aspects, mathematical derivations, and connections to statistical mechanics, including the Boltzmann factor and Maxwell-Boltzmann energy distribution.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions whether simple 2D Ising models have a constant density of states and seeks clarification on how it is calculated.
  • Another participant asks about prior research and findings related to the density of states in the context of the Ising model.
  • A participant discusses the relationship between constant density of states and the use of the Boltzmann factor for calculating probabilities in the Ising model, expressing uncertainty about the justification for a constant DOS.
  • One participant presents a mathematical expression for the Maxwell-Boltzmann energy distribution and seeks guidance on deriving the density of states from it, specifically wanting to eliminate mass and incorporate kB and T.
  • Another participant inquires about a DOS expression for the ideal gas that would align with the Maxwell-Boltzmann probability distribution and yield extensive entropy, questioning if it should be the same quantity.
  • A later reply criticizes the approach of asking multiple questions in the same thread, suggesting that it may hinder the discussion.

Areas of Agreement / Disagreement

Participants express various uncertainties and questions regarding the density of states and its implications in the context of the Ising model and ideal gas. No consensus is reached on the specific properties or calculations related to the density of states.

Contextual Notes

Participants exhibit varying levels of familiarity with the Ising model and related concepts, indicating potential gaps in understanding and assumptions about the density of states. There are unresolved mathematical steps and dependencies on definitions that remain unclarified.

rabbed
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Do simple 2D Ising models have constant density of states?
How is it calculated?
 
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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?
 
Last edited:
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.
 

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