Partition function of modified Ising model

In summary, the conversation discusses the concept of separating particles with different spin states and using this to find the partition function. The equations and constraints for this are given, but there is difficulty in evaluating the sum due to the constrained arguments.
  • #1
LCSphysicist
646
162
Homework Statement
.
Relevant Equations
.
$$H = - J ( \sum_{i = odd}) \sigma_i \sigma_{i+1} - \mu H ( \sum_{i} \sigma_i ) $$
So basically, my idea was to separate the particles in this way::
##N_{\uparrow}## is the number of up spin particles

##N_{\downarrow}## "" down spin particles

##N_1## is the number of pairs of particles close to each other with spin up

##N_2## "" with spin down

##N_3## "" with spin antiparallel
Therefore
$$\hat{H} = - J (N_1 + N_2 - N_3 ) - \mu H (N_{\uparrow} - N_{\downarrow})$$
Subject to the following constraints:
$$\frac{N}{2} = N_1 + N_2 + N_3$$

$$N_{\uparrow} = 2N_1 + N_3$$

$$N_{\downarrow} = 2N_2 + N_3$$

$$N = N_{\uparrow} + N_{\downarrow}$$
But, even make this clarifications, i can't see how to find the partition! If we substitute the above expressions on ##\sum e^{-\hat{H}/kT}##, we will find Z as a function of, for example, ##Z=Z(N_{\uparrow},N_3)##. But i don't know how to evaluate such a sum! I mean, it can't factoriz in the sum of two expoents, because both arguments are constrained! (We can't have ##N_3 = N/2, N_{\uparrow} = 0##, for example)
 
Last edited:
Physics news on Phys.org
  • #2
Done!
 

FAQ: Partition function of modified Ising model

1. What is the partition function of a modified Ising model?

The partition function of a modified Ising model is a mathematical function that describes the statistical mechanics of a system of interacting spins. It is used to calculate the thermodynamic properties of the system, such as the free energy and magnetization.

2. How is the partition function of a modified Ising model different from a regular Ising model?

The partition function of a modified Ising model includes additional parameters or terms that account for modifications to the original Ising model. These modifications can include different types of interactions between spins, external fields, or non-uniform coupling constants.

3. What are some applications of the partition function of a modified Ising model?

The partition function of a modified Ising model is commonly used in physics and statistical mechanics to study phase transitions, critical phenomena, and magnetic properties of materials. It is also used in other fields, such as computer science and social sciences, to model complex systems.

4. How is the partition function of a modified Ising model calculated?

The partition function of a modified Ising model is typically calculated using numerical methods, such as Monte Carlo simulations or mean-field approximations. These methods involve randomly sampling the possible configurations of the system and calculating the partition function based on the energy of each configuration.

5. Can the partition function of a modified Ising model be solved analytically?

In most cases, the partition function of a modified Ising model cannot be solved analytically due to the complexity of the system. However, in some special cases, such as when the system is in a high-temperature regime, analytical solutions may be possible using techniques such as perturbation theory.

Similar threads

Replies
7
Views
2K
Replies
1
Views
1K
Replies
1
Views
2K
Replies
1
Views
1K
Replies
19
Views
2K
Replies
1
Views
2K
Replies
1
Views
1K
Back
Top