In this paper, an algorithm to produce a transition matrix between all states in ideal anti-paramagetic system under the simulated annealing condition that only moves to lower energy states are accepted. The check the accuracy of the transition matrix is confirmed by computer simulation, showing close agreement between model and simulation results.
In this paper we compute all transition probabilities of a system of N isolated particles of spin M to move to any lower energy state. In each move, the current state
This is an unusual use of simulated annealing. Normally, simulated annealing is used to find a solution to a complex problem, such as the optimal path in the traveling salesman problem. In complex magnetic systems, such as spin glasses, simulated annealing is used to find the optimal ground state [
Beginning with two, three and four spin one half particle states, the transition probabilities between all states are enumerated. These lists of probabilities are formed into matrices. This is repeated for spin one particles in section 2 where the transition probabilities are derived from the degeneracy of the states. These degeneracies employ the generalized binomial coefficients, called the multinomials. In Section 3, the spin 3/2 particles are considered. Then, in Section 4, the entries in the transition matrices are derived from the degeneracy of the states, using a simple summation rule. In Section 5, degeneracy vectors are given for various systems. In Section 6, the results of numerical simulations are provided which confirm the results of the matrix calculations.
Our underlying assumptions are:
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In any system N-particles with spin we will assign values. Those values will be relative to the system we are working with. We will assign +1 for spin up, and values −1 for spin down in a spin one half system. +1, 0, −1 for a spin one system, and
Definition 1.1. For any state
Definition 1.2. For any two states
Definition 1.3. For any N-particle system.
Definition 1.4. let
Definition 1.5. The cardinality of the set
This leads us to the following proposition.
Proposition 1.6. There can only be one idealized Ground state;
that is
Poof. The only way to get magnetism number
Definition 1.7. let
is the set of all states that are preferred to a state with magnetism number
Definition 1.8. A system with a finite or countably infinite number of states with the property that given a present state, past states have no influence on the future is known as a Markov chain
Definition 1.9. Markov chains have the property that for a random variable
where
Definition 1.10. Let
From the transition function
P is known as the transition matrix for the Markov chain. We will now apply the concepts of a Markov chain directly to our model.
We will use Markov chains to model the transition states for spin 1/2, spin 1, or spin 3/2 particles. For our model of N-spin particles the state space
We begin by defining the probability function for our model.
Definition 2.1. Let
where N is total number of particles in the system,
and
Since our Markov chain has the property of only transferring to preferred states, our transition probabilities are defined by the following transition function.
This transition function shows how transition probabilities are absorbed when we move to a preferred state from a less desired state. This probability absorption allows us to construct upper triangular square transition matrices of the following form.
Our understanding of the transition function and the transition matrix of our model leads to the following 2 propositions .
Proposition 2.2.
Poof. Staring with
Now
Giving us the desired result.
Proposition 2.3. For the idealized ground state
Poof. By proposition 4.2
We will now look at Markov chains and the spin 1/2 particle.
For a two spin one half particles, the only possibilities of spin configuration of the fermions look like this
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Thus
Giving us the following probability transition table
With transition matrix
For example we would read
our probability matrices upper triangular. Which for our model denotes that system is probabilistically approaching the idealized ground state. Since read
For a system of three spin one half particles, the only possibilities of spin configuration of the fermions look like this
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Thus
Although we can clearly see what the probability of landing in each state is, we can use basic counting arguments to show our results mathematically. Thus
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Giving us the following probability transition table
With transition marix
Example 3.1. Given the transition matrix for the three spin one half particle system, above, we can calculate the probablity of moving to a preferred state.
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With our understanding of the three spin one half particle case and counting arguments, we can develop a generalization for any N spin one half particle system.
Now given our understanding of the probabilities we can build the probability transition matrix for any N-fer- mion system. If
And with
Leading us to the following definition.
Definition 3.2. For any N-fermion system we can calculate
Definition 3.3. For any N-fermion system we can calculate
With these result we can calculate the probability of our system moving to a better state, given any N-fer- mions and
For the spin one particles we will assign
A system with two spin one particles has the following states:
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Thus for a system of two spin one particles, the state space
With the following transition matrix
For a system of three spin one particles the possible states are given by the macro-states
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Thus in terms of probabilities:
These probabilities give us the following transition table.
With the following transition matrix.
Notice that listing out all appropriate micro states gets a bit cumbersome, but unlike the spin 1/2 particles, the spin one particles have 3 possible spins. So to calculate their probabilities with the aid multinomial probability formula.
Now with this formula we are able to calculate the probability of the macro states for a 4 spin one particle system.
For a spin one system with 4 particles we find macro states
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Giving us the following Transition table
Along with the following transition matrix
We now generalize our results with the following formulas
Definition 4.1. For any N-spin one particle system and a given macro state
where
If n-even m-even, or n-odd m-odd
If N-even m-odd, or N-odd m-even
And for the zero case we get the following.
Definition 4.2. Let
Now with this result we can calculate the probability of our system moving to a preferred state, given N-spin one particles and
We now turn our attention to a system of spin 3/2 particles.
Now we turn our attention to spin 3/2 particles. We will de-note all the possible states a particle can take with
the set
To give the spin 3/2 particles integer coefficients we simply represent them in the set
A two spin 3/2 particle system has the following states
・ +3, +3
・ +3, +1
・ +3, −1
・ +3, −3
・ +1, +3
・ +1, +1
・ +1, −1
・ +1, −3
・ −1, +3
・ −1, +1
・ −1, −1
・ −1, −3
・ −3, +3
・ −3, +1
・ −3, −1
・ −3, −3
The two particle system gives us the following states
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Giving us the following probability transition table
With the following transition matrix
For a system of three spin 3/2 particles we calculate the state space in the same manner as we did in the two spin 3/2 particle system. Giving us the following state space
With the following probabilities:
This results in the following transition table
With the following transition matrix
With the aid of the multinomial formula we are able to calculate the probabilities of the state space
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Which gives us the following degeneracy transition table, with
And the transition matrix
With D being the following matrix
With D being the degeneracy matrix. Notice that we can construct any transition probability matrix from just understanding degeneracy matrix D. Thus given our systems it suffices to understand degeneracies in order to construct their transition probabilities.
As we saw in the previous section we can find all the transition probabilities from any j spin-N particle system under assumptions:
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With Probability transition matrix P
To construct D let
With
We will now use the degeneracy vector to construct the degeneracy matrix.
For simplicity we will construct D from our degeneracy vector which we will denote
Thus it follows that
thus we can rewrite
Using
From the preceding section, we see that we can construct the probability transition matrix of a system solely by means of the degeneracy vector. Now here are the degeneracy vectors for n = 5, 6, 7, 8, 9, 10.
・ for n = 5
・ for n = 6
・ for n = 7
・ for n = 8
・ for n = 9
・ for n = 10
・ for n = 5
・ for n = 6
・ for n = 7
・ for n = 8
Two systems are studied: a small system with 10 spin 1/2 particles giving 1024 possible states, and a larger system with 20 spin 1/2 particles, with about a million possible states. The system begins in the least optimal state, all spins pointing up, and at each move, a random trial state is generated. If the trial state is more optimal, meaning it has fewer spins pointing up, it is accepted. It is highly probable the first move will be accepted. After many moves, the likelihood of improvement diminishes. We consider three “times” to illustrate the relative movement slowing at large time: after 9, 81 and 729 moves, so the moves are evenly spaced in log time. The simulation is run one million times for the small system, and one hundred thousand times for the large system. The results are displayed in
Time | ||||
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9 | 8.75 | 9.50 | 91.7 | 93.1 |
81 | 76.1 | 76.6 | 506 | 511 |
729 | 509.5 | 510 | 490 | 489 |
Time | ||||
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9 | 0.86 | 2 | 17.2 | 16 |
81 | 7.72 | 12 | 154 | 156 |
729 | 69.4 | 76 | 1380 | 1348 |
For ideal anti-paramagnetic systems, an algorithm for generating a transition matrix is given, with the constraint that the simulated annealing rule is followed; that only transitions to lower energy are allowed. This matrix is confirmed by direct simulated annealing simulations.
We thank the Editor and the referee for their comments.
Ricardo Suarez,Gregory G. Wood, (2016) Energy-States of Particles with Representational Spin. Applied Mathematics,07,650-664. doi: 10.4236/am.2016.77060