Ising Model in 1-d

from Statistical Physics using Mathematica
© James J. Kelly, 1996-2001

Introduction

The one-dimensional Ising model of a magnetic system consists of a linear chain of N spins with two states each, up or down.  The energy contains two terms:  (1) the nearest neighbor coupling energy is +J for antiparallel or -J or parallel spins; (2) the interaction with the external magnetic field is -μ B when the spin is parallel to B.  Thus, the energy of the system is E = -J(
              N  -  2 
                   (Subscript[N, ↕]))  - μ B ((Subscript[N, ↑]) - 
                (Subscript[N, ↓])) , where (Subscript[N, ↑]) ((Subscript[N, ↓])) is the number of up (down) spins and (Subscript[N, ↕]) is the number of antiparallel pairs (spin flips).  The magnetization, M  =  ((Subscript[N, ↑])  -  
              (Subscript[N, ↓])) μ, in the presence of an external field results from a competition between the effects of the external field and the internal interactions.  In this problem, we use the method of Lagrange multipliers to maximize the entropy with respect to (Subscript[N, ↑]), (Subscript[N, ↓]),  and (Subscript[N, ↕]) subject to the constraints that the particle number N = (Subscript[N, ↑]) + (Subscript[N, ↓]) and the total energy are fixed. The equations are developed and solved for arbitrary J and B.  In addition, we compare to simpler analytic solutions for the special cases that either J or B vanish.

Although the model is too simple to apply directly to real systems, it nevertheless includes some of the most important features of the magnetic response of real materials.  When the spin-spin interaction is negligible, the model describes the paramagnetic response of an ideal (noninteracting) system of spins.  If J > 0, the spin-spin interaction favors alignment and there exists the possibility of ferromagnetic behavior in which there is a spontaneous magnetization generated by the interatomic interactions in the absence of an external field.  Conversely, if J <  0, the spin-spin interaction promotes an alternating pattern of spins for which there is no net magnetization but there is a correlation between neighboring spins that is described as antiferromagnetism.  Although spontaneous magnetization in the absence of an external field does not actually occur in the one-dimensional Ising model, we nevertheless will refer to J > 0 as the ferromagnetic and to J <  0 as the antiferromagnetic cases because in two or three dimensions the Ising model does describe spontaneous magnetization.  The Ising model is historically quite important because it is sufficiently complex to describe a broad range of magnetic phenomena yet is sufficiently simple to be solvable exactly and, hence, provides important tests of approximation methods designed for application to more sophisticated models.

Glossary:
	μ	= elementary magnetic dipole moment
	B	= external magnetic field
	J	= spin-spin interaction energy
	𝒩	= total number of spins
	(Subscript[N, ↑])	= number of up spins
	(Subscript[N, ↓])	= number of down spins
	(Subscript[N, ↕])	= number of spin flips
	M	= net magnetization
	(Subscript[λ, i])	= Lagrange multipliers
	(Subscript[k, B])	= Boltzmann constant
	T	= temperature
	(Subscript[T, B])	= μ B / (Subscript[k, B]) 	= temperature scale for interaction with external field
	(Subscript[T, J])	= J / (Subscript[k, B])	= temperature scale for spin-spin interaction

References

The 1-dimensional Ising model is discussed in considerable detail in the textbook by Pathria.  The present method of solution is based upon a homework problem written by F. Villars for a statistical mechanics course at MIT offered in 1977, which in turn was based upon an analysis used in the first edition of Pathria's book; unfortunately, some of the relevant material was removed from the second edition.  The present implementation using Mathematica is, of course, much more recent and the problem is studied in more detail than the MIT homework problem.

1) F. Villars, MIT course number 8.333: Statistical Mechanics, (MIT, 1977).

2) R. K. Pathria, Statistical Mechanics, First Edition (Pergamon, Oxford, 1972), pp. 414-418.

3) R. K. Pathria, Statistical Mechanics, Second Edition (Butterworth-Heinemann, Oxford, 1996), pp. 366-372.

4) E. Ising, Z. Phys. 31, 253 (1925).

Initialization

Defaults, packages, and symbols

ClearAll["Global`*"];
Off[General::spell, General::spell1];

$DefaultFont = {"Times", 12}; 
$TextStyle = {FontFamily -> "Times", FontSize -> 12};

Needs["Utilities`Notation`"];
Needs["Graphics`Master`"]

Symbolize[(Subscript[k, B])];Symbolize[(Subscript[N, m])_];Symbolize[(Subscript[λ, m])_];Symbolize[(Subscript[T, x])_];Symbolize[(Subscript[U, 0])]

SetAttributes[{μ, (Subscript[k, B]), J}, Constant]

Rules for manipulating logarithms and Stirling's formula

\(logExpand = {Log[a__\ b_] -> Log[a] + Log[b], 
        Log[a_\/b_] -> Log[a] - Log[b], Log[a_\^n_] -> n\ Log[a], 
        Log[Rational[m_, n_]] -> Log[m] - Log[n], 
        Log[a_ + b_\/c_] -> Log[a\ c + b] - Log[c], 
        Log[a_ + b_\/\(c_\^n_. \ d__\^m_. \)] -> 
          Log[a\ c\^n\ d\^m + b] - n\ m\ Log[c\ d]}; 
    logCombine = {c_. \ Log[a_] + c_. \ Log[b_] -> c\ Log[a\ b], 
        c_. \ Log[a_] - c_. \ Log[b_] -> c\ Log[a\/b], 
        Log[a_] + Log[b_] -> Log[a\ b], 
        Log[a_] - Log[b_] -> Log[a\/b], 
        c_Integer\ Log[a_] -> Log[a\^c]};\)

StirlingApprox = {Log[x_!] -> x Log[x] - x};

Establish basic model

Once the number of spin-up atoms (Subscript[N, ↑]) is specified, (Subscript[N, ↓]) = N - (Subscript[N, ↓]) is determined.  Therefore, the combinatorial problem reduces to determining the number of distinct arrangements of (Subscript[N, ↑]) ↑'s and (Subscript[N, ↓]) ↓'s which contain exactly (Subscript[N, ↕]) spin-flips.  For simplicity, suppose that (Subscript[N, ↕]) is odd.  If we then generate a sequence that begins with ↑ it will end with ↓.  For example, the sequence 	    
                                 \(TraditionalForm\`\(\(\(\(\(\(\(\(\(\(\(\(↑\)\(↑\)\(↑\)\(⋮\)\)↓⋮\)↑⋮\)↓↓⋮\)↑⋮\)↓⋮\)↑⋮\)↓↓⋮\)↑↑⋮\)\(↓\)\)\(\ \)\)\) 
is characterized by (Subscript[N, ↑]) = 8, (Subscript[N, ↓]) = 7, and (Subscript[N, ↕]) = 9 where the spin-flips are represented by ⋮.  In this case, (Subscript[N, ↕]) = 
        (Subscript[N, ↑↓]) +  
          (Subscript[N, ↓↑]) with (Subscript[N, ↑↓]) = 5 and (Subscript[N, ↓↑]) = 4.  For odd values of (Subscript[N, ↕]), any sequence of spins can be divided into (1/2) ((Subscript[N, ↕]) + 1) groups that contain at least one ↑ and (1/2) ((Subscript[N, ↕]) + 1) groups containing at least one ↓.  The number of arrangements of (Subscript[N, ↑]) objects in (1/2) ((Subscript[N, ↕]) + 1) groups is then simply a binomial coefficient (({{(Subscript[N, ↑])}, {(1/2) ((Subscript[N, ↕]) + 1)}})) = \(\(N\_↑!\)\/\(\(\((N\_↑ - \(1\/2\) \((N\_↕ + 1)\))\)!\) \(\((\(1\/2\) \((N\_↕ + 
                          1)\))\)!\)\)\).  Similarly, the number of arrangements of (Subscript[N, ↓]) objects among (1/2) ((Subscript[N, ↕]) + 1) groups is another binomial coefficient of similar form obtained by interchanging (Subscript[N, ↑ ])⟶ 
        (Subscript[N, ↓]).  Alternatively, we can generate an additional set of sequences starting with ↓ by taking our sequences starting with ↑ and interchanging all \(TraditionalForm\`\(\(↑\)\(\ \)\(⟷\)\(\ \)\(↓\)\)\) such that \(TraditionalForm\`N\_\(\(\(↑\)\(↓\)\)\(\ \)\)⟶\ N\_\(\(↓\)\(↑\)\)\).  In either case the number of groupings of each type is simply (1/2) ((Subscript[N, ↕]) + 1).  Therefore, the total multiplicity is twice the product of these two binomial coefficients.  Finally, we assume that (Subscript[N, ↕]) ≫ 1 and obtain the multiplicity function below.  Furthermore, for large (Subscript[N, ↕]) one can show that the multiplicity function has this same asymptotic behavior for even as for odd values of (Subscript[N, ↕]), but that derivation is omitted for brevity (surely one spin-flip more or less must be a 1/(Subscript[N, ↕]) effect anyway).

multiplicity = ((2 (Subscript[N, ↑])!  (Subscript[N, ↓])!)/(( (Subscript[N, ↑]) - (( (Subscript[N, ↕]))/2))! ( (Subscript[N, ↓]) - (( (Subscript[N, ↕]))/2))! ((((( (Subscript[N, ↕]))/2)!))^2)))

logMultiplicity =
(Log[multiplicity]//PowerExpand)/.StirlingApprox//Simplify

The energy can be obtained by scanning the spins from the left to the right.  Each time a spin flip is encountered the number of spins that are potentially parallel to their neighbors decreases by one.  If we account for the absence of a left neighbor for the first spin, the number of parallel contributions to the energy is N - 1 - (Subscript[N, ↕]) while the number of flips is (Subscript[N, ↕]).  Therefore, the spin-spin interaction energy is \(TraditionalForm\`\(-\((N - 
                  1 - \(N\_↕\))\)\)\ J\  + \ \(N\_↕\) J ≈ \ \(-\((N - 2 \( N\_↕\))\)\) J\) to order (N^(-1)).

energy = -J (𝒩 - 
              2 (Subscript[N, ↕])) -  μ B ((Subscript[N, ↑]) - 
              (Subscript[N, ↓]))

particleNumber = (Subscript[N, ↑]) + (Subscript[N, ↓]);

magnetization  =  ((Subscript[N, ↑]) -  
              (Subscript[N, ↓])) μ;

Method of Lagrange multipliers

The method of Lagrange multipliers is used to develop equations for the values of (Subscript[N, ↑]), (Subscript[N, ↓]),  and (Subscript[N, ↕]) which maximize the entropy subject to the constraint that the particle number and total energy are fixed.  Thus, we have a basic variational equation and two constraint equations with two associated Lagrange multipliers.  Although the particle number constraint is so simple that either (Subscript[N, ↑]) or (Subscript[N, ↓]) could be eliminated from the outset, reducing the number of Lagrange multipliers to one, we chose to employ the more general method for illustrative purposes even though the equations could be solved more simply the other way.

Construct system of equations

makeTotalDifferential[x_, cList_, dList_] := 
      Simplify[Collect[Dt[x, Constants -> cList], dList]]

cList = {B, 𝒩};
dList = {Dt[(Subscript[N, ↑]), Constants -> cList], 
          Dt[(Subscript[N, ↓]), Constants -> cList], 
          Dt[(Subscript[N, ↕]), Constants -> cList]};

d0 = makeTotalDifferential[logMultiplicity,	cList,dList];
d1 = makeTotalDifferential[energy, cList,dList];
d2 = makeTotalDifferential[particleNumber, cList,dList];

eqs = (0 == 
              Coefficient[(Collect[#1, dList] &)[
                  d0 + (Subscript[λ, 1]) d1 + (Subscript[λ, 2]) d2], #1] &) /@ 
        dList

Solution for J=0

Rather than produce the relatively complicated general solution right away, we first consider the special case in which the spin-spin interaction is neglected.  In that case our model reduces to a collection of indepedent spins in an external field and we expect to find the familiar solution for paramagnetism in an ideal spin-(1/2) system.  The second Lagrange multiplier is superfluous and can be eliminated, whereupon the remaining two equations can be solved symbolically.

noCouplingEqs = 
      Eliminate[
          eqs /. {J -> 0, 
              (Subscript[N, ↓]) -> 𝒩 - 
                  (Subscript[N, ↑])}, (Subscript[λ, 2])] // Simplify

\(noCouplingSolution = 
      Join[
		{N\_↓ -> 𝒩 - N\_↑, 
          J -> 0}, \(Solve[
            noCouplingEqs, {N\_↑, 
              N\_↕}]\)[[1]]]\)

(Subscript[U, 0]) = 
      Simplify[ExpToTrig[
          energy //. noCouplingSolution]]

We now recognize, of course, that (Subscript[λ, 1]) should be interpreted as -β.  Nevertheless, we now obtain this result directly by requiring the Lagrange multiplier to enforce the energy constraint.  We then express the entropy in terms of either the energy or the temperature.

lam1ToUnoJ = Solve[
          U == 
            (Subscript[U, 0]), (Subscript[λ, 1])][[1]]

sNoCoupling[
        U] = ((Subscript[k, B]) logMultiplicity //. noCouplingSolution /. 
                lam1ToUnoJ // TrigToExp) //. logCombine // 
        Simplify

The temperature of the system is obtained from the standard thermodynamic relationship between temperature, entropy, and energy.

dSdU = (∂_U) sNoCoupling[U]  // Simplify

uNoCoupling[
        T] = (U /. Solve[(1/T) == dSdU, 
                  U][[1]])  // 
          ExpToTrig  // Simplify

The physical interpretation of (Subscript[λ, 1]) is now clear.

lam1ToT = {(Subscript[λ, 1]) -> -(1/((Subscript[k, B]) T))};

(energy//.noCouplingSolution//ExpToTrig//Simplify)/.lam1ToT

(magnetization//.noCouplingSolution//ExpToTrig//Simplify)/.lam1ToT

Thus, we recover the familiar result for paramagnetism in a system of noninteracting  spin-(1/2) particles.

(logMultiplicity  //. noCouplingSolution /. lam1ToT) // 
      FullSimplify[#, {𝒩 > 0, (Subscript[k, B]) > 0, T > 0, 
            B ∈ Reals, μ ∈ Reals}] &

Solution for B=0

Once again, the equations can be solved after the superfluous (Subscript[λ, 2]) is eliminated.

noFieldEqs = (Thread[Exp[#1], Equal] &) /@ 
            Eliminate[
              eqs /. B -> 0, (Subscript[λ, 2])] //. {Exp[
                x_ + c_.  Log[y_]] -> 
              (y^c) Exp[
                  x]} /. {(Subscript[N, ↓]) -> 𝒩 -  
              (Subscript[N, ↑])}

noFieldSolution = 
      Join[{(Subscript[N, ↓]) -> 𝒩 -  (Subscript[N, ↑]), 
          B -> 0}, Solve[
            noFieldEqs, {(Subscript[N, ↑]), 
              (Subscript[N, ↕])}][[2]] ]

(energy//.noFieldSolution//Simplify)//ExpToTrig//Simplify

logMultiplicity //. noFieldSolution // 
      Simplify[#, {𝒩 > 0, 
            J ∈ Reals, (Subscript[λ, 1]) ∈ Reals}] &

Express entropy in terms of energy and deduce temperature.

lam1ToU = Solve[
            U == energy //. 
              noFieldSolution, (Subscript[λ, 1])][[1]]  // Simplify

sNoField[
        U] = (Subscript[k, B]) logMultiplicity //. noFieldSolution /. lam1ToU  // 
        Simplify

dSdU  =  D[sNoField[U],  U]  //. logCombine // Simplify

uNoField[
        T]  =  U  /. Solve[ 1/T  ==  dSdU,  
                U][[1]] // 
          ExpToTrig // Simplify

Again, the physical interpretation of (Subscript[λ, 1]) is now clear.

lam1ToT = {(Subscript[λ, 1]) -> -(1/((Subscript[k, B]) T))};

(energy//.noFieldSolution//ExpToTrig//Simplify)/.lam1ToT

magnetization//.noFieldSolution//Simplify

In the absence of an external field, the spin-spin coupling produces a correlation between neighboring spins which reduces the energy for either sign of J from that obtained for a purely random arrangement of spins.  However, without the guidance of an external field  the net magnetization vanishes.

((logMultiplicity  //. noFieldSolution // Simplify) //. 
              logExpand // ExpToTrig // Simplify) /. lam1ToT

General solution

Construct suitable equations and their solutions

There are two natural scales in this problem, which can be conveniently expressed in terms of temperatures that correspond to the magnetic energy with respect to the external field and the spin-coupling energy, such that (Subscript[T, B])  =  μ B/(Subscript[k, B]) and (Subscript[T, J])  =  J/(Subscript[k, B]).  Hence, if the rules {(Subscript[k, B]) → 1,  (Subscript[T, B]) → 1} have the effect of expressing T and (Subscript[T, J]) in units of (Subscript[T, B]).

temperatureScales  =  {J -> (Subscript[T, J]), 
        B -> ((Subscript[T, B])/μ), (Subscript[λ, 1]) -> -β,  
        T -> (1/β)};

temp2 = 
      Eliminate[(Thread[#1 //. logCombine, Equal] &) /@ 
            eqs /. {(Subscript[N, ↓]) -> 𝒩 - 
                (Subscript[N, ↑])}, (Subscript[λ, 2])]

simplifiedEqs = (Thread[Exp[#1], Equal] &) /@ 
            temp2 //. {Exp[x_ + c_.  Log[y_]] -> 
              (y^c) Exp[x]}/.temperatureScales

Symbolic solutions can be obtained, but their algebraic forms are quite complicated and and somewhat difficult to simplify.  Nevertheless, by plotting the results for several interesting situations we can examine their physical meaning without simplifying the formulas themselves. We can also easily show that the limiting cases do reduce to the expected forms  Therefore, we decline to simplify these functions further.

genSol = 
      Join[{(Subscript[N, ↓]) -> 𝒩 - 
                (Subscript[N, ↑])}, Solve[
              simplifiedEqs, {(Subscript[N, ↑]), 
                (Subscript[N, ↕])}][[2]]]  // Simplify

energySolution =  ((energy //. genSol) /. temperatureScales  // 
            PowerExpand) // Simplify

magnetizationSolution =  magnetization //. genSol /. 
            temperatureScales  // PowerExpand // Simplify

Check limiting cases.

Limit[energySolution, (Subscript[T, J]) -> 0]  // PowerExpand  // 
        ExpToTrig  // Simplify

Limit[energySolution, (Subscript[T, B]) -> 0]  // PowerExpand  // 
        ExpToTrig  // Simplify

Limit[magnetizationSolution, (Subscript[T, J]) -> 0]  // PowerExpand  // 
        ExpToTrig  // Simplify

Limit[magnetizationSolution, (Subscript[T, B]) -> 0]

Examine energy and magnetization versus spin coupling

scaledEnergy[β_, (Subscript[T, J]) : _ /; (Subscript[T, J]) == 0] := 
      Tanh[β]
scaledEnergy[β_, (Subscript[T, J]) : _ /; 
          (Subscript[T, J]) ≠ 
            0] := -energySolution/𝒩 /. {(Subscript[T, B]) -> 
            1}

alignment[β_, (Subscript[T, J]) : _ /; (Subscript[T, J]) == 0] := 
      Tanh[β]
\(alignment[β_, \(T\_J\) : _ /; T\_J ≠ 0] := 
	
      magnetizationSolution/\((𝒩\ μ)\) /. {T\_B -> 
            1}\)

Ferromagnetic case: spin-spin interaction favoring alignment

\(\(Plot[
        Evaluate[
          Table[scaledEnergy[β, T\_J], {T\_J, 0, 2, 0.5}]], {β, 
          0.01, 3}, PlotLabel -> "Energy vs. Spin-coupling", 
        GridLines -> Automatic, Frame -> True, 
        FrameLabel -> {\*""μB/((Subscript[k, B])T)"", "U/(-𝒩μB)"}];\)\)

\(\(Plot[
        Evaluate[
          Table[alignment[β, T\_J], {T\_J, 0, 2, 0.5}]], 
	{β, 
          0.01, 3}, PlotRange -> {{0, 3}, {0, 1}}, 
	
        PlotLabel -> "Alignment vs. Spin-coupling", 
	
        GridLines -> Automatic, Frame -> True, 
	
        FrameLabel -> {\*""μB/((Subscript[k, B])T)"", "M/(𝒩μB)"}];\)\)

When the coupling between neighboring spins favors parallel orientation, the coupling tends to reinforce the alignment produced by the external magnetic field.  Therefore, saturation is approached more rapidly as the spin-coupling increases.

Antiferromagnetic case: spin-spin interaction opposing alignment

\(\(Plot[
        Evaluate[
          Table[scaledEnergy[β, T\_J], {T\_J, 
              0, \(-2\), \(-0.5\)}]], {β, 0.01, 3}, 
        PlotLabel -> "Energy vs. Spin-coupling", 
        GridLines -> Automatic, Frame -> True, 
        FrameLabel -> {\*""μB/((Subscript[k, B])T)"", "U/(-𝒩μB)"}];\)\)

\(\(Plot[
        Evaluate[
          Table[alignment[β, T\_J], {T\_J, 
              0, \(-2\), \(-0.5\)}]], 
	{β, 0.01, 3}, 
        PlotRange -> {{0, 3}, {0, 1}}, 
	
        PlotLabel -> "Alignment vs. Spin-coupling", 
	
        GridLines -> Automatic, Frame -> True, 
	
        FrameLabel -> {\*""μB/((Subscript[k, B])T)"", "M/(𝒩μB)"}];\)\)

Heat Capacity

cvSolution = (∂_T)(energySolution /. β -> 
                  (1/T)) /. {(Subscript[T, B]) -> 1};
cvLimit = Limit[cvSolution, (Subscript[T, J]) -> 0] // PowerExpand // 
        Simplify[#, T > 0] &

cv[T_, (Subscript[T, J]) : _ /; (Subscript[T, J]) ≠ 0] := 
      cvSolution /𝒩
cv[T_, (Subscript[T, J]) : _ /; (Subscript[T, J]) == 0] := 
      cvLimit /𝒩
cv[0, (Subscript[T, J]) : _] := 0

\(\(Plot[Evaluate[Table[cv[T, T\_J], {T\_J, 0, 2, 0.5}]], {T, 0.1, 5}, 
        PlotRange -> {{0, 5}, {0, 1}}, 
        PlotLabel -> "Heat Capacity vs. Spin-coupling", 
        GridLines -> Automatic, Frame -> True, 
        FrameLabel -> {\*""T/(Subscript[T, B])"", \*""(Subscript[C, V])/(𝒩(Subscript[k, B]))""}];\)\)

Correlations

It is useful to distinguish between two types of correlations between neighboring spins.  The external magnetic field promotes a net alignment of the spins which can be described as a form of long-range order in the system.  However, without a spin-spin interaction the arrangement of spins is still considered random since the orientation of each spin depends only the probability that it is aligned with the external field and is independent of the orientation of its neighbors.  Conversely, the spin-spin interaction promotes of form of  short-range order in which the orientation of pairs is preferentially parallel for J > 0 or antiparallel for J < 0.  The long-range order is measured by the net magnetization or alignment of the spin system.  To gauge the short-range order we define a correlation parameter which compares the probability for parallel spins to the probability for antiparallel spins.

First, we define the correlation parameter and then we recognize that one of the equations for this model can be expressed directly in terms of this correlation parameter.

pairRules = {(Subscript[N, ↑]) -> 
          (Subscript[N, ↑↑]) + (((Subscript[N, ↕]))/2), (Subscript[N, ↓]) -> 
          (Subscript[N, ↓↓]) + (((Subscript[N, ↕]))/2), 𝒩 -> 
          (Subscript[N, ↑↑]) + 
            (Subscript[N, ↓↓]) + (Subscript[N, ↕])}

\(src = \(N\_\(\(↑\)\(↑\)\)\ N\_\(\(↓\)\(↓\)\)\)\/N\_↕\%2 /. \(Solve[
            simplifiedEqs[[2]] /. pairRules, \(N\_↕\)]\)[[1]]\)

In the absence of spin-spin coupling ((Subscript[T, J])→0), the value of (1/4) arises from a purely random arrangements of spins and is independent of the external field.  Therefore, short-range order corresponds to deviations from the value of (1/4) obtained for random mixing.  The degree if short-range order displays an exponential dependence upon the ratio between the spin-spin and thermal energies with values greater than (1/4) representing ferromagnetic correlations for J>0 and values smaller than (1/4) representing antiferromagnetic correlations for J<0.

\(\(LogPlot[
        Evaluate[src /. {β -> 1}], {T\_J, \(-1\), 
          1}, PlotLabel -> "short-range correlations", 
        PlotRange -> {{\(-1\), 1}, {0.01, 10}}, 
        GridLines -> Automatic, Frame -> True, 
        FrameLabel -> {\*""\!\(T\_J\)/T"", "src"}];\)\)

Problems

Ising ring

Evaluate the multiplicity function for an Ising ring in which the spins are arranged on a closed ring instead of a line.  You should find that the combinatorial problem is actually easier to analyze and that it is not necessary to consider even or odd values of  (Subscript[N, ↕]) separately, only one choice being possible (which?).