# Paramagnetism: Microcanonical Ensemble Formalism

This derivation explores the statistical mechanics of an ideal paramagnet using the Microcanonical Ensemble. Unlike the canonical approach where temperature is fixed, here we fix the total energy and derive temperature as a statistical property.

1. Physical Model and Assumptions

We consider an ideal paramagnetic system defined by the following assumptions:

• The system consists of \(N\) identical, non-interacting spins.
• Each spin carries a magnetic moment \(\mu\).
• An external uniform magnetic field \(\mathbf{H} = H \hat{z}\) is applied.
• Only spin degrees of freedom contribute (orbital angular momentum \(L=0\)).
• Spins are treated as discrete variables.

2. Energy of a Single Spin

The potential energy of the \(i\)-th magnetic moment in the external field is:

\[ \varepsilon_i = -\boldsymbol{\mu}_i \cdot \mathbf{H} \]

For a spin-1/2 system, the spin variable \(\sigma_i\) can take values \(\pm 1\) (aligned or anti-aligned with the field). Thus:

\[ \varepsilon_i = -\mu H \sigma_i \]

3. Energy of the Whole System

The total Hamiltonian is the sum of individual energies:

\[ H(\sigma_1, \dots, \sigma_N) = -\mu H \sum_{i=1}^{N} \sigma_i \]

We define the population numbers:

• \(N_+\): Number of spins with \(\sigma = +1\) (up)
• \(N_-\): Number of spins with \(\sigma = -1\) (down)

The total number of particles is conserved:

\[ N_+ + N_- = N \]

The sum of spins is the difference in populations:

\[ \sum_{i=1}^{N} \sigma_i = N_+ - N_- \]

Therefore, the total energy \(E\) is:

\[ E = -\mu H (N_+ - N_-) \]

4. Total Magnetization

The total magnetization \(M\) is the sum of the magnetic moments:

\[ M(\sigma_1, \dots, \sigma_N) = \mu \sum_{i=1}^{N} \sigma_i \]

Expressing this in terms of populations:

\[ M = \mu (N_+ - N_-) \]

Comparing the expressions for Energy and Magnetization yields the fundamental relation:

\[ E = -H M \]

5. Microcanonical Ensemble Definition

For an isolated paramagnet:

• Total Number \(N\) is fixed.
• Total Energy \(E\) is fixed.

According to the fundamental postulate of statistical mechanics, all microscopic configurations consistent with this fixed energy \(E\) are equally probable.

\[ P(\sigma_1, \dots, \sigma_N) = \frac{1}{\Gamma(E)} \]

where \(\Gamma(E)\) is the total number of microstates (density of states) with energy \(E\).

6. Counting Microstates (Density of States)

For a fixed energy \(E\), the number of up spins \(N_+\) is fixed. Solving the energy equation for \(N_+\):

\[ N_+ - N_- = -\frac{E}{\mu H} \quad \text{and} \quad N_+ + N_- = N \] \[ 2N_+ = N - \frac{E}{\mu H} \implies N_+ = \frac{1}{2}\left(N - \frac{E}{\mu H}\right) \]

The number of ways to choose \(N_+\) spins to be up out of \(N\) total spins is given by the binomial coefficient:

\[ \Gamma(E) = \binom{N}{N_+} = \frac{N!}{N_+! N_-!} \]

This discrete count acts as the phase-space volume in this discrete system.

7. Entropy (Boltzmann Formula)

The entropy is defined by Boltzmann's relation:

\[ S(E) = k_B \ln \Gamma(E) \]

Using Stirling's Approximation for large \(N\) (\(\ln N! \approx N \ln N - N\)):

\[ \ln \Gamma = \ln N! - \ln N_+! - \ln N_-! \] \[ \approx (N \ln N - N) - (N_+ \ln N_+ - N_+) - (N_- \ln N_- - N_-) \]

Since \(N = N_+ + N_-\), the linear terms cancel:

\[ \ln \Gamma \approx N \ln N - N_+ \ln N_+ - N_- \ln N_- \] \[ \ln \Gamma = -N_+ \ln \left(\frac{N_+}{N}\right) - N_- \ln \left(\frac{N_-}{N}\right) \]

Now, we rewrite \(N_\pm\) in terms of Magnetization \(M\):

\[ N_\pm = \frac{N}{2} \left( 1 \pm \frac{M}{N\mu} \right) \]

Substituting this back, we get the Entropy in terms of \(M\):

\[ S = -\frac{k_B}{2} \left[ \left(N+\frac{M}{\mu}\right)\ln\left(1+\frac{M}{N\mu}\right) + \left(N-\frac{M}{\mu}\right)\ln\left(1-\frac{M}{N\mu}\right) \right] \]

8. Statistical Definition of Temperature

From thermodynamics, temperature is defined as:

\[ \frac{1}{T} = \left(\frac{\partial S}{\partial E}\right)_N \]

Since \(E = -H M\), we can use the Chain Rule:

\[ \frac{\partial S}{\partial E} = \frac{\partial S}{\partial M} \frac{\partial M}{\partial E} = \frac{\partial S}{\partial M} \left(-\frac{1}{H}\right) \]

Thus, we define \(\beta\):

\[ \beta \equiv \frac{1}{k_B T} = -\frac{1}{k_B H} \frac{\partial S}{\partial M} \]

9. Evaluating the Derivative

We differentiate the entropy expression with respect to \(M\):

\[ \frac{\partial S}{\partial M} = -\frac{k_B}{2} \left[ \frac{1}{\mu} \ln\left(1+\frac{M}{N\mu}\right) + \dots - \frac{1}{\mu} \ln\left(1-\frac{M}{N\mu}\right) + \dots \right] \]

(Note: The derivative of the terms inside the logs cancels out).

\[ \frac{\partial S}{\partial M} = -\frac{k_B}{2\mu} \ln \left( \frac{1 + \frac{M}{N\mu}}{1 - \frac{M}{N\mu}} \right) \]

Substituting this back into the expression for \(\beta\):

\[ \beta = \frac{1}{2\mu H} \ln \left( \frac{1 + \frac{M}{N\mu}}{1 - \frac{M}{N\mu}} \right) \]

10. Probability Ratio of Spin Orientations

Recall that \(1 + \frac{M}{N\mu} = \frac{2N_+}{N}\) and \(1 - \frac{M}{N\mu} = \frac{2N_-}{N}\). The term inside the logarithm is simply the ratio of populations:

\[ \frac{N_+}{N_-} = \frac{P(\sigma=+1)}{P(\sigma=-1)} \]

Inverting the log equation from Step 9:

\[ 2\mu H \beta = \ln \left( \frac{N_+}{N_-} \right) \implies \frac{N_+}{N_-} = \exp(2\mu H \beta) \]

\[ \frac{P(+1)}{P(-1)} = \exp\left(\frac{2\mu H}{k_B T}\right) \]

This recovers the Boltzmann Factor directly from microcanonical counting.

11. Average Spin and Magnetization

The average value of a single spin is:

\[ \langle \sigma \rangle = \frac{1 \cdot P(+1) + (-1) \cdot P(-1)}{P(+1) + P(-1)} = \frac{\frac{P(+1)}{P(-1)} - 1}{\frac{P(+1)}{P(-1)} + 1} \]

Substitute the exponential ratio \(e^x\) where \(x = 2\mu H \beta\):

\[ \langle \sigma \rangle = \frac{e^x - 1}{e^x + 1} = \frac{e^{x/2} - e^{-x/2}}{e^{x/2} + e^{-x/2}} = \tanh(x/2) \]

Here, \(x/2 = \mu H \beta = \frac{\mu H}{k_B T}\).

Thus, the total magnetization \(\langle M \rangle = N\mu \langle \sigma \rangle\) is:

\[ \langle M \rangle = N\mu \tanh\left(\frac{\mu H}{k_B T}\right) \]

This is the standard Equation of State for an ideal paramagnet.

12. Thermodynamic Consistency

Using the differential of the entropy derived in Step 9:

\[ dS = \frac{\partial S}{\partial M} dM \]

Substituting \(\frac{\partial S}{\partial M} = -k_B \beta H\):

\[ dS = -k_B \beta H \, dM \]

Multiplying by \(T\) (where \(T = 1/k_B \beta\)):

\[ T dS = -\frac{1}{k_B \beta} (k_B \beta H) dM = -H dM \]

\[ T dS + H dM = 0 \]

This matches the Second Law of Thermodynamics for a magnetic system ($dE = TdS - HdM$ with $dE = -HdM - MdH$, leading to magnetic work relations). It confirms that \(\beta = 1/k_B T\) is the correct statistical definition of temperature.

13. Physical Interpretation

• Entropic Origin: Paramagnetism arises purely from spin entropy. The internal energy prefers alignment with the field, while entropy prefers randomness.
• Competition: The magnetic field \(H\) tries to reduce entropy by ordering spins. Temperature \(T\) represents the thermal agitation opposing this order.
• Entropic Force: Magnetization can be viewed as an entropic force response to the external field constraint.