# The Canonical Ensemble

This framework describes a system in thermal equilibrium with a heat reservoir. Unlike the Microcanonical ensemble (fixed \(E\)), here the Temperature \(T\) is fixed, and energy fluctuates.

1. Derivation of the Canonical Probability

System Setup

We consider a composite system \(A^0\) which is isolated (Microcanonical) with total energy \(E^0\). It is divided into two parts:

• System \(A\): The small system of interest (energy \(E_r\)).
• Reservoir \(A'\): A large heat bath (energy \(E'\)).

The conservation of energy requires: \( E_r + E' = E^0 \implies E' = E^0 - E_r \).

The Probability Condition

We ask: What is the probability \(P_r\) of finding system \(A\) in a specific microstate \(r\) with energy \(E_r\)?

According to the principle of equal a priori probability, \(P_r\) is proportional to the number of microstates available to the reservoir when system \(A\) is in state \(r\).

\[ P_r = C' \Gamma'(E^0 - E_r) \]

where \(\Gamma'(E')\) is the density of states of the reservoir.

Taylor Expansion (The "Heat Bath" Limit)

Since the reservoir is much larger than the system (\(E^0 \gg E_r\)), we can expand the logarithm of the density of states around \(E^0\):

\[ \ln \Gamma'(E^0 - E_r) \approx \ln \Gamma'(E^0) - E_r \left( \frac{\partial \ln \Gamma'}{\partial E'} \right)_{E'=E^0} \]

We identify the derivative with the definition of temperature parameter \(\beta\):

\[ \beta \equiv \frac{\partial \ln \Gamma'}{\partial E'} = \frac{1}{kT} \]

Substituting this back:

\[ \ln \Gamma'(E^0 - E_r) \approx \text{const} - \beta E_r \]

Exponentiating gives the probability distribution:

\[ P_r = C e^{-\beta E_r} \]

Normalization & The Partition Function

The total probability must sum to 1: \(\sum_r P_r = 1\).

This allows us to determine the constant \(C\) (or \(1/Z\)). We define the Partition Function (\(Z\)):

\[ Z = \sum_{r} e^{-\beta E_r} \]

Thus, the canonical probability is:

\[ P_r = \frac{e^{-\beta E_r}}{Z} \]

Note: This is the Boltzmann factor normalized by the partition function.

2. Macroscopic Quantities from Z

The partition function \(Z\) acts as a "generating function" for thermodynamic variables.

Mean Energy (\(\overline{E}\))

The average energy is the ensemble average:

\[ \overline{E} = \sum_{r} E_r P_r = \frac{\sum_r E_r e^{-\beta E_r}}{\sum_r e^{-\beta E_r}} \]

We can express this as a derivative of \(Z\):

\[ \overline{E} = -\frac{\partial \ln Z}{\partial \beta} \]

Generalized Force & Work

Suppose the energy levels \(E_r\) depend on an external parameter \(x\) (e.g., Volume \(V\), Magnetic Field \(H\)). The macroscopic work done when \(x\) changes by \(dx\) is:

\[ \Delta W = \sum_r P_r \left( \frac{\partial E_r}{\partial x} dx \right) \]

Substituting \(P_r = \frac{e^{-\beta E_r}}{Z}\) and noting that \(\frac{\partial e^{-\beta E_r}}{\partial x} = -\beta \frac{\partial E_r}{\partial x} e^{-\beta E_r}\):

\[ \Delta W = -\frac{1}{\beta Z} \sum_r \frac{\partial}{\partial x} (e^{-\beta E_r}) dx = -\frac{1}{\beta Z} \frac{\partial Z}{\partial x} dx \]

This defines the Generalized Force (\(\overline{F}_{gen}\)):

\[ \overline{F}_{gen} = \frac{1}{\beta} \frac{\partial \ln Z}{\partial x} \]

Examples:

• Pressure (\(P\)): Parameter \(x = V\). \[ \overline{P} = \frac{1}{\beta} \frac{\partial \ln Z}{\partial V} \]
• Magnetization (\(M\)): Parameter \(x = H\). \[ \overline{M} = \frac{1}{\beta} \frac{\partial \ln Z}{\partial H} \]

3. Application: Ideal Paramagnet

We apply the canonical formalism to evaluate the partition function of an ideal paramagnet.

System Parameters

• Temperature: \(\beta = 1/kT\)
• Magnetic Field: \(H\)
• Energy: \(E = -\mu_B H (N_+ - N_-)\)

Calculating Z

The total partition function sums over all configurations of spins \(\{\sigma_1, \dots, \sigma_N\}\):

\[ Z = \sum_{\{\sigma\}} \exp\left[ -\beta \left( -\mu_B H \sum_{i=1}^N \sigma_i \right) \right] \]

Since the spins are non-interacting, the sum factorizes into a product of \(N\) single-spin sums:

\[ Z = \prod_{i=1}^N \left[ \sum_{\sigma_i=\pm 1} e^{\beta \mu_B H \sigma_i} \right] = \left( e^{\beta \mu_B H} + e^{-\beta \mu_B H} \right)^N \]

Using the identity \(e^x + e^{-x} = 2 \cosh x\):

\[ Z = [2 \cosh(\beta \mu_B H)]^N \]

Macroscopic Observables

Log of Z: \(\ln Z = N \ln [2 \cosh(\beta \mu_B H)]\)

Magnetization:

\[ \overline{M} = \frac{1}{\beta} \frac{\partial \ln Z}{\partial H} = \frac{N}{\beta} \frac{1}{2 \cosh(\dots)} \cdot 2 \sinh(\dots) \cdot (\beta \mu_B) \] \[ \overline{M} = N \mu_B \tanh(\beta \mu_B H) \]

Mean Energy: \(\overline{E} = -H \overline{M} = -N \mu_B H \tanh(\beta \mu_B H)\)

4. Recovering the Laws of Thermodynamics

We verify the consistency of Statistical Mechanics with Thermodynamics by analyzing the total differential of \(\ln Z\).

Consider \(\ln Z\) as a function of \(\beta\) and the external parameter \(x\):

\[ d(\ln Z) = \frac{\partial \ln Z}{\partial x} dx + \frac{\partial \ln Z}{\partial \beta} d\beta \]

Substituting our physical identifications (\(\overline{F} = \frac{1}{\beta}\frac{\partial \ln Z}{\partial x}\) and \(\overline{E} = -\frac{\partial \ln Z}{\partial \beta}\)):

\[ d(\ln Z) = \beta \overline{F} dx - \overline{E} d\beta \]

Rearranging terms to group exact differentials:

\[ d(\ln Z) + \overline{E} d\beta + \beta d\overline{E} = \beta \overline{F} dx + \beta d\overline{E} \] \[ d(\ln Z + \beta \overline{E}) = \beta (dW + d\overline{E}) \]

From the First Law, \(dW + d\overline{E} = dQ\) (Heat). Thus:

\[ d(\ln Z + \beta \overline{E}) = \beta dQ \]

Since \(dQ = T dS\) and \(\beta = 1/kT\), we identify the quantity in parentheses as Entropy \(S/k\):

\[ S = k (\ln Z + \beta \overline{E}) \]

Rearranging gives the definition of Helmholtz Free Energy (\(F\)):

\[ F = \overline{E} - TS = -kT \ln Z \]

We have successfully recovered the Second Law of Thermodynamics purely from Statistical Mechanics.