Chapter 8: Q. 8.18 (page 343)

Starting from the partition function, calculate the average energy of the one-dimensional Ising model, to verify equation 8.44. Sketch the average energy as a function of temperature.

Short Answer

Expert verified

The average energy = $- N ε \tanh β ε$

Sketch for the average energy as function of temperature

Step by step solution

Step 1. Given information

The partition function of the system:

$Z = \sum_{s} e^{- ε_{i} / r}$

Step 2. To find the expression for average energy

We substitute $β for 1 / τ$

$Z = \sum_{s} e^{- ε_{n} β}$

The average energy of the system is,

$U = ⟨ ε ⟩$

$= \frac{1}{Z} \sum_{s} ε_{s} e^{- β c_{s}}$

$= - \frac{1}{Z} ε_{s} \sum_{s} e^{- β s_{s}}$

Substituting the value of $\frac{d Z}{d β} = ε_{s} \sum_{s} e^{- β c_{s}}$

$U = - \frac{1}{Z} \frac{d Z}{d β}$

$= - \frac{\partial}{\partial β} (\ln Z)$

The expression for partition function for final sum of $N$ dipole is,

$Z = (2 \cosh β ε)$

$\ln Z = \ln (2 \cosh β ε)$

$U = - \frac{1}{Z} \frac{d Z}{d β}$

$= - \frac{\partial}{\partial β} (\ln (2 \cosh β ε))$

$= \frac{1}{(2 \cosh β ε)} ((2 \sinh β ε)) α$

$= - N ε \tanh β ε$

The average energy = $= - N ε \tanh β ε$

Step 3. Sketch for the following function

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Starting from the partition function, calculate the average energy of the one-dimensional Ising model, to verify equation 8.44. Sketch the average energy as a function of temperature.

Short Answer

Step by step solution

Step 1. Given information

Step 2. To find the expression for average energy

Step 3. Sketch for the following function

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