Problem 17 - An Introduction to Thermal Physics

Question

Consider an Ising model of just two elementary dipoles, whose mutual interaction energy is \(\pm \epsilon .\) Enumerate the states of this system and write down their Boltzmann factors. Calculate the partition function. Find the probabilities of finding the dipoles parallel and antiparallel, and plot these probabilities as a function of \(k T / \epsilon .\) Also calculate and plot the average energy of the system. At what temperatures are you more likely to find both dipoles pointing up than to find one up and one down?

Step-by-Step Solution

Answer

Parallel states are more likely at lower temperatures. Average energy trends towards \(-\epsilon\) as temperature decreases.

Step 1: Enumerate the States of the System

In an Ising model with two dipoles, the possible states are determined by whether the spins are up (+1) or down (-1). The system can have the following states: \((+1, +1), (-1, -1), (+1, -1), (-1, +1)\). The states \((+1, +1)\) and \((-1, -1)\) are parallel states. The states \((+1, -1)\) and \((-1, +1)\) are antiparallel states.

Step 2: Identify Energy of Each State

Parallel states, \((+1, +1)\) and \((-1, -1)\), have an energy of \(-\epsilon\). Antiparallel states, \((+1, -1)\) and \((-1, +1)\), have an energy of \(+\epsilon\).

Step 3: Write Down the Boltzmann Factors

The Boltzmann factor for each state is calculated using the formula: \(e^{-E/kT}\), where \(E\) is the energy of the state, \(k\) is the Boltzmann constant, and \(T\) is the temperature. For parallel states, the Boltzmann factor is \(e^{\epsilon/kT}\), and for antiparallel states, it is \(e^{-\epsilon/kT}\).

Step 4: Calculate the Partition Function

The partition function \(Z\) is the sum of the Boltzmann factors for all possible states: \[Z = e^{\epsilon/kT} + e^{\epsilon/kT} + e^{-\epsilon/kT} + e^{-\epsilon/kT} = 2(e^{\epsilon/kT} + e^{-\epsilon/kT})\].

Step 5: Find the Probabilities of States

The probability of a state is given by the Boltzmann factor of that state divided by the partition function \(Z\). The probability of parallel states (either of the two) is:\[P_{\text{parallel}} = \frac{2e^{\epsilon/kT}}{Z} = \frac{e^{\epsilon/kT}}{e^{\epsilon/kT} + e^{-\epsilon/kT}}\].The probability of antiparallel states (either of the two) is:\[P_{\text{antiparallel}} = \frac{2e^{-\epsilon/kT}}{Z} = \frac{e^{-\epsilon/kT}}{e^{\epsilon/kT} + e^{-\epsilon/kT}}\].

Step 6: Calculate the Average Energy of the System

The average energy \(\langle E \rangle\) can be calculated using:\[\langle E \rangle = \sum_{i} P_i E_i\],where \(P_i\)s are the probabilities, and \(E_i\)s are corresponding energies. Therefore, we have:\[\langle E \rangle = \left(-\epsilon\right) \frac{e^{\epsilon/kT}}{e^{\epsilon/kT} + e^{-\epsilon/kT}} + \epsilon \frac{e^{-\epsilon/kT}}{e^{\epsilon/kT} + e^{-\epsilon/kT}}\],which simplifies to:\[\langle E \rangle = -\epsilon \tanh(\epsilon/kT)\].

Step 7: Analyze Results and Plots

We plot \(P_{\text{parallel}}\) and \(P_{\text{antiparallel}}\) against \(kT/\epsilon\) to observe the behavior. As \(T\) decreases, \(P_{\text{parallel}}\) becomes larger than \(P_{\text{antiparallel}}\). Plot the average energy \(\langle E \rangle\) also against \(kT/\epsilon\). The temperature at which you are more likely to find parallel spins than antiparallel spins is where \(P_{\text{parallel}} > P_{\text{antiparallel}}\), which occurs when \(\epsilon/kT > 0\).

Key Concepts

Boltzmann factors

In the realm of statistical mechanics, Boltzmann factors play a crucial role in determining the likelihood of a system being in a particular state. For the Ising model of two dipoles, we evaluate these factors based on the energy of each state.
The Boltzmann factor for a state with energy \( E \) at temperature \( T \) is given by \( e^{-E/kT} \), where \( k \) is the Boltzmann constant. This exponential expression emphasizes how the probability of being in a high-energy state diminishes as temperature decreases.

• Parallel states, such as \((+1, +1)\) and \((-1, -1)\), have lower energy (\(-\epsilon\)). Their Boltzmann factor is \( e^{\epsilon/kT} \).
• Antiparallel states, such as \((+1, -1)\) and \((-1, +1)\), are higher in energy (\(+\epsilon\)). Their factor is \( e^{-\epsilon/kT} \).

Thus, Boltzmann factors not only factor energy levels and temperature but are also integral in computing probabilities within a system.

partition function

The partition function, denoted as \( Z \), is a cornerstone in statistical physics. It sums up the Boltzmann factors of all possible states of a system. This function encapsulates the entire statistical behavior of a system. In our Ising model, the partition function is evaluated as follows:
The two parallel states and two antiparallel states contribute their respective Boltzmann factors:

• 2 parallel states contribute \( 2e^{\epsilon/kT} \)
• 2 antiparallel states contribute \( 2e^{-\epsilon/kT} \)

Thus, the overall partition function is \[ Z = 2(e^{\epsilon/kT} + e^{-\epsilon/kT}) \].
The partition function normalizes probabilities and is an essential component in finding the expectation values of various thermodynamic quantities.

average energy

The concept of average energy provides insight into the typical energy level of a system when at thermal equilibrium. The average energy \( \langle E \rangle \) is weighted based on the probabilities of each state, ensuring states with higher probabilities influence \(\langle E \rangle\) more.
For our Ising system, the computation is as follows:

• Each parallel state with energy \(-\epsilon\) has associated probability \( \frac{e^{\epsilon/kT}}{Z} \).
• Each antiparallel state with energy \(+\epsilon\) holds the probability \( \frac{e^{-\epsilon/kT}}{Z} \).

The weighted average energy is then realized as\[\langle E \rangle = -\epsilon \tanh(\epsilon/kT)\].This formula shows how, at high temperatures, the energy is distributed more uniformly, but at low temperatures, the system prefers lower energy states.

parallel and antiparallel states

Dipole interactions in an Ising model can be either parallel or antiparallel. Parallel states align either both up or both down, featuring lower energy \(-\epsilon\), whereas antiparallel states are where one dipole is up and the other is down, presenting higher energy \(+\epsilon\).
The probabilities of these states illuminate the statistical preference of each configuration relative to temperature:

• Probability of parallel states: Given by \( P_{\text{parallel}} = \frac{e^{\epsilon/kT}}{e^{\epsilon/kT} + e^{-\epsilon/kT}} \). This probability rises as temperature drops, favoring stability and lower energy.
• Probability of antiparallel states: Given by \( P_{\text{antiparallel}} = \frac{e^{-\epsilon/kT}}{e^{\epsilon/kT} + e^{-\epsilon/kT}} \). As temperature increases, these states become more likely as thermal energy overrides the dipole interaction strength.

Ultimately, at low temperatures, parallel states dominate, making systems energetically favorable.