Thermodynamic Response Functions from Free Energy

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Hello! Welcome to the fourth lesson in our course on the statistical mechanics of phase transitions.

In our last lesson, we explored the concept of spontaneous symmetry breaking (SSB). We saw how, below a critical temperature $T_{c}$ , the free energy landscape $F (T, m)$ changes its shape, causing the symmetric state (order parameter $m = 0$ ) to become unstable and leading the system to settle into a new, less symmetric ground state with $m \neq = 0$ .

This lesson will make that picture quantitative. We will see how the derivatives of the free energy, which describe the shape and curvature of this landscape, correspond directly to measurable physical quantities called thermodynamic response functions. These functions—like specific heat, compressibility, and magnetic susceptibility—tell us how a system responds to external stimuli. Understanding them is key to connecting the abstract theory of phase transitions to concrete experimental observations.

From Partition Function to Thermodynamics

The central quantity bridging the microscopic world of statistical mechanics and the macroscopic world of thermodynamics is the free energy. For systems at constant temperature $T$ , volume $V$ , and particle number $N$ (the canonical ensemble), the relevant potential is the Helmholtz free energy, $A$ .

As you'll recall from your statistical mechanics course, it is defined in two ways:

Thermodynamically: $A = E - TS$ , where $E$ is the internal energy and $S$ is the entropy.
Statistically: $A = - k_{B} T ln Z$ , where $Z$ is the canonical partition function.

This second relation is the master key. If we can calculate the partition function $Z$ , we can calculate the free energy $A$ and, from it, all other thermodynamic quantities.

The first derivatives of the free energy give us the fundamental state variables. The differential form of the Helmholtz free energy is:
$d A = - S d T - p d V + μ d N$
From this, we can immediately identify:

Entropy: $S = - (\frac{\partial A}{\partial T})_{V, N}$
Pressure: $p = - (\frac{\partial A}{\partial V})_{T, N}$
Chemical Potential: $μ = (\frac{\partial A}{\partial N})_{T, V}$

For problems involving magnetic fields, we simply add a magnetic work term, $- M d H$ . The differential becomes $d A = - S d T - p d V - M d H + μ d N$ , which gives the magnetization:

Magnetization: $M = - (\frac{\partial A}{\partial H})_{T, V, N}$

Response Functions as Second Derivatives

While first derivatives give us the state of the system in equilibrium, second derivatives tell us about the stability of that equilibrium. They measure the system's response to a small change in external conditions. These are the response functions.

Let's look at a concise summary of the key response functions and their relation to thermodynamic potentials.

Response Functions

This document from the University of Rochester provides a clear and efficient summary of the main response functions and how they are derived from thermodynamic potentials. We'll use it as our guide.

Please read the first page of this document, focusing on the definitions in Eq. (1.8.1) and their expressions as second derivatives in Eq. (1.8.2). Also read the final short section (1.8.31) which gives the common expression for C_V.

As you've just read, response functions are second derivatives of the appropriate free energy. Let's walk through the most important examples.

1. Specific Heat at Constant Volume, $C_{V}$

The specific heat measures how much heat is required to change a system's temperature. At constant volume, its fundamental definition is $C_{V} = T (\frac{\partial S}{\partial T})_{V}$ .

Since we know $S = - (\frac{\partial A}{\partial T})_{V}$ , we can substitute this in:
$C_{V} = T \frac{\partial}{\partial T} (- \frac{\partial A}{\partial T})_{V} = - T (\frac{\partial ^{2} A}{\partial T ^{2}})_{V}$
This directly links the specific heat to the curvature of the free energy with respect to temperature.

Alternatively, and perhaps more familiarly, $C_{V}$ is the rate of change of internal energy with temperature. As shown in Eq. (1.8.31) of the notes you just read, these definitions are equivalent:
$C_{V} = (\frac{\partial E}{\partial T})_{V}$

2. Isothermal Compressibility, $κ_{T}$

Compressibility measures how much a system's volume changes in response to pressure. The isothermal compressibility is defined as:
$κ_{T} = - \frac{1}{V} (\frac{\partial V}{\partial p})_{T}$
Calculating this from the Helmholtz free energy $A (T, V)$ is awkward because pressure $p$ is the variable we want to change, but volume $V$ is the natural variable of $A$ . It's more natural to use a potential whose variables are $T$ and $p$ . This is precisely what the Gibbs free energy, $G (T, p, N)$ , is for, defined by the Legendre transform $G = A + p V$ .

The differential of $G$ is $d G = - S d T + V d p + μ d N$ . From this, we see that $V = (\frac{\partial G}{\partial p})_{T, N}$ . Substituting this into the definition of $κ_{T}$ gives:
$κ_{T} = - \frac{1}{V} (\frac{\partial ^{2} G}{\partial p ^{2}})_{T, N}$
This shows that compressibility is proportional to the second derivative of the Gibbs free energy with respect to pressure.

3. Isothermal Magnetic Susceptibility, $χ_{T}$

The magnetic susceptibility is the direct analog of compressibility for a magnetic system. It measures how strongly the magnetization responds to an external magnetic field.
$χ_{T} = (\frac{\partial M}{\partial H})_{T}$
Let's derive its form from the Helmholtz free energy $A (T, H)$ (assuming constant V, N). As we saw, the magnetization is given by the first derivative:
$M = - (\frac{\partial A}{\partial H})_{T}$
Taking one more derivative with respect to $H$ immediately gives us the susceptibility:
$χ_{T} = \frac{\partial}{\partial H} (- \frac{\partial A}{\partial H})_{T} = - (\frac{\partial ^{2} A}{\partial H ^{2}})_{T}$
So, susceptibility is the curvature of the free energy with respect to the magnetic field.

Test your understanding!

Near a continuous phase transition, a system becomes highly susceptible to external influences. For example, a ferromagnet near its Curie temperature will show a very large change in magnetization for a very small applied magnetic field.

How is this physical behavior—a large response to a small stimulus—reflected in the shape of the free energy function $A (T, H)$ near the critical point?

Show answer

A large susceptibility $χ_{T}$ means that the second derivative, $- (\frac{\partial ^{2} A}{\partial H ^{2}})_{T}$ , is large and positive. This implies that the free energy function $A (H)$ becomes very "flat" or has a very small positive curvature near $H = 0$ . A small change in $H$ can then lead to a large change in the system's state (magnetization) because the energetic cost of changing $M$ is small. At the critical point itself, this curvature goes to zero, leading to a divergent susceptibility.

Response Functions and Fluctuations

There is a profound and beautiful connection between response functions and the spontaneous fluctuations of a system at equilibrium. This is a cornerstone of statistical physics, often referred to as the Fluctuation-Dissipation Theorem. It states that the same microscopic forces that cause a system to dissipate energy when driven by an external field also cause it to fluctuate at equilibrium.

Let's see a concrete derivation of this for the specific heat.

Statistical Mechanics Lecture 5

The following segment from a Stanford lecture by Leonard Susskind provides a masterful derivation of the relationship between specific heat and the fluctuations of energy in a system. This is a classic and fundamental result.

Watch from 1:16:58 to 1:30:15. Follow the derivation that connects the mean-square fluctuation in energy, (\langle (\Delta E)^2 angle = \langle E^2 angle - \langle E angle^2), to the specific heat at constant volume, C_V. The key steps involve expressing the average energy and its square as derivatives of the partition function with respect to eta = 1/k_B T.

The final result from the video is one of the most important in statistical mechanics:
$⟨(Δ E)^{2} ⟩ = k_{B} T^{2} C_{V}$
This equation tells us something remarkable: the specific heat $C_{V}$ , a macroscopic quantity you can measure with a calorimeter, is directly proportional to the microscopic variance of the energy distribution of the system at equilibrium.

Similar relationships exist for other response functions:

Susceptibility and Magnetization Fluctuations: $⟨(Δ M)^{2} ⟩ = k_{B} T χ_{T}$
Compressibility and Number Fluctuations: $⟨(Δ N)^{2} ⟩ = k_{B} T \frac{N ^{2}}{V} κ_{T}$ (in the grand canonical ensemble)

These relationships provide the intuition for why response functions diverge at a critical point. As a system approaches criticality, fluctuations occur over all length scales, leading to enormous fluctuations in the order parameter (e.g., magnetization). This, in turn, manifests as a divergence in the corresponding response function (susceptibility).

Magnetic Specific Heat Capacity Near a Critical Point — The magnetic specific heat of Gadolinium Trichloride (GdCl3) as a function of temperature. The sharp, lambda-shaped peak near 3.4 K is the signature of a continuous phase transition. This divergence in a response function is the hallmark of critical behavior, driven by large-scale fluctuations in the system.

Conclusion

This lesson has connected the abstract concept of the free energy landscape to concrete, measurable physical quantities.

Key Takeaways:

Thermodynamic potentials, like the Helmholtz free energy $A$ , are derived from the partition function $Z$ via $A = - k_{B} T ln Z$ .
First derivatives of the appropriate free energy yield macroscopic state variables like pressure ( $p$ ), entropy ( $S$ ), and magnetization ( $M$ ).
Second derivatives of the free energy give the thermodynamic response functions, such as specific heat ( $C_{V}$ ), compressibility ( $κ_{T}$ ), and magnetic susceptibility ( $χ_{T}$ ). They quantify the system's reaction to external changes.
Response functions are directly related to the magnitude of equilibrium fluctuations in the system (Fluctuation-Dissipation Theorem).
The divergence of response functions at a critical point is the experimental signature of a phase transition and is a direct consequence of large-scale critical fluctuations.

Preview of the Next Lesson:

We have now seen that thermodynamic quantities like magnetization and response functions like susceptibility behave in a dramatic and singular way near a critical point. In our next lesson, we will begin to characterize this singular behavior quantitatively by introducing critical exponents. We will define the standard set of exponents (α, β, γ, δ) that describe how these quantities scale as we approach the critical point, paving the way to understanding the powerful concept of universality.