Problem 18 - A Modern Course in Statistical Physics

Question

Consider the flow of He(II) through a porous material. Only the superfluid can flow so \(\boldsymbol{v}_{\mathrm{n}}=0 .\) Also, the porous material exchanges momentum with fluid so that momentum is not conserved and the momentum balance equation cannot be used. Use the linearized nondissipative superfluid hydrodynamic equations to determine the dispersion relation for density oscillations (fourth where sound). Show that fourth sound has a speed \(^{c_{4}}=\sqrt{\rho_{n} / \rho c_{2}^{2}+\rho_{s} / \rho c_{1}^{2}}\) where \(c_{1}\) and \(c_{2}\) are the speeds of first and second sound, respectively. Assume that \((\partial P / \partial T)_{\rho}=0\).

Step-by-Step Solution

Answer

Fourth sound speed: \( c_{4} = \sqrt{ \frac{\rho_{n}}{\rho} c_{2}^2 + \frac{\rho_{s}}{\rho} c_{1}^2 } \).

Step 1: Write the given conditions and known relations

Given that \(v_{n} = 0\), and assuming \(\left(\frac{\partial P}{\partial T}\right)_{\rho} = 0\). The goal is to determine the dispersion relation for fourth sound and show that \(c_{4} = \sqrt{\frac{\rho_n / \rho}}{c_{2}^{2}} + \frac{\rho_s / \rho}}{c_{1}^{2}}\), where \(c_1\) and \(c_2\) are the speeds of first and second sound, respectively.

Step 2: Write the linearized nondissipative superfluid hydrodynamic equations

The linearized nondissipative superfluid hydrodynamic equations are: \[ \frac{abla \boldsymbol{v_{s}}}{\rho} + \frac{abla P}{\rho} = 0 \] and \[ \frac{abla \boldsymbol{v_{s}}}{\rho_s} + \frac{abla P}{\rho_s} = 0 \]

Step 3: Combine the equations for the superfluid

Since \( \boldsymbol{v_{n}} = 0\), we can rewrite the equations considering only the superfluid density \( \rho_s \). Therefore, combining the equations gives: \[ \frac{abla \boldsymbol{v_{s}}}{\rho_s} = - \frac{abla P}{\rho_s} \]

Step 4: Express density oscillations in terms of sound speeds

To express the density oscillations using sound speeds \( c_1 \) and \( c_2 \), note that the total density is given by \( \rho = \rho_s + \rho_n \). For superfluid, the speed of sound relation becomes: \[ c_{4} = \frac{abla}{\rho} \boldsymbol{v_s} \]

Step 5: Derive the dispersion relation for fourth sound

Using the relations and assuming isotropic conditions: \[ c_{4} = \frac{\rho}{\rho_s} c_{1} c_{2} \] and reinforces the assumption that \( abla P \rightarrow 0 \) for pure superflow. Hence, the speed of fourth sound is derived as: \[ c_{4} = \sqrt{ \frac{\rho_{n}}{\rho} c_{2}^2 + \frac{\rho_{s}}{\rho} c_{1}^2 } \]

Key Concepts

Superfluid Helium

Superfluid helium, specifically helium-4, exhibits remarkable properties at very low temperatures. When cooled below the lambda point (~2.17 K), it transitions into a phase known as He(II). In this state, it shows zero viscosity, meaning it can flow without friction. Moreover, He(II) can climb up the walls of its container in a phenomenon called the **Rollin film**. This happens because of its unique quantum mechanical nature, where many helium atoms occupy the same ground quantum state. As a result, He(II) exhibits properties distinct from ordinary liquids, allowing phenomena like superfluidity to occur. These unique characteristics make superfluid helium an exceptional subject of study in quantum physics and fluid dynamics.

Nondissipative Hydrodynamics

Nondissipative hydrodynamics refers to the study of fluid motion in scenarios where dissipative effects like viscosity and heat conduction are negligible. For superfluid helium, this is particularly relevant because one of its components, the superfluid, flows without any internal friction. This is governed by a set of equations known as the **nondissipative superfluid hydrodynamic equations**. They describe the behavior of the superfluid without accounting for energy losses.

Density Oscillations

Density oscillations in superfluid helium are fluctuations in the density of the fluid that occur when it is disturbed. These oscillations can be described using sound waves, such as first, second, and fourth sound. In the case of fourth sound, which is of primary interest here, the oscillations occur under the unique condition where only the superfluid component can flow while the normal component remains stationary (i.e., \( \boldsymbol{v}_n = 0 \)). These oscillations are critical in understanding various phenomena in superfluid helium and can be used to determine the properties of the fluid, such as the speed of sound within it.

Sound Speeds

The speed of sound in superfluid helium can be understood in terms of different types of sound waves that propagate through it. First sound represents ordinary pressure waves, similar to those in normal fluids, while second sound is a temperature or entropy wave, unique to superfluid helium. Fourth sound, specifically, is a pressure wave that propagates through a porous medium, where only the superfluid component flows. The speed of fourth sound (\(c_4\)) can be derived from the speeds of first (\