Problem 19 - A Modern Course in Statistical Physics

Question

Derive the wave equations for first sound and second sound in a nondissipative superfluid for the case when \((\partial P / \partial T)_{p}=0\). Show that only second sound propagates when the momentum density is zero. Show that only first sound propagates when the super fluids and normal fluids move in phase so \(v_{\mathrm{n}}=v_{s}\).

Step-by-Step Solution

Answer

When the momentum density is zero, only second sound propagates. When normal and superfluids move in phase (\backslashphi_{\mathrm{n}}=v_{s}), only first sound propagates.

Step 1: Understanding the Properties of the System

Identify that the superfluid is nondissipative and \((\frac{\backslashpartial P}{\backslashpartial T})_p = 0\), which means that temperature changes do not affect pressure when entropy remains constant.

Step 2: Define the Two Types of Sound Waves

First sound involves the density of the entire fluid, while second sound primarily involves temperature or entropy waves. Let \( v_n \) and \( v_s \) be the velocities of the normal fluid and superfluid components respectively.

Step 3: Write the Basic Equations of Motion

Use conservation equations for mass, momentum, and energy for superfluids. The expressions are: \[ \backslashfrac{\backslashpartial \rho}{\backslashpartial t} + \backslashes \backslashcdot (\rho v) = 0 \]\ and \[ \backslashfrac{\backslashpartial (\rho_n v_n + \rho_s v_s)}{\backslashpartial t} + \backslashbackslashnabla \backslashcdot (\rho_n v_n v_n + \rho_s v_s v_s) = 0 \].

Step 4: Establish Boundary Conditions

Given \( v = 0 \), analyze the implications. First, sound waves rely on divergences of the combined fluid velocities \((v_n + v_s)\). With \(\backslashpartial P/ \backslashpartial T = 0\), equation simplification shows second sound will dominate due to motion of temperature wave only.

Step 5: Demonstrate Second Sound Propagation

For \( \rho_n v_n = -\rho_s v_s \), normal and superfluid densities balance. When \( v = 0 \), re-evaluate momentum equations; only second sound remains since temperature changes dominate. Only the entropy wave persists.

Step 6: Show First Sound Propagation Condition

When \( v_n = v_s \), substitute into motion equations. Normal and super fluid terms move in phase (\backslashpartial v_n = \backslashpartial v_s). This results in the propagation form for first sound —a density wave involving the entire fluid.

Step 7: Final Assessment and Conclusion

Summarize that second sound occurs when momentum density is zero due to separation of temperature effects, and first sound when superfluid and normal move in tandem.

Key Concepts

First Sound

First sound is a fundamental concept in the study of superfluids. It refers to the conventional sound waves, similar to those we hear in everyday life. These waves occur due to compressions and rarefactions in the fluid, altering its density. In a superfluid, first sound involves both the normal fluid and superfluid components.
When studying superfluids, we denote the velocities of the normal fluid and superfluid as \(v_n\) and \(v_s\) respectively. Under the condition where these two velocities are equal (\(v_n = v_s\)), first sound propagates through the fluid. This happens because the components move in unison, causing density variations.
Mathematically, this can be expressed through the conservation of mass and momentum equations, where both components contribute equally to the sound wave. Therefore, first sound represents an overall density wave in the entire fluid.

Second Sound

Second sound is unique to superfluids. Instead of involving density changes, it primarily involves temperature or entropy waves. This type of sound wave is a result of temperature variations propagating through the fluid.
Second sound becomes significant especially in conditions where the momentum density is zero. This means that the overall momentum of the fluid does not change, letting temperature differences move independently from the density changes. In many cases, especially when \((\frac{\partial P}{\partial T})_p = 0\), second sound becomes the dominant wave, with only temperature variations contributing to its propagation.
It's important to remember that second sound reflects how quantum mechanical effects in superfluids allow for separate temperature and density wave modes—a phenomenon not seen in ordinary fluids.

Nondissipative Superfluid

A nondissipative superfluid is a superfluid where there is no dissipation of energy. This implies that there are no losses due to friction or viscosity, allowing the fluid to exhibit extraordinary behaviors. In such systems, various interesting phenomena occur, including the propagation of first and second sound.
In a nondissipative superfluid, energy remains conserved, and disturbances can propagate without losing any energy to internal friction. This is why both first and second sound can travel over long distances within the fluid. The nondissipative nature is crucial in maintaining the pristine conditions required for observing second sound, which can be easily dampened in ordinary, viscous fluids.
Analyses of such fluids often leverage equations that simplify under the assumption of zero dissipation, making it easier to study the wave behaviors.

Momentum Density

Momentum density in the context of superfluids refers to the distribution of momentum within the fluid. It is a vector quantity, combining the contributions from both the normal fluid and superfluid components.
In a superfluid, the total momentum density can be given by the sum of the momentum densities of the two components: \(\rho_n v_n + \rho_s v_s\). Analyzing cases where this momentum density is zero gives critical insights into the behavior of second sound. Specifically, when momentum density is zero, temperature waves dominate the propagation—this is a key condition for second sound.
By understanding momentum density, it becomes easier to identify the mechanisms behind various wave propagations in superfluids and helps in delineating between first and second sound phenomena.

Wave Equations

Wave equations are crucial for describing the propagation of disturbances in superfluids. For first and second sound, these equations help in modeling how waves travel through the fluid. The basic form involves equations derived from the conservation laws of mass, momentum, and energy.
For first sound, the wave equation often includes terms that account for the total density and velocity of the fluid. This reflects the involvement of both normal and superfluid components in the density wave propagation.
On the other hand, the wave equation for second sound mainly involves temperature or entropy terms, given the predominance of thermal waves in this phenomenon. By solving these wave equations under different conditions (like zero momentum density or synchronised fluid velocities), one can derive relationships that dictate when each type of sound will dominate.
Understanding these wave equations provides a core mathematical toolset for analyzing and predicting behaviors in superfluid environments.