# Sound in a gas of non-interacting particles?

• greypilgrim
In summary, the conversation discusses the behavior of particles in an ideal gas and their interactions. The particles are assumed to be non-interacting in statistical approaches, but still collide and exchange momentum and energy. The Hamiltonian used to derive thermodynamic relations is purely kinetic and assumes that collisions occur only occasionally. The relaxation time for small perturbations in the gas is inversely proportional to the particle density, relative velocity, and scattering cross section. The gas can be described as a perfect fluid when the cross section is larger. Sound can still be transmitted in an ideal gas, but its speed is determined by considering small perturbations in density. As the collision cross section approaches zero, the damping of sound increases.

#### greypilgrim

Hi.

In some statistical approaches (e.g. canonical ensemble), the particles of an ideal gas are non-interacting. Still, it's possible to derive the ideal gas law and other thermodynamic relations.

Wikipedia gives an equation for the speed of sound in an ideal gas. How can there be waves in a medium of non-interacting components?

Or do statistical approaches not apply here because we have no thermal equilibrium and we need a kinetic approach?

The particles in an ideal gas interact when they collide and exchange momentum and energy.
Without collisions there will be no thermal equilibrium.

• vanhees71
Generally, when assuming the particles are non-interacting, this doesn't discount collisions. It just means they are being treated as essentially rigid spherical particles that collide but don't repel or attract each other.

• vanhees71 and Delta2
greypilgrim said:
the particles of an ideal gas are non-interacting
Non-interacting means that behave independent from one another, and exchange momentum only upon collision.

At the top of the wiki,
Generally, a gas behaves more like an ideal gas at higher temperature and lower pressure, as the potential energy due to intermolecular forces becomes less significant compared with the particles' kinetic energy, and the size of the molecules becomes less significant compared to the empty space between them
.

• Delta2
nasu said:
The particles in an ideal gas interact when they collide and exchange momentum and energy.
Without collisions there will be no thermal equilibrium.
Generally, when assuming the particles are non-interacting, this doesn't discount collisions. It just means they are being treated as essentially rigid spherical particles that collide but don't repel or attract each other.
256bits said:
Non-interacting means that behave independent from one another, and exchange momentum only upon collision.

But the Hamiltonian used to derive all thermodynamics in the canonical ensemble is purely kinetic:
$$H\left(\left\{\mathbf{q}_i\right\},\left\{\mathbf{p}_i\right\}\right)=\sum_i \frac{\mathbf{p}^2_i}{2m}$$
It's separable, there is no interaction at all.

Check vol. X of Landau and Lifshitz, where the Boltzmann equation is derived. Then you'll see, how this comes about: The particles are assumed to move almost always freely and that only from time to time collisions occur, i.e., the mean free path of the particles is assumed to be much smaller than the interaction range. The relaxation time, i.e., the time scale upon which small perturbations of the gas from equilibrium relaxes to equilibrium is parametrically given by
$$\tau_{\text{eq}} \sim \frac{1}{n v_{\text{rel}} \sigma},$$
where ##n## is the particle-number density, ##v_{\text{rel}}## the average relative velocity between the particles, and ##\sigma## the (elastic) scattering cross section for ##2 \rightarrow 2## collisions.

The gas is the better described by hydrodynamics (i.e., local thermal equibrium) the smaller ##\tau_{\text{eq}}## is. So a gas is the better a perfect fluid the larger the cross section is!

So my assumptions from the start is correct that one needs to take a kinetic approach and the ideal gas as modeled in the canonical ensemble is not capable of transmitting sound?

Sure it is. You have to consider a small perturbation (in density in this case) to get the speed of sound (linear-response theory).

Does it mean that, on the limit as the collision cross-section shrinks to zero, sound becomes impossible as the damping of sound diverges?

## 1. What is sound in a gas of non-interacting particles?

Sound in a gas of non-interacting particles refers to the propagation of pressure waves through a medium, such as air, composed of particles that do not interact with each other. These particles can be atoms, molecules, or other tiny particles that are free to move and collide with each other.

## 2. How is sound produced in a gas of non-interacting particles?

Sound is produced in a gas of non-interacting particles when an object or source, such as a speaker or a vibrating object, creates a disturbance in the medium. This disturbance causes the particles in the medium to vibrate, creating a pressure wave that travels through the medium as sound.

## 3. What factors affect the speed of sound in a gas of non-interacting particles?

The speed of sound in a gas of non-interacting particles depends on the temperature, density, and composition of the gas. In general, sound travels faster in gases with higher temperatures and lower densities. The composition of the gas can also affect the speed of sound, as different gases have different molecular weights and properties that can affect the propagation of sound waves.

## 4. How does sound travel through a gas of non-interacting particles?

Sound travels through a gas of non-interacting particles in the form of longitudinal waves, which means that the particles in the medium vibrate parallel to the direction of the wave's motion. As the disturbance or source continues to vibrate, the pressure waves travel outward in all directions, carrying the sound energy with them.

## 5. How does the behavior of sound in a gas of non-interacting particles differ from that in other mediums?

The behavior of sound in a gas of non-interacting particles is different from that in other mediums because the particles in a gas are not bound to each other and can move more freely. This allows sound waves to travel faster and with less attenuation compared to other mediums, such as solids or liquids, where the particles are more closely packed and have stronger interactions with each other.