# Dispersion relation statistical mechanics

• Luca2018
In summary, the conversation discusses the general case of the Einstein and Debye Model, which involves considering wave-excitations with a dispersion relation of the form \omega(\vec(k)) =c |\vec(k) |^(\gamma) in d dimensions (d=1,2,3) with wave-vector \vec(k). The task at hand is to determine the density of states g(\omega) for each dimension and arbitrary \gamma>0, as well as the corresponding power-law for energy and specific heat at low temperatures. The use of delta distribution is suggested for solving part a) and a reference to Ziman's "Theory of Solid" is recommended for further understanding.
Luca2018
Homework Statement
Consider general wave-excitations with a dispersion relations \omega(\vec(k)) =c |\vec(k) |^(\gamma) canpropagateind-dimensions(d=1,2,3)with wave-vector \vec(k).
a) Determine the density of states g(\omega) for each of the dimensions d =1,2,3and arbitrary γ>0.
b) What is the corresponding power-law at low temperatures for the energy
E \propto T^(\alpha) and the specific heat? (Hint: The independent modes of the wave- excitations can be described by the single-particle partition function for a harmonic oscillator).

Relevant Equations
I think you have to use the delta distribution to solve part a).
I know that this is the general case of the Einstein and Debye Model.

Luca2018 said:
Homework Statement:: Consider general wave-excitations with a dispersion relations \omega(\vec(k)) =c |\vec(k) |^(\gamma) canpropagateind-dimensions(d=1,2,3)with wave-vector \vec(k).
a) Determine the density of states g(\omega) for each of the dimensions d =1,2,3and arbitrary γ>0.
b) What is the corresponding power-law at low temperatures for the energy
E \propto T^(\alpha) and the specific heat? (Hint: The independent modes of the wave- excitations can be described by the single-particle partition function for a harmonic oscillator).

Relevant Equations:: I think you have to use the delta distribution to solve part a).

I know that this is the general case of the Einstein and Debye Model.

vanhees71
It seems to me like a generalization of the Debye Model. If you study the Debye Model (for example check Ziman's "theory of solid" chapter 2, but you can find it everywhere) you will see that it is assumed a dispersion relation of the form ##\omega = ck##; just substitute that relation with the one you are given and do the exactly same calculations.

## 1. What is a dispersion relation in statistical mechanics?

A dispersion relation in statistical mechanics is a mathematical relationship that describes the relationship between the energy and momentum of a particle or system. It is often used to study the behavior of particles in a material or the properties of a system at the microscopic level.

## 2. How is a dispersion relation determined experimentally?

A dispersion relation can be determined experimentally by measuring the energy and momentum of particles in a material or system and plotting them on a graph. The slope of the resulting curve is the dispersion relation, which can then be used to make predictions about the behavior of the particles or system.

## 3. What is the significance of a dispersion relation in statistical mechanics?

The dispersion relation is significant in statistical mechanics because it provides a way to understand the behavior of particles in a material or system. It can also be used to calculate important properties such as the speed of sound, thermal conductivity, and electrical conductivity.

## 4. How does the dispersion relation relate to the concept of energy bands?

The dispersion relation is closely related to the concept of energy bands, which describe the allowed energy states of particles in a material. The shape of the dispersion relation determines the width and spacing of energy bands, which in turn affects the properties of the material.

## 5. Can a dispersion relation be used to study non-particle systems?

Yes, a dispersion relation can be used to study non-particle systems such as waves or fields. In these cases, the dispersion relation describes the relationship between the frequency and wavelength of the wave or the strength and direction of the field. This allows for the prediction of wave behavior or the properties of the field.

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