# Express entropy density in terms of energy density (Ashcroft/Mermin)

• EE18
In summary, the starting point is the identity $$\left(\frac{\partial u}{\partial T}\right)_n = T\left(\frac{\partial s}{\partial T}\right)_n.$$I integrate both with respect to ##T## after dividing through by ##T##, finding$$\int_0^T \left(\frac{\partial s}{\partial T'}\right)_n = s(T) - s(0) = s(T)$$I then use an integration by parts to combine the integrals together and find$$\frac{1}{2} \int \ldots = x + \frac{1}{2} EE18 Homework Statement I am trying to solve Problem 2a in Chapter 2 of Ashcroft and Mermin. I'm not asking about the whole problem, but just the start. Essentially, one needs to expression the entropy density ##s## in terms of the energy density ##u## and, as far as I can tell from other methods to solving this problem, I should arrive at ##s(T) = u(T)/T##. Relevant Equations ##s(T) = u(T)/T## The starting point is the identity$$\left(\frac{\partial u}{\partial T}\right)_n = T\left(\frac{\partial s}{\partial T}\right)_n.$$I then try to proceed as follows: Integrating both with respect to ##T## after dividing through by ##T##, we find$$ \int_0^T \left(\frac{\partial s}{\partial T'}\right)_n = s(T) - s(0) = s(T)  = \int_0^T \frac{1}{T'}\left(\frac{\partial u}{\partial T'}\right)_n = \frac{u}{T'}\biggr\rvert_0^T + \int_0^T \frac{1}{T'^2}u ...$$where we have used an integration by parts, the fundamental theorem of calculus, and the third law of thermodynamics (in (1)), and where we recall that u_0 is the energy density in the ground state (see AM (2.79)). But I can't seem to go any further. I found a solution which does something I totally can't understand in the last line, and I'm hoping someone can either clarify that solution and/or show me how to go further. The problem is obviously the last integral on the right-hand side. #### Attachments • FullSol.png 27.2 KB · Views: 84 Last edited: EE18 said: I found a solution which does something I totally can't understand in the last line, and I'm hoping someone can either clarify that solution and/or show me how to go further. The problem is obviously the last integral on the right-hand side. Compare to the line ##s=\ldots##: you have the same integral on the RHS (with a factor 1/2), so you can combine the integrals together:$$
\int \ldots = x + \frac{1}{2} \int \ldots
$$so$$
\frac{1}{2} \int \ldots = x
$$DrClaude said: Compare to the line ##s=\ldots##: you have the same integral on the RHS (with a factor 1/2), so you can combine the integrals together:$$
\int \ldots = x + \frac{1}{2} \int \ldots
$$so$$
\frac{1}{2} \int \ldots = x
$$Thanks for helping out! Where is that factor of ##1/2## coming from though? EE18 said: Thanks for helping out! Where is that factor of ##1/2## coming from though? From the integration by parts (integral of ##T^{-2} dT##). Edit: To be clear, I moved the factor in front of the integral, but it appears in ##1/2T## in the image you posted. DrClaude said: From the integration by parts (integral of ##T^{-2} dT##). Edit: To be clear, I moved the factor in front of the integral, but it appears in ##1/2T## in the image you posted. Maybe I'm going crazy, but isn't ##-\int 1/x^2 dx = 1/x##? I don't see where the 1/2 comes from? DrClaude and vanhees71 Of course! DrClaude Sorry, I was sloppy when I tried to reproduce what was is the image. I can't make sense of it now. vanhees71 DrClaude said: Sorry, I was sloppy when I tried to reproduce what was is the image. I can't make sense of it now. I begin to think that image is rubbish. It's essentially doing 2 successive integrations by parts (the 2nd one incorrectly, imho). When done correctly, it should just return (uselessly) to its starting point. Btw, since at least one other variable is being held constant, maybe you just treat the first line as if it were$$ \frac{du}{dT} ~=~ T \; \frac{dS}{dT} ~,~~~~~~ \Rightarrow~~~
du ~=~ T dS ~.$$Of course, you've got to be careful about what depends on what, and what's being held constant. PhDeezNutz, DrClaude and vanhees71 I think the authors were lazy in their notation of partial derivatives. You were given the following. ##\left(\frac{\partial U}{\partial T} \right)_n = T \left( \frac{\partial S}{\partial T} \right)_n## It should read (I think)$$\left(\frac{\partial U}{\partial T} \right)_{n,total} = T \left( \frac{\partial S}{\partial T} \right)_{n,total}U\left(S\left(U,T\right), T\right)S\left(U\left(S,T\right), T\right)$$There are more than one way that ##U## or ##S## depend on T$$\left( \frac{\partial U}{\partial T} \right)_{n,total} = \frac{\partial U}{\partial S} \left(\frac{\partial S}{\partial T} \right)_{n,direct} + \left(\frac{\partial U}{\partial T} \right)_{n,direct}T\left( \frac{\partial S}{\partial T} \right)_{n,total} = T \frac{\partial S}{\partial U} \left(\frac{\partial U}{\partial T} \right)_{n,direct} + T\left(\frac{\partial S}{\partial T} \right)_{n,direct}$$Setting them equal$$\frac{\partial U}{\partial S} \left(\frac{\partial S}{\partial T} \right)_{n,direct} + \left(\frac{\partial U}{\partial T} \right)_{n,direct} = T \frac{\partial S}{\partial U} \left(\frac{\partial U}{\partial T} \right)_{n,direct} + T\left(\frac{\partial S}{\partial T} \right)_{n,direct}$$Comparing like terms$$\frac{\partial U}{\partial S} = T \Rightarrow U = STT\frac{\partial S}{\partial U} = 1 \Rightarrow S = \frac{U}{T}

This is my take on it. I could be wrong of course. I know in principal there are more terms since ##n## might not be constant.

## What is entropy density and how is it related to energy density?

Entropy density is a measure of the amount of disorder or randomness per unit volume in a system. Energy density, on the other hand, is the amount of energy stored in a given system per unit volume. The relationship between entropy density and energy density can be derived from thermodynamic principles, often involving the use of thermodynamic potentials such as the Helmholtz free energy. In the context of Ashcroft and Mermin, this relationship is typically explored in the framework of statistical mechanics and solid-state physics.

## How do Ashcroft and Mermin approach the concept of entropy density in their book?

Ashcroft and Mermin approach entropy density from a statistical mechanics perspective. They discuss how the entropy of a system can be derived from the microscopic states that the system can occupy. By considering the distribution of these states and their respective probabilities, they derive expressions for entropy and, consequently, entropy density. They also explore how these concepts apply to various physical systems, including solids and electrons in metals.

## Can you provide a basic formula that relates entropy density to energy density?

A basic formula that relates entropy density $$s$$ to energy density $$u$$ can be derived using the thermodynamic relation:$s = \frac{1}{T} (u + p - \mu n)$where $$T$$ is the temperature, $$p$$ is the pressure, $$\mu$$ is the chemical potential, and $$n$$ is the number density. This formula encapsulates the interplay between different thermodynamic quantities and provides a way to express entropy density in terms of energy density and other relevant parameters.

## What assumptions are typically made when expressing entropy density in terms of energy density?

When expressing entropy density in terms of energy density, several assumptions are often made to simplify the derivation. These assumptions may include treating the system as being in thermal equilibrium, assuming a specific form of the equation of state, and considering idealized models of the physical system (e.g., ideal gas, free electron gas). Additionally, it is common to assume that the system is homogenous and isotropic, meaning that properties do not vary with position or direction.

## How does the concept of entropy density apply to real-world materials, according to Ashcroft and Mermin?

Ashcroft and Mermin apply the concept of entropy density to real-world materials by examining specific cases such as metals, semiconductors, and insulators. They discuss how the electronic structure of these materials influences their thermodynamic properties, including entropy and energy densities. For instance, in metals, the free electron model is used to describe how electron states contribute to the overall entropy density. They also explore how lattice vibrations (phonons) contribute to the

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