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I am having trouble with the following physics exercise:

In the case of a paramagnet in a magnetic field $H$, show that from the requirement that $F(E)$ (the Helmholtz free energy) be minimal in thermodynamic equilibrium, one can produce the known expression for the average energy $\overline{E}$ of this system.

I am really not sure about how to proceed. I have an example in my book labeled "minimum property of the free energy of a paramagnetic system," in which it they assert that the entropy of the paramagnetic system with $N$ magnets and spin excess $2s$ is approximately equal to

$$-\left(\frac{1}{2}N + s\right)\log\left(\frac{1}{2} + \frac{s}{N}\right) - \left(\frac{1}{2}N - s\right) \log\left(\frac{1}{2} - \frac{s}{N}\right).$$

It then asserts that the energy in a magnetic field $B$ is $-2smb$, where $m$ is the magnetic moment of an elementary magnet. Through some computation, it is then shown that $\partial F/\partial s$ (the derivative of the Helmholtz free energy with respect to spin excess) occurs when $2s = N\tanh(mB/t)$. I'm not sure if this is relevant though, because, from my understanding, the energy $-2smb$ is the total internal energy rather than the average energy?

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Although this question arises from an exercise, you have already done almost all of the work to solve it. Your question is basically "should I be using the energy $E$ or the average energy $\overline{E}$ here?" or "am I justified in solving the problem this way?". This is a conceptual question, connected with the transformation of ensembles between microcanonical and canonical, which I believe this exercise is trying to help illustrate. So this answer will concentrate on that context.

You are correct that the energy $E$ of your paramagnet system is a simple linear function of the spin excess $2s$. So, go ahead, as you imply in your question: use that equation to substitute for $s$ in your entropy equation, to give you an entropy function $S(E)$. Use that to calculate a Helmholtz function, at a given $T$, which for the moment I'll call $\mathcal{F}(E)=E-TS(E)$. Do as the exercise asks: differentiate $\mathcal{F}(E)$ to find the energy $E=\hat{E}$ which minimizes $\mathcal{F}(E)$ and compare with the average energy $\overline{E}$ which comes from analyzing this model in the canonical ensemble (at temperature $T$). I'm leaving all of that to you.

The conversion between microcanonical and canonical ensembles is found in many statistical mechanics books. In the canonical ensemble at temperature $T$, the probability distribution function for energy may be written $$ \mathcal{P}(E) \propto \Omega(E)\exp(-E/k_BT) $$ where $\Omega(E)$ is the density of states (number of states per unit energy), and $\exp(-E/k_BT)$ is the Boltzmann factor for energy $E$. NB, here I am glossing over the fact that $E$ only takes discrete values in this spin model: I am treating it as a continuous variable (as the exercise expects you to do, when you differentiate with respect to $E$).

$\Omega(E)$ is a very rapidly increasing function of $E$, for large $N$. The Boltzmann factor is a very rapidly decreasing function of $E$ (which is, after all, an extensive variable). The consequence is that $\mathcal{P}(E)$ has a very sharp peak at some value $E=\hat{E}$. Moreover, because of the sharpness, this value will be very close to the average energy $\overline{E}$. Before differentiating to find the maximum of $\mathcal{P}(E)$, it is convenient to take logs: $$ -k_BT \ln \mathcal{P}(E) = E- k_BT\ln\Omega(E) + \text{const} = E-TS(E) + \text{const} = \mathcal{F}(E) + \text{const}, $$ where we recognize $S(E)=k_B\ln\Omega(E)$, the formula which was used to obtain your entropy expression. Differentiating $\mathcal{F}(E)$ with respect to $E$, setting to zero, and hence finding $\hat{E}$, is the key step. In the process of doing that, one gets to match up the energy $E$ of the microcanonical ensemble with the temperature of the canonical ensemble, by requiring $$ \left . \frac{\partial S(E)}{\partial E} \right|_{E=\hat{E}}=\frac{1}{T} $$

One can go further and discuss the width of the $\mathcal{P}(E)$ distribution, the link between $\mathcal{F}(\hat{E})$ and the thermodynamic free energy $F$, and so on, but that would go beyond the scope of the question. The main point is that, provided $N$ is large, we get the desired result: the temperature $T$ in the canonical ensemble is chosen to make $\overline{E}\approx \hat{E}=E$ in the microcanonical ensemble.

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