Conventions for chemical potentials

General
1

In a conversation with Mateus we came across this issue: how to define the various chemical potentials that we have in our EoSes, particularly when the system might be out of beta equilibrium.

Chemical potentials are associated with conserved (or “almost conserved”, i.e. slowly equilibrating) quantities. So, for example, one should write the chemical potential of the electron in terms of its conserved charges which are electric charge Q (-1) and electron lepton number L_e (+1),

\mu_e = -\mu_Q + \mu_{Le} = \mu_{NQ} + \mu_{Le}

where NQ is negative charge (Note that \mu_{NQ}=-\mu_Q; we should not use the notation “\mu_Q” for the negative charge chemical potential!) and L_e is electron lepton number.
In neutrino-free matter the neutrinos carry lepton number away, so \mu_{Le}=0. But in neutrino-trapped matter \mu_{Le} will not be zero.

The nucleons carry baryon number (+1), electric charge (0 or +1), isospin (\pm\frac{1}{2}), and strangeness (0). So one would write

\mu_n = \mu_B - \frac{1}{2}\mu_I
\mu_p = \mu_B - \mu_{NQ} + \frac{1}{2}\mu_I
etc

When the system is out of cold beta equilibrium this means there is a nonzero \mu_I.
But \mu_e is always equal to \mu_{NQ} because the electron has charge Q=-1 (or NQ=+1).

2

To add some context, our conversation began from a discussion on which variables are necessary for the Flavor Equilibration module, and whether we should pass the electron chemical potential.

Currently, both CMF and CEFT define the charge and baryon chemical potentials from the relations
\mu_p= \mu_B + \mu_Q
\mu_n= \mu_B
(@dfriede1 correct me if I’m wrong about CEFT)

@alford If we completely neglect \mu_I (i.e. set \mu_I=0), would your module not work?

@vdexheim @jakinh your opinion would be very much welcome on how to reconcile the definition from CMF and CEFT with Flavor.

1 Like
3

Our module needs to explore away from cold beta equilibrium so it needs to explore \mu_I\neq 0. This is because at T\gtrsim 1 MeV true beta equilibrium is no longer at \mu_I=0.

4

@alford Correct me if I am wrong, but this seems only a matter of definition.

The nuclear modules currently produce a table with np matter, which has two chemical potentials (\mu_p, \mu_n). The charge potential is defined at this stage in our current formalism, that is why we do not relate it to the electron charge, but to the proton.

If we added \mu_I as well, we’d have 3 charge potentials related to the two of neutrons and protons, so there would be an ambiguity in their definition.

Then, the table goes through the Lepton module, which neutralizes the system and adds the extra electron chemical potential \mu_e, which is independent of the others out of equilibrium.

Now we have the three independent potentials (\mu_n, \mu_p, \mu_e), which we redefine as
\mu_n= \mu_B
\mu_p=\mu_B + \mu_Q

You need another set of independent potentials, which I will call (\tilde \mu_B, \tilde \mu_Q, \tilde \mu_I), which are defined from
\mu_n= \tilde \mu_B - \tilde \mu_I /2
\mu_p= \tilde \mu_B + \tilde \mu_I /2 - \tilde \mu_{NQ}
\mu_e= \tilde \mu_{NQ}

In both cases we have 3 independent potentials, so we can just map them (\mu_B, \mu_Q, \mu_e) \rightarrow (\tilde \mu_B, \tilde \mu_{NQ}, \tilde \mu_I) as

\tilde \mu_B= \mu_B + \frac{\mu_Q + \mu_e}{2}
\tilde \mu_I= \mu_Q + \mu_e
\tilde \mu_{NQ} = \mu_e.

Could this be done internally in your module? I think that’s the easiest solution.

5

Exec summary: Hadronic modules should just deal with \mu_n and \mu_p, and not talk about \mu_B etc.

We have two ways to talk about chemical potentials: assigning them to particles, or assigning them to conserved charges.

There is no ambiguity about the particle chemical potentials. The hadronic EoS modules can calculate \mu_n and \mu_p for non-neutral, non-beta-equilibrated hadronic matter. No problem there, and I think that’s all the hadronic EoS modules should do. It is better for them to avoid talking about the charge chemical potentials like \mu_B, \mu_Q, \mu_I, \mu_{Le} because these have complications, which I will now describe.

In order to determine the charge chemical potentials we have to make a choice. Neutral non-beta-equilibrated matter is described by 3 particle chemical potentials, and we have 2 obvious charges that are conserved, baryon number and electric charge, so we have an incomplete mapping

(\mu_n,\mu_p,\mu_e) \leftrightarrow (\mu_B,\mu_Q,\mu_X)

What do we adopt as the mysterious X charge, which gets equilibrated away by weak interactions?
I think the natural answer is isospin, since it is already part of our nuclear physics framework, and is known to be violated by weak interactions.

But it seems that you are proposing a different X charge, where \mu_e = \mu_{NQ} + \mu_X. So your X is like lepton number, but it is not actually lepton number because neutrinos (if we added them) would not carry it.
So it doesn’t correspond to any charge that one finds in textbooks.

Would it be simpler to just stick to \mu_n,\mu_p,\mu_e, and let user modules infer the charge chemical potentials, i.e. make their own choice of X?

6

When we created MUSES a long time ago, we decided to use \mu_B, \mu_S, and \mu_Q as the independent chemical potentials, mainly due to lattice QCD preferences. But in my opinion, we can easily move between definitions, since \mu_Q=\mu_I and Y_Q, Y_I, and Y_S relate. The CompOSE instruction manual (https://compose.obspm.fr/download/pdf/manual_v3.00.pdf) has a nice section discussing how to define distance from beta equilibrium using what they call the lepton chemical potential, defined as the difference between \mu_Q a \mu_{electron} (with some specific signs). I suggest we continue this discussion on the Monday NS MUSES meeting, becuase I think I might be missing something (I am extremely jetlagged!).