# Chemical potential equilibrium cosmology

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In summary, the conversation discusses the concept of chemical potential in thermodynamics. The equilibrium number density has the same form as the non-equilibrium number density, but with a chemical potential of zero. This is because in equilibrium, there is no preferred direction of the reaction, and by definition, the chemical potential must vanish. The chemical potential represents the energy added to the system when changing the particles at constant entropy and volume, and in equilibrium, the chemical potentials of two subsystems must be equal. However, the chemical potential does not necessarily have to vanish in equilibrium, as it is related to the free energy and the free energy should be minimum in equilibrium to prevent work from being produced in a certain direction.
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In Dodelson's "Introduction to Modern Cosmology" at p. 61 he introduces a non- equilibrium number density
$$n_i = g_i e^{\mu_i/T} \int \frac{d^3p}{(2\pi)^3} e^{-E_i/T}$$
and an equilibrium number density
$$n_i^{(0)} = g_i \int \frac{d^3p}{(2\pi)^3} e^{-E_i/T},$$
from which it follows that the equilibrium number density, has the same form as number density of non-equilibrium just with a chemical potential of zero.

Question: Why does the chemical potential vanish for the equilibrium case?

ask yourself: What is the chemical potential? (that's a thermodynamics question).

As a fast answer: that's because in equllibrium there is not preferred direction of the reaction. So by its definition the chemical potential will vanish.

Last edited:
ChrisVer said:
ask yourself: What is the chemical potential? (that's a thermodynamics question).

As a fast answer: that's because in equllibrium there is not preferred direction of the reaction. So by its definition the chemical potential will vanish.

The chemical potential arises in the first law of thermodynamics as
$$dE = TdS -pdV + \mu dN$$
and hence it represents the energy added to the system when one changes the particles in the system at constant entropy and volume. I don't see why the chemical potential has to vanish in equilibrium: if two subsystems of a larger system is in equilibrium, their chemical potentials must equal, but not vanish.

The chemical potential is gives you the change of the free energy as you change the number of particles:
$\mu := \frac{\partial E}{\partial N}\Big|_{S,V}$
Now in equlibrium, the free energy should be minimum (otherwise work can be produced along a reaction's direction- so you get a preferable arrow).

## 1. What is chemical potential equilibrium cosmology?

Chemical potential equilibrium cosmology is a theoretical model that attempts to explain the origins and evolution of the universe through the lens of chemical potential. It proposes that the early universe was in a state of chemical equilibrium, where the chemical potential of different particles was balanced, leading to the formation of complex structures and eventually the universe as we know it.

## 2. How does chemical potential equilibrium cosmology differ from traditional cosmological models?

Traditional cosmological models, such as the Big Bang theory, focus on the role of gravity and expansion in the evolution of the universe. In contrast, chemical potential equilibrium cosmology places a greater emphasis on the role of chemical reactions and interactions between particles in shaping the early universe.

## 3. What evidence supports the idea of chemical potential equilibrium in the early universe?

Some evidence for chemical potential equilibrium in the early universe comes from observations of the cosmic microwave background radiation. These measurements show a distribution of particles that is consistent with a state of chemical equilibrium, providing support for this model.

## 4. How does chemical potential equilibrium cosmology explain the formation of complex structures in the universe?

According to this model, the balancing of chemical potentials in the early universe allowed for the formation of complex structures, such as galaxies and stars, through the interaction and fusion of different particles. This process would have continued to occur over billions of years, leading to the diverse and intricate structures we see in the universe today.

## 5. Are there any current challenges or criticisms of chemical potential equilibrium cosmology?

One criticism of this model is that it relies heavily on hypothetical particles and interactions that have not yet been observed or confirmed by experiments. Additionally, some researchers argue that the assumptions and equations used in chemical potential equilibrium cosmology may not accurately reflect the actual conditions of the early universe.

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