# Maximum entropy implies linear expansion?

• johne1618
In summary, the Beckenstein bound states that the entropy inside a sphere of radius R containing a mass M must be given by the inequality S <= (2*pi*k*c*R*M)/hbar. In order to maximize the entropy, the sphere must be filled with as much mass as possible, reaching the limit of a black hole when the relationship G*M/R = c^2/2 holds. However, in the case of an expanding sphere, this relationship still holds and the mass must continuously increase with radius, leading to a creation of matter. This matter is thought to enter the observable universe as the edge of the universe expands. The maximum entropy condition for the expanding sphere can be written as S = (k*A)/4, where A
johne1618
Imagine we are at the center of a sphere of radius R containing a mass M.

The Beckenstein bound states that the entropy inside that sphere, S, must be given by the inequality:

$\large S \leq \frac{2 \pi k c R M}{\hbar}.$

In order to maximise the entropy we need to fill the sphere of radius R with as much mass M as possible. The limit is reached when we have created a black hole. This occurs when the following relationship holds:

$\large \frac{G M}{R} = \frac{c^2}{2}. \ \ \ \ \ \ \ \ \ \ (1)$

Now in the case of a black hole the event horizon is at a constant radius R and we are sitting on the singularity at the center.

But instead let us assume that the sphere is expanding.

Let us also assume that Equation (1) always holds so that the expanding sphere always has a maximum entropy. This of course means that the mass M must increase with radius R. Thus matter is being continuously created.

If we assume flat space then Equation (1) implies that the mass density, $\rho$, is given by

$\large \rho = \frac{3 c^2}{8 \pi G R^2}. \ \ \ \ \ \ \ \ (2)$

If we assume that the radius $R(t)$ is expanding with the Universal scale factor $a(t)$ then we can say:

$R(t) = R_0 a(t) \ \ \ \ \ \ \ \ (3)$

where t is the cosmological time, $R_0$ is the radius at the present time $t_0$ and $a(t_0) = 1$.

Now let us consider the Friedmann equation for flat space with no cosmological constant:

$\large (\frac{\dot{a}}{a})^2 = \frac{8 \pi G}{3} \rho \ \ \ \ \ (4)$

Substituting Equations (2) and (3) into Equation (4) we obtain

$\large (\frac{\dot{a}}{a})^2 = \frac{c^2}{R_0^2 a^2}.$

As the Hubble parameter at the present time, $H_0$, is given by

$H_0 = c / R_0,$

we finally arrive at

$\large (\frac{\dot{a}}{a})^2 = \frac{H_0^2}{a^2},$

which has the simple linear solution

$a = H_0 t.$

This solution to the Friedmann equation is thus the maximum entropy solution.

It is very close to what is observed.

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Let us also assume that Equation (1) always holds so that the expanding sphere always has a maximum entropy. This of course means that the mass M must increase with radius R. Thus matter is being continuously created.

Where is this matter being created? At the edge of our observable universe? What's creating it?

If it's at the edge of our Observable Universe, it may mean it's not created but simply enters our Observable Universe as its edge is moving further away.
Just a thought, I might be wrong.

Constantin said:
If it's at the edge of our Observable Universe, it may mean it's not created but simply enters our Observable Universe as its edge is moving further away.
Just a thought, I might be wrong.

That is my understanding of how it works, not that it is being created.

Drakkith said:
Where is this matter being created? At the edge of our observable universe? What's creating it?

The maximum entropy condition for the expanding sphere can also be written as

$\large S = \frac{k A}{4}$

where A is the surface area of the sphere in units of Planck area $\hbar G / c^3$.

Thus, following the Holographic principle, we can think of all the entropy of the sphere as residing on its surface.

Now we also have the following thermodynamic relationship between the change in the mass/energy of the system and the change in its entropy

$\large dM = \frac{T}{c^2} dS,$

where T is the Unruh temperature at the surface of the sphere due to the gravity of the mass inside it.

This shows that the mass change of the expanding sphere occurs due to the increase of the entropy in its expanding surface area.

The horizon radiates Hawking radiation, but I guess the universe is way too big to be kept in balance by this -- it might be a fun exercise to check if there exists some radius where this is true. If I remember correctly, the radiation power depends on M like ~M^-2, so there might be a "sweet spot".

clamtrox said:
The horizon radiates Hawking radiation, but I guess the universe is way too big to be kept in balance by this -- it might be a fun exercise to check if there exists some radius where this is true. If I remember correctly, the radiation power depends on M like ~M^-2, so there might be a "sweet spot".

In this model the radius of the spherical horizon R increases with cosmological time t according to the formula:

$R = c t.$

I presume that the expanding sphere will not radiate Hawking radiation outside itself as the radiation could not "outrun" the receeding horizon.

johne1618 said:
In this model the radius of the spherical horizon R increases with cosmological time t according to the formula:

$R = c t.$

I presume that the expanding sphere will not radiate Hawking radiation outside itself as the radiation could not "outrun" the receeding horizon.

It's a borderline case, but maybe if ct is slightly larger than R, the horizon starts radiating causing you to expand slower and pushing you towards ct -> R.

## 1. What is maximum entropy?

Maximum entropy is a principle in statistical mechanics that states that the most likely distribution of microstates in a system is the one with the highest entropy. In other words, it is the state with the most disorder or randomness.

## 2. How does maximum entropy relate to linear expansion?

In the context of thermodynamics, maximum entropy implies that when a substance undergoes a linear expansion, the resulting change in entropy is minimized. This means that the substance will expand in a way that maximizes its disorder or randomness.

## 3. Why does maximum entropy imply linear expansion?

This is because in order to reach a state with maximum entropy, a substance will expand in a way that distributes its molecules as evenly as possible. This results in a linear expansion, rather than an expansion in a specific direction.

## 4. What are the practical applications of maximum entropy and linear expansion?

Maximum entropy is a fundamental principle in thermodynamics and has many practical applications, such as in the study of phase transitions, the development of efficient engines and refrigeration systems, and the prediction of chemical reactions. Linear expansion, on the other hand, has practical applications in fields such as construction and engineering, where materials may expand or contract due to changes in temperature.

## 5. Are there any limitations to the principle of maximum entropy?

While maximum entropy is a powerful and widely applicable principle, it does have some limitations. For example, it assumes that a system is in thermal equilibrium and neglects any external forces or constraints that may affect its behavior. Additionally, it may not be applicable to systems with high levels of nonlinearity or complexity.

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