The Poincaré Recurrence Theorem is a foundational result in dynamical systems and ergodic theory, which asserts that a system in a finite phase space will, after sufficient time, return arbitrarily close to its initial state. Mathematical derivation and proof:

Statement of the Theorem

Let (X, 𝕌, μ, T) be a measure-preserving dynamical system where:

• X is the phase space.
• 𝕌 is a σ-algebra of measurable sets.
• μ is a finite measure with μ(X) < ∞.
• T: X → X is a measurable, measure-preserving transformation (μ(T^-1(A)) = μ(A) for all A ∈ 𝕌).

Then for any measurable set A ⊆ X with μ(A) > 0, almost every point in A will return to A infinitely often under iteration of T. That is, for almost every x ∈ A, there exist infinitely many n ∈ ℕ such that Tⁿ(x) ∈ A.

Proof

1. Setup and Definitions

Define the return time set for a point x ∈ A:

R_A(x) = { n ∈ ℕ : Tⁿ(x) ∈ A }.

The goal is to show that for almost every x ∈ A, R_A(x) is infinite.

2. Measure Preservation

Since T is measure-preserving, the measure of T^-n(A) is the same as the measure of A:

μ(T^-n(A)) = μ(A),   ∀ n ≥ 1.

3. Construct the Escape Set

Define the escape set E as the set of points in A that leave A and never return:

E = { x ∈ A : Tⁿ(x) ∉ A,   ∀ n ≥ 1 }.

We aim to show μ(E) = 0.

4. Decompose A

The set A can be decomposed as:

A = ⋃_n=0^∞ (T^-n(A) − ⋃_k=0^n-1 T^-k(A)) ∪ E.

This decomposition divides A into disjoint subsets of points that first return to A at specific times n, along with the escape set E.

5. Measure of the Escape Set

The union of pre-images ⋃_n=0^∞ T^-n(A) is an invariant set (since T is measure-preserving), and thus its measure cannot exceed μ(X). Because μ(A) > 0 and measure-preservation ensures recurrence in finite measure spaces, any measure assigned to E would contradict this preservation unless μ(E) = 0.

6. Almost Sure Recurrence

For any point x ∈ A outside of E, R_A(x) must be infinite since the measure-preserving dynamics of T ensure that the measure “flows” back into A repeatedly over time.

Conclusion

Thus, for almost every x ∈ A, the point x returns to A infinitely often, completing the derivation of the Poincaré Recurrence Theorem.

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 Thermal Equilibrium and the Early Universe

In the first few seconds after the Big Bang, the universe was extremely hot and dense. During this period, the universe was in a state of **thermal equilibrium**, where particles (photons, quarks, electrons, neutrinos, etc.) interacted frequently, allowing them to reach thermal distributions.

– **Temperature**: The temperature of the universe decreased as it expanded. Initially, temperatures were extremely high (on the order of \(10^{12}\) K or higher), allowing particles to have relativistic velocities.
– **Maxwell-Boltzmann Distribution**: For non-relativistic particles, the energy distribution follows the Maxwell-Boltzmann distribution. However, at high temperatures (when particles are relativistic), the distribution functions for particles need to follow quantum statistics:
– **Fermions** (particles like electrons, neutrinos): Fermi-Dirac distribution.
– **Bosons** (particles like photons): Bose-Einstein distribution.

Equation of State

The relationship between pressure \(P\), energy density \(\rho\), and temperature \(T\) is governed by the **equation of state**. In the early universe, different particle species contribute to the total energy density and pressure, depending on whether they are relativistic or non-relativistic:
– **Radiation-dominated universe**: When the temperature is very high, most particles are relativistic (moving near the speed of light). In this case, the energy density is proportional to \(T^4\), and the pressure is related to the energy density as \(P = \frac{1}{3} \rho\).
– **Matter-dominated universe**: As the universe expands and cools, particles like protons and neutrons become non-relativistic. In this regime, the energy density is dominated by the mass of particles, and pressure is much smaller compared to energy density (\(P \approx 0\)).

Phase Transitions in the Early Universe

As the universe cooled, it underwent several important phase transitions, which are essential in understanding its statistical mechanics:

Quark-Gluon Plasma to Hadrons (~10 microseconds)
At extremely high temperatures, matter existed as a **quark-gluon plasma**, where quarks and gluons were not confined into protons and neutrons. As the universe cooled, quarks and gluons became confined in hadrons (protons, neutrons), in a phase transition similar to the condensation of gas into liquid.

– This phase transition likely had significant implications for baryogenesis (the creation of matter over antimatter).

Electroweak Phase Transition (~10^-12 seconds)
The **electroweak phase transition** occurred when the temperature dropped to around 100 GeV. At higher temperatures, the electromagnetic and weak forces were unified, but as the universe expanded and cooled, they separated into distinct forces. This transition also set the stage for the **Higgs mechanism**, giving particles mass.

Decoupling of Neutrinos (~1 second)
At temperatures around 1 MeV, neutrinos decoupled from the rest of the matter in the universe. Before this, neutrinos were in thermal equilibrium with electrons, protons, and neutrons. After decoupling, they stopped interacting significantly and freely streamed through space.

Recombination (~380,000 years)
As the universe continued to cool (down to ~3000 K), electrons combined with protons to form neutral hydrogen atoms in a process known as **recombination**. This is a significant event because, before recombination, the universe was opaque to radiation due to free electrons scattering photons (via Thomson scattering). After recombination, photons decoupled from matter, leading to the **Cosmic Microwave Background (CMB)** radiation.

Boltzmann Equation and Expansion
The evolution of the number density of particles, their momentum distributions, and their interactions are governed by the **Boltzmann equation** in an expanding universe. The expansion of the universe is described by the **Friedmann equations**, and the Boltzmann equation incorporates the effects of cosmic expansion.

– **Collision term**: Represents particle interactions, ensuring thermal equilibrium when collisions are frequent.
– **Hubble term**: Represents the effect of the expanding universe, diluting the number density of particles and redshifting the momentum.

This equation helps in describing how different particle species freeze out or decouple as the universe expands and cools.

Particle Freeze-Out

As the universe cooled, certain particle species could no longer interact frequently enough to maintain thermal equilibrium, leading to **freeze-out**. This is important for understanding the relic abundances of particles like neutrinos and dark matter.

– For instance, **neutrinos** decoupled when the temperature fell below a few MeV, and their distribution since then has been largely unaffected by further interactions.
– **Dark matter** particles, if they are weakly interacting massive particles (WIMPs), would have also frozen out, leaving behind a relic density that depends on their interaction cross-section and mass.

Entropy and Expansion
One of the key concepts in the statistical mechanics of the early universe is the near-conservation of **entropy**. As the universe expands adiabatically, the total entropy remains constant. This is crucial because the entropy per unit volume decreases as space expands, but the total entropy in a comoving volume (a volume that expands with the universe) remains constant.

– The **entropy density** in the early universe is dominated by the relativistic species (photons, neutrinos), and its evolution provides important clues about the thermal history of the universe.

Cosmic Microwave Background (CMB)
The CMB is a direct remnant of the early universe and provides a snapshot of the universe when it was about 380,000 years old. The nearly perfect blackbody spectrum of the CMB is a direct consequence of the statistical mechanics of the early universe, which was in thermal equilibrium before recombination.

– The small fluctuations in the CMB provide information about the density perturbations in the early universe, which later grew into galaxies and large-scale structure due to gravitational collapse.

Summary of the Statistical Mechanics of the Early Universe
The statistical mechanics of the early universe revolves around thermal equilibrium, phase transitions, particle interactions, and the gradual decoupling of different species as the universe expanded and cooled. The early universe was governed by quantum statistics (Bose-Einstein and Fermi-Dirac distributions), phase transitions such as quark-gluon plasma to hadrons, and the electroweak transition. Over time, particles froze out of equilibrium, leaving relic densities that still influence the universe today, and the CMB remains as a crucial observable from this epoch.

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