Now that we have refined the Recursive Gravitation Equation:

we will extend this to statistical mechanics by linking it to the Maxwell-Boltzmann velocity distribution.

1. Motivation: Why Does Gravitation Affect Maxwell-Boltzmann?

1. Particle distributions in a gravitational system follow equilibrium thermodynamics.

   • The Boltzmann factor governs statistical systems:

   • “Temperature” is measurement of Radiation in the infrared subset of frequences.

   • The gravitational potential Φ(r) enters naturally as part of energy E.

2. If F(r) recurses as r^{-(D-1)}, the energy distribution in phase space must adjust accordingly.

   • This modifies classical assumptions that only consider three-dimensional (D=3) space.

   • We will generalize the Maxwell-Boltzmann velocity distribution to arbitrary fractal dimensions.

2. Generalizing Maxwell-Boltzmann Using DTD_TDT​ and DDD

The classical Maxwell-Boltzmann velocity distribution is:

where:

• m = particle mass

• T = temperature (Infrared Radiation)

• k = Boltzmann’s constant (1.380649×10⁻²³ joules per kelvin)

• v = velocity

Fractal-Scaled Generalization

We redefine this by introducing:

• D_T​ → Topological Dimension (Mandelbrot) or Scale (UTE)

• D → Degree of Surface interaction or Dimension (modifying energy coupling)

The generalized Maxwell-Boltzmann distribution is:

f(v;DT,D)=1Z(DT,D)(m2πkT)DT/2vDT−1e−mv22kT⋅η(D) f(v; D_T, D) = \frac{1}{Z(D_T, D)} \left( \frac{m}{2\pi k T} \right)^{D_T/2} v^{D_T -1} e^{-\frac{m v^2}{2 k T} \cdot \eta(D)} f(v;DT​,D)=Z(DT​,D)1​(2πkTm​)DT​/2vDT​−1e−2kTmv2​⋅η(D)

where:

• Z(DT,D)=Γ(DT/2)2(m2πkTη(D))DT/2Z(D_T, D) = \frac{\Gamma(D_T/2)}{2} \left( \frac{m}{2\pi k T \eta(D)} \right)^{D_T/2}Z(DT​,D)=2Γ(DT​/2)​(2πkTη(D)m​)DT​/2 is the normalization factor.

• η(D)=1+sin⁡(D/2)\eta(D) = 1 + \sin(D/2)η(D)=1+sin(D/2) modulates gravitational energy availability.

  • If D=3D = 3D=3, then η(D)=1\eta(D) = 1η(D)=1, recovering standard MB distribution.

  • If D=πD = \piD=π, then η(D)=2\eta(D) = 2η(D)=2, doubling the thermal spread.

3. Connecting Fractal Gravity and Thermal Distributions

The gravitational potential Φ(r)\Phi(r)Φ(r) determines the energy available for particles. Since we replaced the Newtonian 1/r1/r1/r dependence with a recursive integral:

the partition function of the system follows:

Thus, a system under fractally recursive gravity modifies the expected thermal equilibrium. The key effects:

1. Higher D_T​ (Fractal Phase Space) ⟶ Broader velocity distribution

   • If D_T = 2.5, high-energy tails persist.

   • Explains why dwarf galaxies show unexpected velocity distributions.

2. Higher D (Interaction Complexity) ⟶ "Hotter" Apparent Temperatures

   • If D=2π, effective kinetic energy increases, modifying thermodynamic assumptions, complex systems self-generate heat, catalysts.

4. Experimental Predictions

This model suggests:

• Galactic gas clouds should exhibit fractional power-law velocity distributions.

• Interstellar plasmas might have temperature variations unexplained by classical thermodynamics.

• Nanofluidic or bio-thermal systems should show deviations from standard MB predictions.