Theory¶

revised_2025_01_19

Introduction¶

This document outlines a neutrino mechanics (NeuMekan) framework that aims to provide a mathematical and numerical methodology for computing strong nuclear forces between two neutrons. The framework is based on a set of experimental facts, fundamental assumptions, lemmas, propositions, and a theorem that collectively describe the origin of atomic‑scale forces arising from neutrino dynamics. Similar principles can be applied to compute gravitational forces on larger scales.

In this framework, neutrinos form a pervasive dynamical medium that interacts mechanically with other neutrinos and with neutrons. These interactions are modeled using Newtonian mechanics and, more specifically, the kinetic theory of gases. Neutrinos are assumed to flow from regions of higher neutrino pressure to regions of lower neutrino pressure, where neutrino pressure is defined as a scalar quantity proportional to the neutrino number density and the square of the neutrino velocity.

Within this picture, both gravitational and nuclear forces can be interpreted as the net unbalanced neutrino pressure acting on the surfaces of neutrons. The pressure difference is induced by spatial variations in neutrino flow velocity, analogous to Bernoulli‑type relationships between flow speed and pressure.

A numerical example involving two neutrons, based on this framework, is implemented in the NeuMekan simulation code repository. The simulations demonstrate neutrino density distributions that resemble the electric potential distributions around nuclei obtained in quantum mechanics. Instead of solving the Schrödinger equation directly, the NeuMekan code uses solid harmonic functions to compute neutrino density distributions, which can then be related to nuclear forces between neutrons. A similar methodology can in principle be extended to compute gravitational forces.

Experimental Facts¶

Fact 1 (Neutrino Mass).
(F1) Neutrinos possess nonzero mass, as measured by experiments.

Fact 2 (Neutrino Translational Velocity).
(F2) Neutrinos possess translational velocity close to the speed of light.

Fact 3 (Neutrinos Produced by Sources).
(F3) Neutrinos are produced by nuclear fusion processes, such as those occurring in the Sun and in the Earth's core.

Fundamental Assumptions¶

Assumption 1 (Neutrinos Consumed by Neutrons).
(A1) Neutrinos are consumed by neutrons, leading to a reduction in neutrino number density in the vicinity of neutrons.

Assumption 2 (Neutrino Rotational Velocity).
(A2) Neutrinos possess rotational velocity (spin).

Assumption 3 (Neutrino–Neutrino Interaction).
(A3) Neutrinos interact mechanically with other neutrinos through collisions, following the laws of Newtonian mechanics.

Lemmas¶

Lemma 1 (Neutrino Pressure).
(L1) Neutrino pressure is defined as a scalar quantity proportional to the neutrino number density (A1) and the square of the neutrino velocity (A2).

Lemma 2 (Neutrino Flow).
(L2) Neutrinos flow from regions of higher neutrino pressure (F3) toward regions of lower neutrino pressure (A1), with flow magnitude proportional to the pressure gradient.

Lemma 3 (Neutrino Non-uniform Pressure).
(L3) The neutrino pressure distribution (L1) on the surface of neutrons is non-uniform as a result of differences in neutrino flow velocity (L2) on the surface of neutrons, consistent with Bernoulli-type relationships between flow speed and pressure.

Propositions¶

Proposition 1 (Laplace Equation).
(P1) The neutrino density distribution in space obeys the Laplace equation, derived from the conservation of neutrino number and the conservation of energy.

Proposition 2 (Boundary Conditions of Neutrino Density).
(P2) The neutrino density distribution on the surface of neutrons (L1) is assumed to be a known boundary condition, determined by the neutrino consumption rate of neutrons.

Proposition 3 (Boundary Conditions of Neutrino Flow Velocity).
(P3) The neutrino flow velocity (L2), in the direction normal to the surface of neutrons, is treated as a steady-state boundary condition.

Theorem¶

Theorem.
The nuclear force between two neutrons is the result of unbalanced neutrino pressure (L3) on the surfaces of the neutrons, arising from pressure differences induced by variations in neutrino flow velocity (L2) in accordance with Bernoulli-type relationships between flow speed and pressure. The neutrino density distribution in space is computed by solving the Laplace equation (P1) with boundary conditions specified by (P2) and (P3). The resulting non-uniform neutrino pressure distribution on the surfaces of the neutrons leads to a net force acting on each neutron, which constitutes the strong nuclear force between them.

Computational Methodology¶

To compute the neutrino density distribution in space, we solve the Laplace equation (P1) using boundary conditions specified by (P2) and (P3). The solution involves the following steps:

Define Geometry and Boundary Conditions:
Specify the geometry of the neutrons and the corresponding boundary conditions for neutrino density and flow velocity on their surfaces.
Define Boundary Conditions on the Surfaces of Neutrons:
Specify the neutrino density distribution on the surfaces of the neutrons (P2) and the neutrino flow velocity normal to the surfaces of the neutrons (P3). These boundary conditions are essential for solving the Laplace equation for neutrino density distribution in space. This step requires knowledge of certain coefficients that are currently unknown (see the "Missing Coefficients" section below).
Superposition of Solid Harmonic Functions: The neutrino density distribution in space is represented as a superposition of a few lower \((l,m)\) modes of solid harmonic functions (solutions of the Laplace equation) centered at each neutron.
Determine Coefficients:
The coefficients of these few lower \((l,m)\) modes are determined by enforcing the boundary conditions on the surfaces of the neutrons. Since we are still lacking the boundary conditions on the surfaces of neutrons (C2, C3), these coefficients cannot yet be fully determined.
Calculate the Neutrino Pressure
Calculate the (non-uniform) neutrino pressure distribution on the surfaces of the neutrons from the computed neutrino density distribution (L1, L3).
Calculate the Resulting Force on Neutrons
Calculate the resulting force acting on each neutron by integrating the non-uniform neutrino pressure distribution over the surface area. This resulting force is the strong force between the neutrons as described in the Theorem.

Missing Coefficients¶

To fully implement the computational methodology outlined above, several key parameters need to be determined. These parameters are currently unknown and require experimental measurement or further theoretical analysis:

Friction Coefficient
(C1) The friction coefficient between neutrinos and neutrons is required to calculate the neutrino flow velocity toward the surfaces of neutrons. This coefficient is currently unknown and needs to be determined experimentally or through further theoretical analysis.

Neutrino Density Distribution on the Surface of Neutrons
(C2) The neutrino density distribution on the surfaces of neutrons is required as a boundary condition to solve the Laplace equation for the neutrino density distribution in space. This coefficient is currently unknown and needs to be determined experimentally or through further theoretical analysis.

Neutrino Density Distribution in the Cosmological Background
(C3) The neutrino density distribution in the cosmological background is required as a boundary condition to solve the Laplace equation for the neutrino density distribution in space. This coefficient is currently unknown and needs to be determined experimentally or through further theoretical analysis.