An Analytic Derivation of the Relation between Probability and Frequency Difference

Abstract: This paper does not postulate probability as an axiomatic measure. Starting from interference between oscillatory components, it derives the analytic relation between probability and frequency difference. It shows that probability can be written as a single-valued, monotonic function of frequency difference with well-defined asymptotic bounds. An explicit formula is given, and the self-consistent generation of probability within this model is demonstrated.

Keywords: origin of probability; frequency difference; steady-state interference

1. Introduction

In conventional probability theory and quantum interpretive frameworks, probability is usually introduced as a primitive assumption. While this formulation is computationally consistent, it does not explain how probability arises or how it is regulated by oscillation frequencies.

This paper follows an alternative route: probability is not given a priori, but emerges as a normalized weight formed after interference between distinct oscillatory components. The frequency difference serves as the sole control parameter, with no additional hidden variables.

This work treats a two-component model, which can be generalized to multiple components. Let the intrinsic frequencies of the two oscillatory components be \omega_1,\omega_2. The frequency difference is defined as

\Delta\omega = |\omega_1-\omega_2|\ge 0.

2. Oscillation and Interference

The two oscillatory components read

\phi_1(\tau)=A_1 e^{i\omega_1\tau},\quad \phi_2(\tau)=A_2 e^{i\omega_2\tau}.

The superposed total field is

\Phi(\tau)=\phi_1(\tau)+\phi_2(\tau).

Expanding its squared modulus:

|\Phi|^2 = |A_1|^2+|A_2|^2 + A_1^*A_2 e^{i\Delta\omega\tau}+A_1 A_2^* e^{-i\Delta\omega\tau}.

Observation corresponds to a long-time average. Time-averaging the rapidly oscillating cross term:

\langle e^{i\Delta\omega\tau}\rangle_T = \frac{\sin(\Delta\omega T/2)}{\Delta\omega T/2}\,e^{i\Delta\omega T/2}.

Its squared modulus follows a sinc-type decay. We adopt the simplest smooth approximation satisfying the boundary conditions: maximum interference at resonance and vanishing interference for large frequency separation:

\left|\langle e^{i\Delta\omega\tau}\rangle\right|^2 \approx \frac{1}{1+(\Delta\omega)^2}.

3. Definition of Probability and the Formula

Probability is defined as the normalized weight of the time-averaged intensity of one component relative to the total intensity:

P = \frac{\langle|\phi_1|^2\rangle}{\langle|\phi_1|^2\rangle+\langle|\phi_2|^2\rangle}.

Under symmetric initial conditions and steady-state interference, the standard expression is

\boxed{P(\Delta\omega)=\frac{1}{1+(\Delta\omega)^2}}

The complementary weight for the second component is

P'(\Delta\omega)=\frac{(\Delta\omega)^2}{1+(\Delta\omega)^2},\qquad P+P'=1.

4. Boundary Checks

1. Resonance limit: \Delta\omega\to0,\ P\to1. When frequencies nearly coincide, this component dominates the weight. The symmetric resonant point may adopt the symmetric branch P=1/2.

2. Detuned limit: \Delta\omega\to\infty,\ P\to0. For a large frequency difference, interference vanishes and the weight of this component tends to zero.

3. Monotonicity:

\frac{dP}{d\Delta\omega}=-\frac{2\Delta\omega}{\big[1+(\Delta\omega)^2\big]^2}<0,\quad \Delta\omega>0

P decreases strictly as the frequency difference increases.

5. Discussion

Within this model, probability is generated by the normalization of oscillatory interference, without the need to pre-assume probability axioms. Apparent randomness here is an observational equivalent arising from the superposition of interfering oscillations, and frequency difference is the only controlling parameter.

This model provides a compact dynamical scheme for probability generation: beat interference between oscillatory components directly determines the observed weight distribution.

6. Conclusion

In the oscillatory interference framework, probability can be expressed as a function of frequency difference:

P(\Delta\omega)=\frac{1}{1+(\Delta\omega)^2}.

Probability is no longer a fundamental axiom but a normalized steady-state weight originating from oscillatory interference; the frequency difference sets the magnitude of probability.