Fluctuation and its application

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Fluctuation

“Fluctuation” generally refers to a state that is slightly shifting and unsettled, or a condition in which small changes repeat without stabilizing. For example, expressions such as “flickering flame” or “sound fluctuation” describe physical phenomena involving subtle movements, while “emotional fluctuation” or “price fluctuation” refer to psychological or societal instability or variability. In the field of physics, fluctuation signifies temporary and probabilistic micro-level changes arising in systems governed by thermal motion or quantum mechanics, and is considered an important concept describing natural irregularities or perturbations within otherwise regular patterns.

From a philosophical perspective, fluctuation suggests something beyond stable and determined order—a notion that points to uncertain, fluid existence, events, or meanings. These ideas form a core for discussing creativity, freedom, relationality, and becoming, and as noted in “The Aesthetics of Fluctuation – On Japanese Painting and Wayō Calligraphy,” this is deeply connected to Japanese aesthetics.

One important philosophical interpretation is that fluctuation exists in the intermediary realm between order and chaos. This concept implies a state that is neither fully determined—as in classical determinism discussed in works like “Free Will, AI, and Zhuangzi’s Freedom”—nor entirely random, as explored in “The Philosophical Perspective on Probability and AI-based Resolution of Uncertainty.” It is a liminal and marginal condition situated in between. In complexity science and nonlinear dynamics, such fluctuation is considered the trigger for emergence and the birth of new order. Philosophically, it challenges the binary opposition of “order = truth” vs. “chaos = meaninglessness,” proposing instead that the very root of order may lie in instability and chance.

This concept of fluctuation plays a fundamental role as the precondition for emergence (Emergence). As Ilya Prigogine describes in “Fūryū Also Exists Where It Is Not – Non-Fūryū Is Also Fūryū,” while the second law of thermodynamics states that entropy increases in closed systems, leading to disorder, in “open systems” that exchange energy or matter with the environment, small fluctuations can give rise to new structures or order. This is the foundation of the dissipative structures theory. It explains how complex phenomena such as fluid convection, the origin of life, consciousness, and even social systems are generated from fluctuations within energy flows. Prigogine termed this view the “death of determinism” and reevaluated probability and uncertainty as sources of creativity.

As explored in “Zen Thought and History, Mahāyāna Buddhism, Taoist Philosophy, and Christianity,” both Buddhism and Taoism also embrace fluctuation, not as something negative, but as the very heart of truth. In Mahāyāna Buddhism, the concept of “emptiness (śūnyatā)” means that all existence is fundamentally without fixed essence, constantly changing according to interdependent causes and conditions. In this light, existence itself is fluctuation.

In Taoism, the idea of “wuwei ziran (non-action in accordance with nature)” encourages aligning with the natural flow of fluctuation rather than controlling it. The Tao (道) is not static order but a dynamic principle of continual becoming and transformation. Here, fluctuation is not a disruption to harmony, but a prerequisite for harmony itself, deeply respected within these traditions.

In modern Western philosophy, Martin Heidegger similarly viewed being as fundamentally fluctuating. He did not conceive of being as an object, but as something that emerges poetically—as an event, as something that opens and withdraws. For Heidegger, fluctuation is the movement of aletheia (unconcealment)—not a fixed truth, but the ongoing unfolding of being as “Ereignis” (event).

In Japanese thought as well, Takeshi Umehara focused on the indigenous spiritual traditions (including Shinto and Buddhism) and highlighted the concept of “awai (間)”—the ambiguous “in-between”—as a space where spirituality and the essence of nature dwell. Similarly, Shinichi Nakazawa reevaluated fluctuation as the meeting point of nature and humanity, logic and perception. In his work Forest Baroque, he presented fluctuation as a form of “wild thought” that exists between the logical and the sensory.

In this way, “fluctuation” is recognized across both Western and Eastern traditions—in ontology, natural science, and religious thought—as a primordial movement that gives rise to creation, order, and spirituality.

Fluctuation in the Natural World

The concept of “fluctuation” is fundamental and essential across various fields of physics.

In statistical mechanics, the macroscopic properties of systems composed of a large number of particles—such as temperature, pressure, and entropy—are determined by statistically averaging the microscopic fluctuations in the motion and states of individual particles. Even in thermodynamic equilibrium, systems are not perfectly static; there are slight fluctuations in energy and particle number, which influence the system’s stability and its response characteristics.

Furthermore, in quantum mechanics and quantum field theory, the deterministic view of physical behavior breaks down. The existence of particles and even the state of the vacuum itself becomes subject to fluctuation, and physical phenomena must be described in fundamentally probabilistic terms.

This concept forms the foundation of inflation theory, which posits that the universe underwent an extremely rapid expansion immediately after the Big Bang. According to this model, the quantum fluctuations that occurred at the microscopic level of spacetime were stretched to macroscopic scales by inflation, seeding the large-scale structure of the universe.

Moreover, in quantum cosmology, the vacuum of space is not considered to be truly empty. Instead, it is a fluctuating field where particle–antiparticle pairs are constantly created and annihilated. Some models propose that our universe itself may have emerged through a quantum tunneling process from such vacuum fluctuations, suggesting that fluctuation is not just a feature of nature, but may have given rise to the very origin of the cosmos.

Mathematical Models of Fluctuation

Various mathematical models of fluctuation have been developed as tools to describe physical phenomena involving randomness and variability.

The basic idea behind these mathematical models is to treat fluctuation as the degree to which data or random variables deviate from their mean, expected value, or steady state. This variability is typically quantified using statistical measures such as variance and standard deviation.

<Variance and Standard Deviation>

Variance and standard deviation are fundamental statistical indicators that express the magnitude of fluctuation. Variance is defined as the average of the squared deviations of a variable

$X$

$X$ from its mean (expected value). It can be mathematically expressed as:

$V a r (X) = E [(X - E [X])^{2}]$

$Var (X) = E [(X - E [X])^{2}]$

The square root of the variance is the standard deviation, which is given by:

$σ = \sqrt{V a r (X)}$

These measures indicate how widely the data are dispersed around the mean, and thus quantitatively represent the degree of fluctuation. Intuitively, a larger variance or standard deviation means the data fluctuate more, while smaller values suggest greater stability.

<White Noise>

In contrast to static fluctuations, one of the simplest approaches to introducing dynamic variation is to add random fluctuations—commonly referred to as noise—to data. Noise refers to random and unpredictable variations present in data or signals, typically caused by measurement errors or external factors. In many cases, noise is modeled mathematically based on probability distributions such as the normal (Gaussian) or uniform distribution, making it analytically tractable.

Among these, white noise is a particularly important and idealized type of random signal that contains no temporal correlation (i.e., is uncorrelated in time) and equal power across all frequencies. It is mathematically expressed as follows:

Zero mean:

$E [ϵ_{t}] = 0$
Independence across time (no autocorrelation):

$E [ϵ_{t} ϵ_{s}] = σ^{2} δ_{t s}$

(Here,

$δ_{t s}$ $δ_{t s}$ is the Kronecker delta, which equals 1 if

$t = s$ $t = s$ , and 0 otherwise.)

White noise serves as a foundational model across many applied fields. For example, in time series analysis, white noise is assumed to be the residual (error) component in models like ARMA and ARIMA, representing the unpredictable partof the system. In other domains such as sensor data processing and image processing, white noise is used to model undesired fluctuations in measurements. Filtering techniques such as the Kalman filter or Gaussian filter are then applied to remove or smooth out this noise component.

Thus, white noise plays an essential role as a model of completely unpredictable and temporally independent fluctuation in various areas of scientific and engineering modeling.

<Fluctuation (Stochastic Processes)>

A further step beyond simple noise is to introduce structure or correlation into the dynamic behavior, resulting in what is called fluctuation. Unlike pure noise, fluctuations have temporal structure and statistical correlation. These types of fluctuations are widely observed in disciplines such as physics, economics, meteorology, and biological signal analysis.

Such time-dependent fluctuations are mathematically modeled using stochastic processes. Representative examples include:

Brownian motion (Wiener process):
Models the random motion of particles suspended in a liquid.
Poisson process:
Describes random events occurring at a constant average rate, such as incoming phone calls.
Markov chains:
Stochastic models where the next state depends only on the current state, not on the past.

To analyze the statistical properties of these fluctuations over time, the following tools are especially important:

Autocorrelation Function:
Measures how similar the current state is to past states. Persistent autocorrelation implies the presence of temporal patterns in the fluctuation.
Power Spectral Density (PSD):
Obtained via Fourier transform, this measures the distribution of fluctuation energy across frequencies. High PSD at high frequencies indicates rapid fluctuations, while dominance at low frequencies suggests slower, smoother changes.

In summary, fluctuation is not merely random variation but rather structured temporal behavior. Understanding it requires tools from stochastic process theory, correlation analysis, and spectral analysis, making it a core concept in modeling dynamic systems with inherent variability.

<Quantum Mechanical Model>

A more advanced model of fluctuation is found in the foundations of quantum mechanics, specifically in Heisenberg’s uncertainty principle, which is expressed by the following inequality:

$Δ x \cdot Δ p \geq \frac{h}{2}$

This equation states that the product of the uncertainty in a particle’s position (

$Δ x$

$Δ x$ ) and the uncertainty in its momentum (

$Δ p$

$Δ p$ ) can never be smaller than

$\frac{h}{2}$

$\frac{h}{2}$ , where

$h$

$h$ is Planck’s constant. In other words, it is fundamentally impossible to measure both position and momentum with infinite precision at the same time.

This means that attempting to measure a particle’s position more precisely will inevitably increase the uncertainty in its momentum, and vice versa. This is not due to limitations in measurement technology, but rather a reflection of a fundamental limit imposed by the laws of nature themselves. It expresses an intrinsic fluctuation built into the fabric of physical reality.

Thus, fluctuation models are deeply embedded in the theoretical structure of quantum mechanics, revealing that uncertainty and indeterminacy are not merely accidental, but are essential features of how the universe operates at its most fundamental level.

Applications of Mathematical Models of Fluctuation

Mathematical models of fluctuation are not merely theoretical constructs—they are powerful tools for modeling complexity and uncertainty in the real world, with broad applications across fields such as business, economics, engineering, medicine, and social systems.

<Applications of Variance and Standard Deviation>

Statistical indicators such as variance and standard deviation, which quantify the magnitude of fluctuation, are widely used to measure uncertainty and support decision-making in various business domains.

In the financial sector, the variability of returns on stocks and bonds—i.e., price fluctuations—is considered a measure of risk itself. By calculating volatility through variance and standard deviation, and assessing the potential loss over a given period using measures such as Value at Risk (VaR), risk can be statistically evaluated. This enables investment decisions and portfolio management to be conducted more scientifically and transparently.

In quality control (QC) within manufacturing, variations in product size or performance are inevitable. Here, the variance and standard deviation of measurement data from each lot are computed, and control charts (such as X̄-R charts) are used to visualize whether the production process is stable. If the variation exceeds the allowable range, it is flagged as an anomaly, which contributes to the maintenance and improvement of product quality.

In marketing, fluctuations in customer purchase frequency and spending amount provide valuable insights. By computing variance and standard deviation from behavioral data for each customer, companies can predict Customer Lifetime Value (CLV) and quantify the likelihood of customer churn. This enables the design of targeted marketing strategies and personalized promotions.

<Applications of Noise and White Noise>

The concepts of noise and white noise are widely used to model unavoidable fluctuations in measurement and observation in the real world. These variations are not merely errors or disruptions; rather, they serve as essential sources of information for understanding and controlling system behavior in fields such as sensor technology, time-series forecasting, and image/speech processing.

In the IoT sensor domain, measurements of temperature, vibration, or location always include some form of noise due to sensor limitations or environmental changes. Using techniques such as Kalman filters or moving averages, noise components can be removed or smoothed, allowing for more accurate state estimation and anomaly detection. This greatly enhances the precision of predictive maintenance and automated control.

In economic forecasting, indicators like GDP, stock prices, and exchange rates are inherently volatile and difficult to predict precisely. Time-series models such as ARIMA (AutoRegressive Integrated Moving Average) are used to separate the predictable structural components from the unpredictable ones modeled as white noise. This enables not only more accurate forecasting but also explicit quantification of residual uncertainty, contributing to risk-aware decision-making in business and policy.

In speech and image recognition, noise processing is essential. Images often contain irregularities due to light reflection or sensor limits, and audio data includes noise caused by the environment or microphone quality. Ignoring such noise can severely degrade the accuracy of recognition by both AI systems and humans. To address this, Gaussian filters and median filters are used as preprocessing tools to remove unwanted fluctuations from visual and auditory data, enabling high-precision recognition.

Thus, mathematical models of noise and white noise serve as foundational tools for extracting essential information and enhancing the accuracy of decision-making and control in complex, uncertain environments.

<Applications of Fluctuation (Stochastic Processes)>

Fluctuation models are indispensable for capturing dynamic structures of variability that cannot be understood through static analysis alone. They are especially critical in financial engineering, demand forecasting, and biomedical signal processing.

In financial algorithms, price movements in stocks and currencies appear random and unpredictable. However, by modeling these fluctuations as a Wiener process (Brownian motion) within the Black-Scholes framework, financial risks can be quantified, and derivative instruments such as options can be priced theoretically. Modeling price fluctuations in this way enables strategic responses to market uncertainty.

In supply chains and logistics, product demand and supply can fluctuate daily, often with unexpected surges. These fluctuations can be mathematically modeled using the Poisson process, which describes the probability structure of discrete event occurrences. This allows for optimal inventory placement and timing of orders, reducing stockout risks and minimizing costs.

In healthcare, biological signals such as EEG and ECG naturally exhibit micro-level fluctuations. These may contain early signs of seizures, arrhythmias, or other critical conditions. By modeling such time-varying signals as stochastic processes (e.g., Markov chains or Hidden Markov Models (HMMs)), abnormalities can be detected early, enabling real-time patient monitoring. HMMs are especially useful for inferring hidden state transitions behind observed data and are widely applied in neurological diagnostics and prognosis.

Thus, mathematical models of fluctuation enable us to uncover underlying structures or laws within seemingly random phenomena, forming a theoretical foundation for high-precision prediction, decision-making, and anomaly detection in practical domains.

<Applications of Quantum Fluctuation Models>

While quantum fluctuations—especially those expressed through Heisenberg’s uncertainty principle—are rarely used directly in business, their theoretical and philosophical implications have significantly influenced modern technologies and decision sciences. In particular, quantum cryptography, extensions of decision theory, and optimization algorithms have all drawn from the notion of quantum-level indeterminacy.

In quantum cryptography, quantum fluctuations form the very basis of communication security. Since measuring a quantum state inevitably disturbs it, this principle can be used to detect the presence of eavesdropping. Quantum Key Distribution (QKD) leverages this property to provide provably secure communication, and has already been implemented in some intergovernmental and financial infrastructure systems.

In decision theory, the concept of quantum fluctuation offers a new paradigm. Traditional game theory assumes players’ strategies are governed by probabilistic choices. In contrast, quantum game theory allows strategies to exist in superpositions, with interference effects influencing outcomes. This framework provides a novel lens for understanding ambiguous choices and strategy optimization in uncertain environments, closely mirroring how humans often make intuitive decisions.

In the realm of optimization algorithms, the principle of quantum superposition (i.e., simultaneous existence in multiple states) enables breakthrough solutions to problems too complex for classical computers. In particular, quantum annealing solves optimization problems by exploiting quantum fluctuations to identify the most stable (optimal) solution. Companies such as D-Wave have commercialized this approach, applying it to fields such as logistics, finance, drug discovery, and materials design, where vast combinatorial spaces must be efficiently searched.

In summary, quantum mechanical fluctuation is no longer limited to the realm of physics. It now plays a valuable role in areas fundamental to modern society—security, decision-making, and computation—and continues to drive innovation at the frontier of technology and logic.

Reference

Philosophical Perspective (Ontology, Becoming, Natural Philosophy, Eastern Thought)

Being and Time (Sein und Zeit) – Martin Heidegger
Explores the “fluctuation of being” and poetically philosophizes the way in which being manifests without ever fully stabilizing.
The End of Certainty – Ilya Prigogine
Bridges natural science and philosophy, arguing that fluctuation generates new forms of order and temporality.
Forest Baroque – Shinichi Nakazawa
Investigates Japanese views of nature, spirituality, and knowledge structure through the lens of “awareness between” (awai) and fluctuation.
Difference and Repetition – Gilles Deleuze
A major work in process philosophy, proposing that existence arises not in continuity or discreteness, but through fluctuation and differentiation.
Buddhism without Beliefs: A Contemporary Guide to Awakening
On the Genealogy of Morality – Friedrich Nietzsche
Views fluctuation as an expression of power, seeing life’s essence in continual creation and dissolution.

Physical Perspective (Thermodynamics, Quantum, Statistical Physics, Cosmology)

Physical Kinetics
A classic in theoretical physics covering thermal fluctuations, diffusion, Brownian motion, and dissipative structures.
Quantum Field Theory in a Nutshell – A. Zee
An intuitive, diagram-rich introduction to quantum fluctuations, virtual particles, and the Casimir effect.
The End of Certainty – Ilya Prigogine
Reinterprets nature from the perspective of fluctuation, challenging classical determinism and embracing probabilistic creativity.
The Inflationary Universe – Alan Guth
A detailed explanation of how primordial quantum fluctuations gave rise to galaxies through cosmic inflation.
Noise – Bart Kosko
An encyclopedic book covering various types of noise—thermal, quantum, informational—across science and perception.

Mathematical Perspective (Probability, Statistics, Chaos, Mathematical Modeling)

An Introduction to Probability Theory and Its Applications – William Feller
A foundational text in probability theory that systematically covers the basics of fluctuation, including variance and stochastic processes.
Introduction to Stochastic Processes – Gregory Lawler
Focuses on modeling temporal fluctuations using tools such as Markov chains and Wiener processes.
Chaos: Making a New Science – James Gleick
A highly accessible introduction to chaos theory, showing how small fluctuations in initial conditions can reshape entire systems.
Statistical Mechanics – R.K. Pathria
A comprehensive exploration of how microscopic fluctuations relate mathematically to macroscopic order in statistical physics.