1. Introduction

2. Literature Review

3. Theoretical Basis

3.1. Quantization of Time

Figure 3. Quantum Time Wind Concept Diagram.

Figure 3. Quantum Time Wind Concept Diagram.

In the financial markets, candlestick charts, known as "K lines," serve as a graphical representation of price action, with each candlestick representing the price behavior of an asset over a specific time frame. The analogy of comparing $\mathrm{K}$ lines to "time particles" vividly emphasizes the observation of price behavior at discrete points in time. Each candlestick carries crucial information about the price within that period, including the opening price, the highest price, the lowest price, and the closing price. Expressed in mathematical language, if we consider a series of time intervals $\mathsf{\Delta }t\_1$, $\mathsf{\Delta }t\_2,\dots ,\mathsf{\Delta }t\_n$, with each interval corresponding to a candlestick, then a series of candlesticks can be represented as:

$$K_{-}\left(\mathsf{\Delta }{t}_{-}1\right),K_{-}\left(\mathsf{\Delta }{t}_{-}2\right),\dots ,K_{-}\left(\mathsf{\Delta }{t}_{-}n\right)$$

Each candlestick $K_{\_}\left(\mathsf{\Delta }{t}_{-}i\right)$ can be represented by a tuple:In the analysis of financial time series, the opening price S_("open", i), the highest price S_("high", i), and the closing price S_("close", i) are considered key indicators of the price movement of an asset within the i-th time interval. These indicators collectively form the candlestick chart, providing investors and analysts with a powerful tool for interpreting market trends, identifying levels of support and resistance, and recognizing price patterns. Each candlestick represents the price volatility within a time period, and the aggregated information allows for a complete presentation of the price history, enabling predictions about future price behavior based on historical data.

The log-normal distribution is a continuous variable X's probability distribution, where the natural logarithm of $X,\mathrm{l}\mathrm{n}\left(X\right)$, follows a normal distribution. If a random variable $X$ is log-normally distributed, its probability density function (PDF) is given by the following formula:

$$f(x\mid \mu ,\sigma )=\frac{1}{x\sigma \sqrt{2\pi }}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{(\mathrm{l}\mathrm{n}x-\mu {)}^2}{2{\sigma }^2}\right),\mathrm{for}x>0,$$

where $\mu$ and $\sigma$ are the mean and standard deviation of the variable's logarithm, respectively.The probability density function (PDF) for a log-normally distributed variable $X$ is given by:

$$f_X(x;\mu ,\sigma )=\frac{1}{x\sigma \sqrt{2\pi }}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{(\mathrm{l}\mathrm{n}x-\mu {)}^2}{2{\sigma }^2}\right),\mathrm{for}x>0.$$

Here, $\mu$ and $\sigma$represent the mean and standard deviation of the natural logarithm of $X$, respectively. For an asset price $s$, if the logarithm of $s$, denoted as $\mathrm{l}\mathrm{o}\mathrm{g}\left(\mathrm{S}\right)$, follows a normal distribution, then $\mathrm{s}$ itself follows a log-normal distribution. In discrete-time simulations, a time step $\mathsf{\Delta }t$ is typically defined, and price changes are considered at each step. If $s_{-}t$ is the asset price at time $t_{,\mathrm{then}}{s}_{-}\{t+\setminus$ Delta $t\}$ is the price at $t+\mathsf{\Delta }t$. The logarithmic return of the asset price can be expressed as:

$$\mathrm{l}\mathrm{o}\mathrm{g}\left(\frac{S_{t+\mathsf{\Delta }t}}{S_t}\right)\sim N\left(\left(\mu -\frac{{\sigma }^2}{2}\right)\mathsf{\Delta }t,{\sigma }^2\mathsf{\Delta }t\right)$$

In financial modeling, the simulation of asset price paths is an iterative process. Initially, the asset price $s_{-}0$, expected return rate ${\mu }_{,}$ volatility ${\sigma }_{,}$ime step $\mathsf{\Delta }t$, and the total number of steps $\mathrm{N}$ are established. For each time step $\mathrm{n}=\mathrm{0,1},2$, ..., N - 1, a standard normally distributed variable $z_{-}n$ is generated to simulate the randomness of price movements. The asset price is updated according to the following formula:

$$S_{(n+1)\mathsf{\Delta }t}=S_{n\mathsf{\Delta }t}\mathrm{e}\mathrm{x}\mathrm{p}\left(\left(\mu -\frac{{\sigma }^2}{2}\right)\mathsf{\Delta }t+\sigma \sqrt{\mathsf{\Delta }t}{Z}_n\right)$$

This reflects the expected return and random volatility at each time step. The process is repeated until the $N$ iterations are completed. Subsequently, the generated paths are subject to statistical analysis to estimate key financial metrics such as the average price path, volatility, and the probability of extreme events. These simulation techniques have extensive applications in options pricing, risk management, portfolio optimization, and scenario analysis.

3.2. Space Quantization

3.2.1. Einstein Thought Experiment

Einstein's pollen experiments typically refer to his studies concerning Brownian motion. In 1905, Einstein published a series of revolutionary scientific papers, one of which elaborated in detail on Brownian motion—the random movement of pollen particles suspended in water. This discovery had a profound impact on the physics community of the time because it provided direct evidence for the existence of atoms and molecules. Prior to this, many scientists still harbored doubts about the actual existence of microscopic particles.

Figure 4. Einstein's Pollen Thought Experiment.

Figure 4. Einstein's Pollen Thought Experiment.

The central premise is that macroscopically observable random motion is caused by the incessant collisions with numerous microscopic particles within the fluid. Einstein further developed a mathematical model to describe this phenomenon, which is based on the assumption that the movement of pollen particles is induced by their collisions with the water molecules in the fluid. He derived an equation for the mean displacement of pollen particles, $⟨D(t)⟩=\sqrt{2At}$, where $⟨D\left(t\right)⟩$ represents the mean displacement within time $t$, and $A$ is a constant that involves the temperature and the viscosity of the fluid. This equation not only clarified the possibility of revealing the properties of the microscopic world through macroscopic phenomena but also provided a quantitative method for measuring the existence of microscopic particles. Einstein's theory was subsequently confirmed by experimental physicists such as Jean Perrin, whose experimental results not only agreed with Einstein's predictions but also provided direct evidence for the existence of atoms, thereby deepening our understanding of the fundamental structure of matter.

3.2.2. Space Quantization Analysis

Minimum Unit under Uncertainty Perspective

The financial market is a real-time dynamic interaction system involving tens of thousands of participants from around the world. The price fluctuation within an extremely short time frame is inherently uncertain and ambiguous, making it exceedingly difficult to accurately predict every price movement. This is illustrated by the underlying transaction mechanism of financial markets, as depicted in Figure 5, where the number of buy and sell orders for each price can change at any moment.

The principle of uncertainty in financial markets can be interpreted as follows: a specific transaction establishes the price of a particular asset, while the decision by investors to buy or sell the asset at a higher or lower price determines the asset's price trend. At a given moment, it is impossible to know both the precise price of a trading asset and the direction of its price change in the next moment. The uncertainty principle is frequently observable in the market. For example, at a certain point in time, if an individual does not know the exact price of a stock, they certainly cannot anticipate the speed and direction of the next price change. In other words, the uncertainty of trends seems infinite.

However, in the real financial market, one always has information beyond the price of the stock itself at any given time, thus allowing for the prediction of localized price fluctuations within a certain range.

Figure 5. Transaction mechanism behind financial markets.

Figure 5. Transaction mechanism behind financial markets.

Just as in quantum mechanics, the position of a particle is indeterminate. The uncertainty principle describes the relationship between two non-commuting variables. For instance, the product of the uncertainties in position and momentum is greater than or equal to a specific constant, which means it is impossible to simultaneously obtain precise values for both position and momentum. According to Heisenberg's uncertainty principle, there exists the following relationship between energy $\mathsf{\Delta }E$ and time $\mathsf{\Delta }t$ :

$$\mathsf{\Delta }E\mathsf{\Delta }t\ge \frac{\hslash }{2}$$

If time is quantized, then there exists a minimum time unit $\mathsf{\Delta }t$, which defines the lower bound of the time interval that can be measured in this context. Within this minimum time unit, any measurement of energy will have an uncertainty $\mathsf{\Delta }E$. Based on the uncertainty principle, it can be deduced that during the time $\mathsf{\Delta }t$, the uncertainty in energy is at least:

$$\mathsf{\Delta }E\ge \frac{\hslash }{2\mathsf{\Delta }t}$$

This implies that if time is quantized, then during the duration of the minimum time particle $\mathsf{\Delta }t$, there is a fixed lower bound to the uncertainty in energy, which could have significant implications for the energy states of quantum systems. Within a quantized time framework, physical processes (such as the transition of electrons, emission of photons, etc.) would be restricted to occur at discrete points in time. This means that the occurrence of these processes is no longer possible within arbitrarily small time intervals but must occur in multiples of $\mathsf{\Delta }t$. Therefore, the duration $T$ of any process can be expressed as:

$$T=n\mathsf{\Delta }t$$

where $n$ is an integer. This notion of quantized time could lead to new characteristics of physical processes, such as the rate of energy level transitions potentially exhibiting new quantized patterns.

3.3. The Smallest Spatial Unit in the Financial Market

The method of analysis based on Einstein's thought experiments can likewise be applied to financial markets, treating market price fluctuations as a form of financial "Brownian motion." In this model, each transaction is akin to an impact on the asset price, causing random fluctuations and thus aiding our understanding of the complex dynamics of financial markets. Applying Einstein's approach to financial markets allows us to view the price movements of financial assets as phenomena analogous to Brownian motion. In this metaphor, the current price of financial markets is comparable to pollen particles suspended in a fluid, while the orders of global investors are akin to the collisions of water molecules.

Each order, regardless of size, impacts the asset price similarly to how water molecules strike pollen particles. Although the effect of a single order might be negligible, the collective effect of all orders causes price fluctuations on a macro level, resulting in the stock market's "Brownian motion."

In the field of financial market analysis, the mathematical expression of Einstein's Brownian motion, $⟨D(t)⟩=\sqrt{2At}$, can be reinterpreted as a model for asset price volatility, where $⟨D\left(t\right)⟩$ represents the average magnitude of change in asset price over time $t$, and the constant $A$ in this context is associated with market liquidity and trading activity. This model provides a new perspective for understanding the fundamental nature of financial market price movements, revealing how external factors and investor behavior jointly impact asset prices. This theoretical framework not only offers insights into the deeper analysis of financial market fluctuations but also provides solid theoretical support for the formulation of investment strategies and the practice of risk management.

Figure 6. Brownian motion of orders behind the K-line.

Figure 6. Brownian motion of orders behind the K-line.

3.3.1. Argument Basic Assumptions

The minimal unit of stock price movement: Let us assume the existence of a minimal price movement unit, $\mathsf{\Delta }P$, which can be regarded as the "quantum of space" for the stock market. The randomness of transactions: The price change caused by each transaction can either increase by one $\mathsf{\Delta }P$ or decrease by one $\mathsf{\Delta }P$, akin to a random walk model.

In stock markets, Brownian motion is commonly applied to the mathematical model known as Geometric Brownian Motion (GBM), which is utilized to simulate the fluctuations of asset prices, such as stocks or indices. This model is based on key assumptions where the price movements are considered continuous and exhibit both a determinable trend and stochastic volatility. The mathematical expression for geometric Brownian motion can be described by the following stochastic differential equation:

$$dS=\mu Sdt+\sigma SdW$$

Here, $S$ represents the asset price, $\mu$ denotes the expected rate of return of the asset, also known as the drift rate; $\sigma$ represents the volatility of the asset price; $dW$ is a Wiener process term representing random shocks, and $dt$ is the time increment. The model assumes that the logarithmic returns of asset prices are normally distributed, which simplifies the calculation of probabilities at different price levels. The geometric Brownian motion model is an essential tool in financial engineering for risk management and derivative pricing, with the Black-Scholes option pricing model being one notable application based on this framework. Nevertheless, the actual behavior of stock markets can be more complex than this model describes, as it may also be influenced by a variety of factors including market psychology, informational asymmetry, and changes in supply and demand. Analogous to Einstein's analysis of Brownian motion, this paper attempts to construct a simplified model from a statistical physics perspective to describe this phenomenon.

3.3.2. Mathematical Description of Argumentation

Consider a simplified model where stock price movements follow the rules of a onedimensional random walk: within a time interval $\mathsf{\Delta }t$, the price change $\mathsf{\Delta }S$ can be either $+\mathsf{\Delta }P$ or $-\mathsf{\Delta }P$. Assuming that the probabilities for upward or downward movements are equal, that is, $P(\mathsf{\Delta }S=+\mathsf{\Delta }P)=P(\mathsf{\Delta }S=-\mathsf{\Delta }P)=0.5$.

The derivation process is as follows: We define the expected value $E\left[\mathsf{\Delta }S\right]$ and variance $\mathrm{V}\mathrm{a}\mathrm{r}\left[\mathsf{\Delta }S\right]$ of stock price movements. Since each movement is independent and the probabilities of rising or falling are equal, we have:

$$\begin{array}{ll}& E\left[\mathsf{\Delta }S\right]=0\\ & \mathrm{V}\mathrm{a}\mathrm{r}\left[\mathsf{\Delta }S\right]=E\left[\mathsf{\Delta }{S}^2\right]-\left(E\right[\mathsf{\Delta }S]{)}^2=(\mathsf{\Delta }P{)}^2\end{array}$$

Considering the stock price movements over a long period $t$, one can invoke the Central Limit Theorem to predict that the sum of stock price movements approximately follows a normal distribution with a mean of 0 and a variance of $n(\mathsf{\Delta }P{)}^2$, where $n=t/\mathsf{\Delta }t$ represents the number of trades that occur over time $t$. Therefore, the model for stock price movements can be represented as:

$$\mathsf{\Delta }S(t)\sim \mathcal{N}\left(0,n(\mathsf{\Delta }P{)}^2\right)$$

In this model, the stock price changes are modeled as a stochastic process that, over a long time scale, tends to exhibit Gaussian fluctuations around the initial price, which is consistent with the empirical distributions observed in the financial markets under certain conditions.

The normal distribution $\mathcal{N}\left(0,n(\mathsf{\Delta }P{)}^2\right)$ indicates that, over time $t$, the summation of stock price movements is distributed as a normal distribution with a mean of 0 and a variance of $n(\mathsf{\Delta }P{)}^2$. Here, $n$ independent changes in $\mathsf{\Delta }S$ (each with a mean of 0 and a variance of $(\mathsf{\Delta }P{)}^2$ ) are summed to produce the total change $\mathsf{\Delta }S\left(t\right)$. This is just a theoretical model and such a simplified model usually requires further adjustments to accommodate the complexities of real markets. For instance, in actual markets, the probabilities of price increases and decreases may not always be equal, and the magnitude of price change $\mathsf{\Delta }P$ may vary with changing market conditions. Additionally, factors such as transaction costs, market impact, and the speed of information dissemination must be considered for their effects on stock prices. Despite these limitations, the random walk model remains one of the very important theoretical tools in the field of finance and has had a significant impact on the development of modern financial mathematics.

4. Turbulence Models in Financial Markets

5. Demonstration

Table 1. Empirical procedure steps.

5.4. Power Spectrum Analysis

In this study, we calculated and compared the power spectrum $S\left(f\right)$, where $f$ represents frequency. Power spectral analysis is a key technique for exploring the energy distribution of time series in the frequency domain. By analyzing the power spectra of the financial time series $Y\left(t\right)$ and the physical turbulent time series $V\left(t\right)$, we can identify similarities and differences in frequency responses between market volatility and physical turbulent behavior. This method not only reveals the periodicity and randomness of the time series but also aids in a deeper understanding of the universality and specificity of dynamic behaviors across different systems.

Figure 7. Analysis of Financial Market Time Series and Fluid Time Series.

Figure 7. Analysis of Financial Market Time Series and Fluid Time Series.

5.6. Difference Analysis

Figure 8. Power Spectrum Difference Diagram.

Figure 8. Power Spectrum Difference Diagram.

By simulating time series for financial markets and fluid dynamics and comparing their power spectra, this paper is able to observe differences in their frequency responses. In the figure presented, the solid line represents the power spectrum of the financial market time series, while the dashed line corresponds to the power spectrum of the simulated fluid dynamics time series.

Power spectral analysis is a method for quantifying the distribution of energy across various frequencies in a signal or time series. It can be observed from the figure that the power spectra of the two time series are similar in the low-frequency region, suggesting that both may exhibit similar dynamic behaviors over long-term trends. However, as the frequency increases, the power spectra begin to display noticeable differences, which may reflect dissimilarities in behavior between financial market data and fluid dynamics data on shorter time scales.

Specifically, the power spectrum of the financial market time series may show peaks at certain frequencies, reflecting periodic behaviors in the market data or the influence of specific events. In contrast, the simulated fluid dynamics time series exhibits a more uniform distribution of energy across a wider range of frequencies, consistent with the turbulent behavior of fluids where flows at various scales contribute to the overall behavior.

This analysis reveals differences in the statistical physical properties between financial market models and real fluid dynamics models, particularly in their time series dynamic behaviors and frequency responses. Although both financial markets and fluid dynamics can be described using similar mathematical frameworks, they exhibit fundamental differences in their specific behavioral characteristics. This underscores the complexity and nuance that must be considered when applying physical theories to explain economic phenomena.

Figure 9. Fluctuation chart of price and Bessel curvature.

Figure 9. Fluctuation chart of price and Bessel curvature.

In physics, Bézier curves are utilized to simulate and analyze the pathlines of turbulent flow fields. These curves are recognized for their mathematical smoothness and the flexibility of their control points adjustment, offering an efficient means of approximating complex turbulent structures. As such, they provide valuable support in numerical simulations and graphical visualizations for understanding and predicting turbulent behavior. In the analysis conducted here, this paper employs a strategy based on constructed financial Bézier curves to assess the changes in market volatility over time. Specifically, we will focus on the relationship between the time series (X-axis), price fluctuations (Y-axis), and the curvature of the Bézier curve (Z-axis).

Table 2. Empirical Celler of Bessel Curve.

Table 2. Empirical Celler of Bessel Curve.

Step	Program execution content
1. Generate Bessel curve	Find price inflections. Build Bézier curve for trends
2. Monitor price fluctuations	Above Bézier: check for turbulence. Below: same check.
3. Check turbulence	Compute MA_short. Compute MA_long. Large MA_short/MA_long deviation signals turbulence.
4. Execute transactions	Above Bézier: sell. Below Bézier: buy.
5 Main programs	Get financial price data. Create Bézier curve from inflections. Monitor price vs. Bézier. On touch, if turbulent, trade.
6 Record Results	Log trades for review and strategy refinement.

Time Series (X-axis) Analysis: The time series represents the temporal dimension of a dataset of stock prices. Within this dimension, we can observe the evolution of price fluctuations and the curvature of the Bézier curve over time. Typically, time series analysis reveals long-term trends, cyclical patterns, and potential seasonal factors in price behavior. During the analysis, the trends in the time series can reflect changes in market sentiment, as well as potential impacts from economic cycles and macroeconomic events.

Price Fluctuation (Y-axis) Analysis: The price dimension reflects the market value of a stock at specific time points. The magnitude and direction of price fluctuations provide essential information about market activity, investor sentiment, and stock liquidity. Significant price volatility may indicate substantial divergence in market participants' views on the value of a stock or the market's reaction to sudden events. Conversely, stable price movements generally imply market consensus or a lack of significant news events.

Bézier Curve Curvature (Z-axis) Analysis: Within this analytical framework, the curvature of the Bézier curve is characterized by the width of Bollinger Bands, quantifying the range of price volatility. This property of the Bézier curve is used here as an indicator to measure the intensity of price volatility. High curvature of the Bézier curve generally corresponds with increased market volatility, which may be due to inconsistent interpretations of market information by investors or responses to changes in macroeconomic indicators. On the other hand, a lower curvature of the Bézier curve, with a reduced width of Bollinger Bands, reflects decreased market volatility, which may indicate more stable market information and consistent investor expectations, leading to more limited price movements.

Integrating the analyses of these three dimensions provides a dynamic view of how market volatility changes over time. By observing these changes, investors and analysts can make more informed judgments about market trends and risk levels. For instance, during periods of high Bézier curve curvature, the market may exhibit instability, prompting investors to adopt more cautious investment strategies to mitigate potential risks from market fluctuations. Conversely, in periods of low curvature, the market's stability may be higher, and investors might seek to increase investments to capture potential gains. In this way, the Bézier curve curvature becomes a powerful tool in helping us understand and predict dynamic market changes.

In essence, by observing the time series (X-axis), we can track the trends and patterns in price (Y-axis) as they evolve over time. These patterns may result from the market's assimilation of new information or its reaction to macroeconomic events. For example, if the market anticipates a significant policy change, this may be reflected in the time series as a notable rise or fall in stock prices.

On the Y-axis, a detailed analysis of price fluctuations reveals the immediate changes in market supply and demand dynamics. Periods of high price volatility may indicate a strong divergence between buyers and sellers in the market, or the market's rapid response to breaking news. For investors, understanding these price fluctuations is crucial, as they can provide immediate signals about market sentiment and trends.

The curvature of the Bézier curve on the Z-axis provides us with a quantitative measure of market volatility. When the curvature of the Bézier curve increases, it indicates an expansion of the Bollinger Bands width, which is typically accompanied by increased market volatility. During these phases, prices may exhibit dramatic fluctuations, reflecting the possibility of rapidly changing market information or differing expectations and interpretations of the future among market participants. For risk-averse investors, this may be a signal to reduce holdings or seek hedging strategies.

Conversely, when the curvature of the Bézier curve decreases, indicating a narrowing of the Bollinger Bands width, this usually suggests a reduction in market volatility. During these phases, price movements may be more stable, and investor expectations may align more closely. This could signify a period of relatively high market confidence, where investors might consider expanding their investments to capture gains.

In summary, by integrating time, price, and Bézier curve curvature into a three-dimensional view, we gain a deeper understanding of market dynamics. Such a multidimensional analytical approach provides a complex yet comprehensive perspective for investment decisions, helping investors to identify cyclical patterns in the market, predict changes in volatility, and thereby make more informed investment choices.

6. Conclusion

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