A Comprehensive Study of Statistical Mechanics in Financial Thermodynamic Systems

Introduction

Literature Review

According to Perez et al., 2010, the size distribution of business firms using an analogy with Bose–Einstein statistics, annual revenues associated by selected firms with energy levels [1]. The cumulative distribution functions of firm rankings across various countries are well-fitted by BE statistics, with the temperature- like parameter correlating positively with economic development as measured by the Human Development Index. In developing economies, top performing firms tend to cluster at low revenue levels, resembling BE condensation at low temperatures. Distribution functions suggest the potential of extending this analogy to developed nations and to other distributions like income and wealth, thereby reinforcing the proposed link between economic "temperature" and development. According to Staliunas et al., 2003, financial markets as partially coherent bosonic systems, using ideas from Bose Einstein condensation [2]. It assumes that participants tend to follow the majority and seek profit. The first behaviour leads to a Bose–Einstein distribution, while the second introduces exponentially truncated Pareto distributions. These better match empirical data, especially for price changes and wealth distributions. Exponent models are consistent with observed values in financial markets and align with results from kinetic finance models.

According to Kurten 2012, the financial crisis can be avoided if the measures to stop the BEC or to remove the Bose-Condensed fraction will be taken. Therefore, it explains that the financial movements as per time when the majority of population are excluded from trading for the money exchange process. It measures what to do to avoid financial downfall. by making active all existing economic agents. According to Lapinski 2013, statistical mechanics methods and precise estimates the entropy approximation, partition functions, and a custom Laplace-type integral approach [3]. The work separates two cases of resulting distributions, expanding theorem suggested by V.P. Maslov. Distribution functions derived to economic systems, suggesting that different statistical forms may correspond to the economic states of countries by BE statistics. This supports the idea of a shared underlying structure across economies.

According to Kusmartev, 2011, Statistical mechanics is applied to economics by modeling market snapshots as ensembles to derive probability distributions [4]. Distributions of capital, income, and debt are found to follow the Bose–Einstein form, revealing key market relationships. Stock market has been is analysed to observe the financial crises between 1996 to 2008, has been indicated by peaks in activity coefficients and future ones predicted. According to Xu, 2015, a modified kinetic exchange model can be presented by drawing a parallel between economic crises and Bose–Einstein condensation [5]. There will be time dependent parameter ð?¿ representing technological advancement, whose growth widens inequality and induces crises despite rising living standards. ð?¿– ð??ð??– 𝐸ð??ð?¡ð??ð??ð?ð?¦ results in dilemma reflects the second law of thermodynamics, suggesting crises arise in isolated economies, while open economies avoid this fate raising concerns about a globally isolated economy under full globalisation.

Methodology

The loss function measures how far off predictions are from actual values. In your case, it quantifies the error between S&P BSE 100 and the predicted values from global indices. It guides the model during training, helping adjust weights to improve accuracy.

Bose Einstein Statistics

The behaviour of stock market indices can be considered as particles whose collective behaviour is governed by certain probabilistic rules. In this approach, market states are viewed as energy levels, while stocks or trading events are treated as bosons that can occupy these states with no restriction on the number of occupants per state. This is in contrast to classical statistics, where such occupancy restrictions are generally assumed. The Bose- Einstein distribution allows for the emergence of condensation effects, which have been interpreted as the clustering of trades or price movements in specific market conditions. This phenomenon explains the high concentration of trades around particular price points or the formation of strong trends. Such clustering is difficult to model using conventional Gaussian or Poisson statistics but is naturally captured through Bose-Einstein formalism.

Bose-Einstein statistics has been utilized to enhance the understanding of collective behaviour and complex dynamics in stock markets. This approach provides a novel theoretical framework to explain anomalies and non-linear phenomena frequently observed in financial indices. The Bose-Einstein distribution describes the statistical distribution of indistinguishable bosons over energy states in thermal equilibrium

The Bose-Einstein distribution has a minus one in its denominator, unlike the Fermi-Dirac distribution, which has a plus one. This difference exists because bosons, unlike fermions, do not follow the Pauli exclusion principle and can share the same energy state. At low temperatures, bosons tend to gather in the lowest energy state, causing Bose-Einstein condensation. This phenomenon leads to unique quantum effects, as many particles occupy a single state simultaneously, which is not possible for fermions.

Analysis

The analysis is interpreted using the top 100 firms from various countries listed in their respective domestic stock indices, such as the BSE 100, FTSE 100, NASDAQ 100, Euronext 100, TSX 100, and ASX 100. A total of 261 days has been taken into consideration for the last financial year which is FY24-25.

                                                                     Table 1: Descriptive Statistics   

The analysis of top 100 firms across various global stock indices for FY24-25, spanning 261 trading days, reveals generally low positive average daily returns but varying volatility. The FTSE 100 exhibited the lowest volatility with lowest  σ= 0.0061 suggesting stable price movements whereas, NASDAQ 100 displayed the highest volatility  σ = 0.0125 alongside µ = 0.003, indicating  significant daily price swings, typical of growth-oriented stocks. Euronext NV 100 offered the highest µ= 0.0017 but also high σ = 0.0113, signifying a dynamic market with higher risk-reward but S&P ASX 100 notably showed µ = 0, implying flat average daily performance with σ = 0.0073. S&P BSE 100 µ = 0.0003 and σ= 0.0092 and S&P TSX 100 with σ = 0.0005 and µ = 0.007 presented moderate returns and volatility. Overall, the period was characterized by marginal average gains, with notable differences in daily price stability across markets.

The above figure provides a comprehensive overview of several global stock market indices, illustrating the historical price movements of FY24-25. This allows for a visual comparison of their performance trends and relative magnitudes over the observed period.

Table 2: Regression Analysis

The regression analysis has been conducted to analysis the relationship between BSE and other international indices. The model yielded an uncentered R-squared of 0.711, indicating minimal explanatory power, and the overall model found to be statistically significant, as evidenced by a Prob (F-statistic) of 0.212. Critical diagnostic tests, including the Omnibus and Jarque-Bera, strongly indicated a severe violation of the normality assumption for residuals, with respective p-values of 0.001 and an extremely low 6.75e-156, alongside notable skewness and kurtosis. While the Durbin-Watson statistic of 1.950 suggested an absence of significant autocorrelation and the compromised residual normality significantly impacts the reliability of the inferential findings.

The provided visualizations offer insights into the predictive performance of different models for BSE Daily Return. The "Actual vs Predicted BSE Daily Return" time series plot, likely representing one of the regression models given its flat prediction line, demonstrates a significant inability to capture the actual daily fluctuations, showing a largely static predicted value against highly volatile actual returns. In contrast, the "DNN Prediction Performance" scatter plot, showing BSE Daily Return versus Predicted BSE Daily Return, indicates the DNN model to align predictions with actual values along the ðÂÃÃÃÂ???ÂÃÃÂ??ÂÃÂ??¦ = ðÂÃÃÃÂ???ÂÃÃÂ??ÂÃÂ??Â¥ line. While there's a wider scatter indicating prediction errors, the DNN appears to capture a range of movements (both positive and negative), suggesting it may offer a more nuanced, albeit still imperfect, relationship compared to the very static predictions of the regression model shown in the time series plot. Both visualizations collectively highlight the inherent difficulty in accurately predicting shortterm market movements.

The model exhibits excellent predictive accuracy, with very low Mean Squared Error value of 0.000083, Sum of Squared Errors value of 0.020513, and Mean Absolute Error value of 0.028167. Therefore, loss function leads to minimal deviation between predicted and actual values across the dataset.

Table-3 presents an analysis of daily returns for various financial markets during FY24-25, employing a framework seemingly analogous to Bose-Einstein statistics, characterized by parameters like ð?½ Ã°Â??ð??ð?£ð??ð??ð? ð?? ð?¡ð??ð??ð?ð??ð??ð??ð?¡ð?¢ð??ð?? and  z Ã°Â??ð?¢ð??ð??ð?ð??ð?¡ð?¦ .For most markets, a positive  Ã°Â?½ exhibit distributions where ð?ð??ð??ð?¡ð??ð?ð??ð??ð?  are the market activity which tend to aggregate towards lower returns, akin to a cooling effect. The z values, lies within the theoretical range of 0 to 1, indicate plausible occupancy or activity levels within this statistical-mechanical analogy. But the negative ð?½  observed for ENX NV is an anomaly, possibly signifying a highly unusual or non-equilibrium state for its return distribution or a limitation of the model's direct applicability to that specific market. Despite the conceptual alignment, the R2 values across all markets are notably low which is ranging from 0.027 to 0.1303, indicating that this Bose-Einstein model explains only a very small fraction of the observed variance in daily returns. This suggests that while the shape of the return distributions might be consistent with a Bose-Einstein aggregation process, as supported by the Not  rejected outcome of the Kolmogorov-Smirnov test, the framework offers a descriptive lens for the distributional form but possesses limited explanatory or predictive capability for the daily dynamics of these financial markets.

 

 

Bose-Einstein distribution visualization model demonstrates that the fitted distribution, parameterized by ðÂÃÃÃÂ???ÂÃÃÂ??ÂÃÂ??½ = 2.84 and ðÂÃÃÃÂ???ÂÃÃÂ??ÂÃÂ??§ = 0.9960, effectively models the empirical histogram of absolute BSE daily returns. The histogram itself reveals a high concentration of small absolute changes and a rapidly decaying, heavy tail for larger movements. The fitted Bose-Einstein curve successfully captures this right-skewed, heavy-tailed characteristic, indicating that this statistical framework provides a plausible descriptive model for the observed distribution of market volatility. The high z value suggests a tendency for "market activity" to cluster in states of minimal absolute return, aligning with the observed frequency of small daily fluctuations. Bose-Einstein distribution for absolute returns, helps in understanding the error of the model structure. Deviations from the normal curve, as potentially indicated by the visual dissimilarity, would suggest that the errors of the Bose- Einstein model are not Gaussian, impacting the interpretation of statistical inferences if a normal error distribution is implicitly assumed.

 

 

Figure-5 presents two scatter plots comparing the true with predicted values of ð?½ and z across six financial markets: BSE, FTSE, TSX, ASX, NASDAQ, and Euronext NV. In the ð?½  plot, most predicted values align well with the diagonal reference line, Ã°ÂÃÃÃÂ???ÂÃÃÂ??ÂÃÂ??¦ = ðÂÃÃÃÂ???ÂÃÃÂ??ÂÃÂ??Â¥, indicating good predictive performance, although a few outliers suggest occasionally underestimation. Whereas ðÂÃÃÃÂ???ÂÃÃÂ??ÂÃÂ??§ plot shows predicted values clustered tightly around 1, reflecting lower variability and high precision in the predictions. The consistent proximity of the data points to the diagonal lines in both plots suggests that the model performs reliably across different markets. Therefore, the plots demonstrate the effectiveness of the BES model in predicting both ðÂÃÃÃÂ???ÂÃÃÂ??ÂÃÂ??½ and ðÂÃÃÃÂ???ÂÃÃÂ??ÂÃÂ??§ parameters with reasonable accuracy. The Bose-Einstein framework effectively models market distributions globally, as supported by KS test results to capture the high concentration of small absolute changes and heavy tails. This statistical mechanism lens primarily aids in understanding the error structure and volatility distribution, showing a tendency for market activity to cluster in states of minimal absolute return which is high ðÂÃÃÃÂ???ÂÃÃÂ??ÂÃÂ??§ values.

Conclusion

The application of regression to ascertain the relationship between daily return of top 100 firms within international indices demonstrated limited explanatory power. Bose-Einstein statistics were employed to model stock market indices, conceptualizing market states as energy levels and trading events as bosons. This framework effectively captured clustering effects in trading or price movements. However, despite providing a plausible descriptive fit for observed market volatility distributions, its explanatory power was limited, as reflected in notably low Rsquared values across all markets.

This research also underscores the inherent complexities associated with accurately forecasting short-term market movements. Bose- Einstein statistical model while offering analytical insights into stock market behaviour and pattern identification, demonstrated limitations in their predictive capacities within the scope of this study. Efficiency of the regression model was constrained by issues of residual normality and predictive precision, whereas the BoseEinstein framework, despite its descriptive utility for return distributions, exhibited restricted explanatory variance. These findings collectively highlight the formidable challenges confronting financial predictive modelling in dynamic market environments [6,7].

Further Scope of the Research

This paper consists of FY24-25 data and find the nature of the international stock markets. Other researchers can go for further statistical modelling by using various statistical tools and techniques.

Limitations of the Study

Stock market analysing may face a strong challenge to predicting the future stock market and it is a foolish game for traders and investors for a short-term period. But researchers also have to use proper mathematical modelling to examine the nature and predict the future outcome.

Ethical Standards

This stock market analysis was conducted with unwavering adherence to ethical principles. By maintaining the highest standards of research integrity, valuable insights were aimed to be contributed to the field of financial analysis, and the trust of the readers and the wider research community was upheld.

Funding

Author did not receive any funding for support for this work.

Declaration of Interest

No conflict of interest is declared by the author. And this paper is not for any commercial usage, author wrote it for non-commercial and pure education purpose.

Data Sharing Statement

The dataset is available in investment india official website.

References

1. Hernández-Pérez, R. (2010).An analogy of the size distribution of business firms with Bose–Einstein statistics. Physica A: Statistical Mechanics and its Applications, 389(18), 3837- 3843.
2. Staliunas, K. (2003). Bose-Einstein condensation in financial systems. arXiv preprint cond-mat/0303271.
3. Lapinski, T. M. (2013). Limit theorems leading to Bose- Einstein, Maxwell-Boltzmann statistics and Zipf-Mandelbrot law (Doctoral dissertation, University of Warwick).
4. Kusmartsev, F. V. (2011). Statistical mechanics of economics I. Physics Letters A, 375(6), 966-973.
5. Xu, J. (2015). Bose-Einstein condensation mechanism in economic system. Europhysics Letters, 110(5), 58002.
6. Shelankov, A., & Rammer, J. (2008). Counting statistics of interfering Bose-Einstein condensates. Europhysics letters, 83(6), 60002.
7. Wang, J., Lin, C., & Wang, Y. (2019). Thermodynamic entropy in quantum statistics for stock market networks. Complexity, 2019(1), 1817248.