Multifractal Behavior Research Articles

Multifractality Signatures in Lensed Quasars

Abstract Variations in scaling behavior in the flux and emissions of gravitational lensed quasars can provide valuable information about the dynamics within the sources and their cosmological evolution with time. Here, we study the multifractal behavior of the light curves of 14 lensed quasars with multiple images in the r band, with redshift ranging from 0.657 to 2.730, in the search for potential differences in nonlinearity between the signals of the quasar multiple images. Among these lensed systems, nine present two images, two present three images, and three present four images. To this end, we apply the wavelet transform-based multifractal analysis formalism called Wavelet Transform Modulus Maxima (WTMM). We identify strong multifractal signatures in the light curves of the images of all analyzed lensed quasar systems, independently of the number of images, with a significant difference between the degree of multifractality of all the images and combinations. We have also searched for a possible connection between the degree of multifractality and the characteristic parameters related to the quasar source and the lensing galaxy. These parameters include the Einstein ring radius and the accretion disk size and the characteristic timescales related to microlensing variability. The analysis reveals some apparent trends, pointing to a decrease in the degree of multifractality with the increase of the quasar’s source size and timescale. Using a larger sample and following a similar approach, the present study confirms a previous finding for the quasar Q0957+561.

Dynamic speculation and efficiency in European natural gas markets during the COVID-19 and Russia-Ukraine crises

Efficiency and dynamic speculation literature focused on financial and crude oil markets whereas empirical studies on natural gas markets are limited. This study examines market efficiency and dynamic speculation in major European natural gas hubs (the Zeebrugge (ZEE) hub of Belgium, Title Transfer Facility (TTF) hub of the Netherlands, and National Balancing Point (NBP) hub of the UK) during the COVID-19 pandemic and Russia-Ukraine conflict. Using the Asymmetric multifractal detrended fluctuation analysis (A-MFDFA) and the power law exponent (PLE) approach, we show that European natural gas markets exhibit an asymmetric multifractal behavior and long-memory effects, with variations observed during crises. The ZEE hub in Belgium and the NBP hub in the UK demonstrate a higher multifractality under different trends during the COVID-19 pandemic and the Russia-Ukraine conflict, respectively. Notably, the TTF hub in the Netherlands emerges as the most efficient market during all subperiods. Moreover, our results, using the PLE approach, suggest that European gas market inefficiencies during crises can be attributed to speculative strategies that create self-perpetuating dynamics and diminish market transparency, leading to heightened risk and volatility in gas prices. Our findings underscore the need for regulatory intervention to mitigate excessive speculation and enhance transparency in gas markets. Diversifying gas supplies and fortifying European gas hubs emerge as pivotal strategies to bolster resilience and stability during crises.

Multifractality in monitored single-particle dynamics

We study multifractal properties in time evolution of a single particle subject to repeated measurements. For quantum systems, we consider circuit models consisting of local unitary gates and local projective measurements. For classical systems, we consider models for estimating the trajectory of a particle evolved under local transition processes by partially measuring particle occupations. In both cases, multifractal behaviors appear in the ensemble of wave functions or probability distributions conditioned on measurement outcomes after a sufficiently long time. While the nature of particle transport (diffusive or ballistic) qualitatively affects the multifractal properties, they are even quantitatively robust to the measurement rate or specific protocols. On the other hand, multifractality is generically lost by generalized measurements allowing erroneous outcomes or by postselection of the outcomes with no particle detection. We demonstrate these properties by numerical simulations and also propose several simplified models, which allow us to analytically obtain multifractal properties in the monitored single-particle systems. Published by the American Physical Society 2024

Multifractal Analysis of the Impact of Fuel Cell Introduction in the Korean Electricity Market

This study employs multifractal detrended fluctuation analysis to investigate the impact of fuel cell introduction in the Korean electricity market via the lens of multifractal scaling behavior. Using multifractal analysis, the research delineates discrepancies between peak and off-peak hours, accounting for the daily cyclicity of the electricity market, and proposes a crossover point detection method based on the Chow test. Furthermore, the impacts of fuel cell introduction are evidenced through various methods that encompass multifractal spectra and market efficiency. The findings initially indicate a higher degree of multifractality during off-peak hours relative to peak hours. In particular, the crossover points emerged solely during off-peak hours, unveiling short- and long-term dynamics predicated on a near-annual cycle. Additionally, the average Hurst exponent for the short-term was 0.542, while the average for the long-term was 0.098, representing a notable discrepancy. The introduction of fuel cells attenuated the heterogeneity in the scaling behavior, which is potentially attributable to the decreased volatility in both the supply and demand spectra. Remarkably, after the introduction of fuel cells, there was a discernible decrease in the influence of long-range correlation within multifractality, and the market exhibited an increased propensity toward random-walk behavior. This phenomenon was also detected in the market deficiency measure, from an average of 0.536, prior to the introduction, to an average of 0.267, following the introduction, signifying an improvement in market efficiency. This implies that the introduction of fuel cells into the market engendered increased supply stability and a consistent increase in demand, mitigating volatility on both the supply and demand sides, thus increasing market efficiency.

Detailed analysis of local climate at the CTAO-North site on La Palma from 20 yr of MAGIC weather station data

ABSTRACT The Observatorio del Roque de los Muchachos will host the northern site of the Cherenkov Telescope Array Observatory (CTAO), in an area about 200 m below the mountain rim, where the optical telescopes are located. The site currently hosts the MAGIC (Major Atmospheric Gamma-ray Imaging Cherenkov) telescopes, which have gathered a unique series of 20 yr of weather data. We use advanced profile-likelihood methods to determine seasonal cycles, the occurrence of weather extremes, weather downtime, and long-term trends correctly taking into account data gaps. The fractality of the weather data is investigated by means of multifractal detrended fluctuation analysis. The data are published according to the Findable, Accessible, Interoperable, and Reusable (FAIR) principles. We find that the behaviour of wind and relative humidity show significant differences compared to the mountain rim. We observe an increase in temperature of $0.55\pm 0.07\mathrm{(stat.)}\pm 0.07\mathrm{(syst.)}$$^{\circ }$C decade−1, the diurnal temperature range of $0.13\pm 0.04\mathrm{(stat.)}\pm 0.02\mathrm{(syst.)}$$^{\circ }$C decade−1 (accompanied by an increase of seasonal oscillation amplitude of $\Delta C_m=0.29\pm 0.10\mathrm{(stat.)}\pm 0.04\mathrm{(syst.)}$$^{\circ }$C decade−1), and relative humidity of $4.0\pm 0.4\mathrm{(stat.)}\pm 1.1\mathrm{(syst.)}$ per cent decade−1, and a decrease in trade wind speeds of $0.85\pm 0.12\mathrm{(stat.)}\pm 0.07\mathrm{(syst.)}$ (km h−1) decade−1. The occurrence of extreme weather, such as tropical storms and long rains, remains constant over time. We find a significant correlation of temperature with the North Atlantic Oscillation Index and multifractal behaviour of the data. The site shows a weather-related downtime of 18.5 per cent–20.5 per cent, depending on the wind gust limits employed. No hints are found of a degradation of weather downtime under the assumption of a linear evolution of environmental parameters over time.

Comparative analysis of aggregate and sectoral time-varying market efficiency in the Russian stock market during the COVID-19 outbreak and the Russia–Ukraine conflict (RUC)

PurposeThe primary purpose of this research is to identify and compare the multifractal behavior of different sectors during these crises and analyze their implications on market efficiency.Design/methodology/approachWe used multifractal detrended fluctuation analysis (MF-DFA) to analyze stock returns from various sectors of the Moscow Stock Exchange (MOEX) in between two significant periods. The COVID-19 pandemic (January 1, 2020, to December 31, 2021) and the Russia–Ukraine conflict (RUC) (January 1, 2022, to June 30, 2023). This method witnesses multifractality in financial time series data and tests the persistency and efficiency levels of each sector to provide meaningful insights.FindingsResults showcased persistent multifractal behavior across all sectors in between the COVID-19 pandemic and the RUC, spotting heightened arbitrage opportunities in the MOEX. The pandemic reported a greater speculative behavior, with the telecommunication and oil and gas sectors exhibiting reduced efficiency, recommending abnormal return potential. In contrast, financials and metals and mining sectors displayed increased efficiency, witnessing strong economic performance. Findings may enhance understanding of market dynamics during crises and provide strategic insights for the MOEX’s investors.Practical implicationsUnderstanding the multifractal properties and efficiency of different sectors during crisis periods is of paramount importance for investors and policymakers. The identified arbitrage opportunities and efficiency variations can aid investors in optimizing their investment strategies during such critical market conditions. Policymakers can also leverage these insights to implement measures that bolster economic stability and development during crisis periods.Originality/valueThis research contributes to the existing body of knowledge by providing a comprehensive analysis of multifractal properties and efficiency in the context of the MOEX during two major crises. The application of MF-DFA to sectoral stock returns during these events adds originality to the study. The findings offer valuable implications for practitioners, researchers and policymakers seeking to navigate financial markets during turbulent times and enhance overall market resilience.

Isles of regularity in a sea of chaos amid the gravitational three-body problem

Context. The three-body problem (3BP) poses a longstanding challenge in physics and celestial mechanics. Despite the impossibility of obtaining general analytical solutions, statistical theories have been developed based on the ergodic principle. This assumption is justified by chaos, which is expected to fully mix the accessible phase space of the 3BP. Aims. This study probes the presence of regular (i.e. non-chaotic) trajectories within the 3BP and assesses their impact on statistical escape theories. Methods. Using three-body simulations performed with the accurate, regularized code TSUNAMI, we established criteria for identifying regular trajectories and analysed their impact on statistical outcomes. Results. Our analysis reveals that regular trajectories occupy a significant fraction of the phase space, ranging from 28% to 84% depending on the initial setup, and their outcomes defy the predictions of statistical escape theories. The coexistence of regular and chaotic regions at all scales is characterized by a multi-fractal behaviour. Integration errors manifest as numerical chaos, artificially enhancing the mixing of the phase space and affecting the reliability of individual simulations, yet preserving the statistical correctness of an ensemble of realizations. Conclusions. Our findings underscore the challenges in applying statistical escape theories to astrophysical problems, as they may bias results by excluding the outcome of regular trajectories. This is particularly important in the context of formation scenarios of gravitational wave mergers, where biased estimates of binary eccentricity can significantly impact estimates of coalescence efficiency and detectable eccentricity.

Structures Associated with the Borromean Rings’ Complement in the Poincaré Ball

Guided by physical needs, we deal with the rotationally isotropic Poincar´e ball, when considering the complement of Borromean rings embedded in it. We consistently describe the geometry of the complement and realize the fundamental group as isometry subgroup in three dimensions. Applying this realization, we reveal normal stochastization and multifractal behavior within the examined model of directed random walks on the rooted Cayley tree, whose sixbranch graphs are associated with dendritic polymers. According to Penner, we construct the Teichm¨uller space of the decorated ideal octahedral surface related to the quotient space of the fundamental group action. Using the conformality of decoration, we define six moduli and the mapping class group generated by cyclic permutations of the ideal vertices. Intending to quantize the geometric area, we state the connection between the induced geometry and the sine-Gordon model. Due to such a correspondence we obtain the differential two-form in the cotangent bundle of the moduli space.

Delving into the Exchange-Traded Funds (ETFs) Market: Understanding Market Efficiency

Exchange-traded funds (ETFs) are the most popular products in the financial sector today. There is extensive literature on the multifractal analysis of some stock markets, but not about the multifractal behaviour of the ETF market. This study examines the efficiency of stock index ETFs worldwide from an Efficient Market Hypothesis (EMH) perspective, using the ETFs: Ishares Msci World ETF (URTH), Ishares Russell 1000 ETF (IWB), SPDR S&P 500 ETF TRUST (SPY), Ishares Global Clean En. ETF (ICLN), Ishares USD Green Bond ETF (BGRN), from 1 January 2021 to 24 May 2024. It analyses a pre-conflict and a geopolitical conflict to uncover distinct patterns of behaviour reflecting significant changes in market conditions. Before the conflict, the Ishares MSCI World, Ishares Russell 1000, SPDR S&P 500 and Ishares USD Green Bond ETFs showed signs of anti-persistence in returns, indicating a lack of strong relationship or predictability between short-term price movements. The Ishares Global Clean Energy ETF did not reject the random walk hypothesis, suggesting that returns follow a pattern closer to random, where market prices already efficiently reflect all available information. During the conflict, there was a transition in the ETFs' behaviour patterns, as evidenced by the increases in slope values for Ishares MSCI World, Ishares Russell 1000, SPDR S&P 500, Ishares Global Clean Energy and Ishares USD Green Bond. Thus, the possible transition from anti-persistence to long-term memories in ETF returns during the conflict. For portfolio managers, these findings highlight the need to continually adapt investment strategies to manage risks better and take advantage of opportunities in a dynamic and complex investment environment.

Investigation of Temperature Multifractrality According to Zugspitze Weather Station Data

The main multifractal properties of time series of mean, maximum and minimum daily temperatures are analyzed using the method of multifractal fluctuation analysis. As initial data, we used the results of instrumental temperature observations made at the Zugspitze meteorological station in the period from August 1, 1900 to January 31, 2023. In general, variations in the mean, maximum and minimum daily temperatures demonstrate multifractal behavior, especially for small time scales, up to about 90 days An analysis of the generalized Hurst exponent found that the considered time series have a long-term positive correlation and that the multifractality is weaker with large fluctuations. The singularity spectrum for all time series is truncated to the left, which means that the time series have a multifractal structure that is insensitive to local fluctuations of large values.

Quantum logarithmic multifractality

Through a combination of rigorous analytical derivations and extensive numerical simulations, this work reports an exotic multifractal behavior, dubbed “logarithmic multifractality,” in effectively infinite-dimensional systems undergoing the Anderson transition. In contrast to conventional multifractality observed in finite dimensions, logarithmic multifractality at infinite dimension introduces an algebraic behavior with respect to the logarithm of system size or time. We demonstrate this phenomenon across eigenstate statistics, spatial correlations, and wave packet dynamics. Our findings offer crucial insights into strong finite-size effects and slow dynamics in complex systems undergoing the Anderson transition, such as the many-body localization transition. Published by the American Physical Society 2024

Modeling Riverbed Elevation and Bedload Tracer Transport Resting Times Using Fractional Laplace Motion

AbstractRiverbed elevations play a crucial role in sediment transport and flow resistance, making it essential to understand and quantify their effects. This knowledge is vital for various fields, including river engineering and stream ecology. Previous observations have revealed that fluctuations in the bed surface can exhibit both multifractal and monofractal behaviors. Specifically, the probability distribution function (PDF) of elevation increments may transition from Laplace (two‐sided exponential) to Gaussian with increasing scales or consistently remain Gaussian, respectively. These differences at the finest timescale lead to distinct patterns of bedload particle exchange with the bed surface, thereby influencing particle resting times and streamwise transport. In this paper, we utilize the fractional Laplace motion (FLM) model to analyze riverbed elevation series, demonstrating its capability to capture both mono‐ and multi‐fractal behaviors. Our focus is on studying the resting time distribution of bedload particles during downstream transport, with the FLM model primarily parameterized based on the Laplace distribution of increments PDF at the finest timescale. Resting times are extracted from the bed elevation series by identifying pairs of adjacent deposition and entrainment events at the same elevation. We demonstrate that in cases of insufficient data series length, the FLM model robustly estimates the tail exponent of the resting time distribution. Notably, the tail of the exceedance probability distribution of resting times is much heavier for experimental measurements displaying Laplace increments PDF at the finest scale, compared to previous studies observing Gaussian PDF for bed elevation.

Hydrogen adsorption and dissociation, together with a multifractal transition into graphene-based surfaces

Ab-initio molecular dynamics simulations, together with multifractal detrended fluctuation analysis (MF-DFA) were used to study hydrogen adsorption into a five-vacancy graphene surface. This surface was chosen because it presents active bonds, due to a lack of carbon ions and may thus be appropriate for molecular hydrogen adsorption and storage. Hydrogen dissociation and chemical hydrogen adsorption, together with a multifractal transition were observed at the beginning of molecular dynamics simulations. Multifractal phase transition is a consequence of hydrogen dissociation and adsorption and was observed in the multifractal spectrum, as a fishtail-like behavior. Once the molecular hydrogen was adsorbed and dissociated, a weak multifractal spectrum was observed in time-series. Weak multifractal behavior was adduced by comparing to fractional Brownian motion (FBM). Multifractal characterization indicates irregular signals and anti-persistence, caused by temperature. A fractional Brownian motion region for forecasting and benchmarking, based on statistical indicators and multifractal characterization of the test region is proposed. The results obtained in this investigation may be significant in the search for surfaces and strategies for hydrogen storage.

Multifractal characterization of fried foods pore microstructure

This study aimed to unravel the multifractal properties of internal microstructure in relation to textural evolution of meat-analog (MA) based coated fried foods. Wheat and rice flour-based batters were used to coat MA model that was deep fried at 180 °C for 2, 4 and 6 min in canola oil. Internal pore microstructures were investigated by X-ray microtomography based multifractal analysis (MFA) and texture were assessed by mechanical texture analyzer. Results showed that batter coatings greatly influenced the formation of internal microstructure during frying as well as post-frying mass redistribution and textural evolution. The frying time impacted the evolution of micropores, and porosity showed a positive correlation with the fat content of MA-based fried products. Porous, low-moisture and high-fat containing coated fried MA-based product demonstrated higher textural stability. Internal microstructure (pore distribution) of fried MAs were heterogenous and possessed multifractal scaling behavior, as depicted by MFA outcomes (Singularity & Rényi spectra). Strong correlation observed between moisture-fat, texture, porosity and selected multifractal parameters (Δα, Δfα, R-L, D1, D2, ΔD).

Structured multifractal scaling of the principal cryptocurrencies: Examination using a self‐explainable machine learning

AbstractThis paper introduces a novel statistical testing technique known as segmented detrended multifractal fluctuation analysis (SMF‐DFA) to analyze the structured scaling properties of financial returns and predict the long‐term memory of financial markets. The proposed methodology is applied to assess the efficiency of major cryptocurrencies, expanding upon conventional approaches by incorporating different fluctuation regimes identified through a change‐point detection test. A single‐factor model is employed to characterize the endogenous factors influencing scaling behavior, leading to the development of a self‐explanatory machine learning approach for price forecasting. The proposed method is evaluated using daily data from three major cryptocurrencies spanning from April 2017 to December 2022. The analysis aims to determine whether the digital market has experienced significant changes in recent years and assess whether this has resulted in structured multifractal behavior. The study identifies common periods of local scaling among the three prices, with a noticeable decrease in multifractality observed after 2018. Furthermore, complementary tests on shuffled and surrogate data are conducted to explore the distribution, linear correlation, and nonlinear structure, shedding light on the explanation of structured multifractality to some extent. Additionally, prediction experiments based on neural networks fed with multi‐fractionally differentiated data demonstrate the utility of this new self‐explanatory algorithm for decision‐makers and investors seeking more accurate and interpretable forecasts.

Multifractal Behavior of Cryptocurrencies During Periods of Economic Uncertainty

Background: In recent years, investors' interest in cryptocurrencies has increased due to their notable price volatility and rapid price increases. These investors view cryptocurrencies as suitable financial assets for portfolio rebalancing strategies. Purpose: The main objective of this study is to examine the multifractality of the cryptocurrencies Bitcoin (BTC), Lisk (LSK), Quantum (QUA), Litecoin (LTC), Ripple (XRP), Augur (REP), Darkcoin (DASH), EOS, IOTA (MIOTA). Methods: The Detrended Fluctuation Analysis (DFA) econophysics model supports the methodology. Results: The results suggest that during the 2020 pandemic period, the digital currencies LSK, QUA, MIOTA, XRP, REP, BTC, ETH, LTC and DASH showed very significant persistence, indicating that price formation is not random. However, validating that cryptocurrency prices are predictable based on historical time series was impossible. On the other hand, the digital currency EOS proved to be in equilibrium; in other words, price formation follows the random walk pattern, suggesting that prices are not autocorrelated over time. During the 2022 geopolitical conflict, long-term memory patterns shifted significantly towards short-term memories, i.e. anti-persistence. The digital currencies ETH, MIOTA, EOS, LTC, REP, LSK and DASH showed anti-persistence slopes, indicating that prices were less influenced by past events and more by recent events. On the other hand, the cryptocurrencies BTC (0.50), QUA (0.50), and XRP (0.50) demonstrate that prices contain a significant random component and that the residuals are independent and identically distributed (i.i.d.), supporting the idea that white noise might be present. Conclusion: From a risk management perspective, these findings are highly relevant to investors, traders and market participants.

Volatility Analysis of Financial Time Series Using the Multifractal Conditional Diffusion Entropy Method

In this article, we introduce the multifractal conditional diffusion entropy method for analyzing the volatility of financial time series. This method utilizes a q-order diffusion entropy based on a q-weighted time lag scale. The technique of conditional diffusion entropy proves valuable for examining bull and bear behaviors in stock markets across various time scales. Empirical findings from analyzing the Dow Jones Industrial Average (DJI) indicate that employing multi-time lag scales offers greater insight into the complex dynamics of highly fluctuating time series, often characterized by multifractal behavior. A smaller time scale like t=2 to t=256 coincides more with the state of the DJI index than larger time scales like t=256 to t=1024. We observe extreme fluctuations in the conditional diffusion entropy for DJI for a short time lag, while smoother or averaged fluctuations occur over larger time lags.

On the sensitivity of sea ice deformation statistics to plastic damage

Abstract. We implement a plastic damage parametrization, distinct from the elastic damage in the elasto-brittle framework, in the standard viscous–plastic (VP) sea ice model to disentangle its effect from resolved model physics (visco-plastic with and without damage) on its ability to reproduce observed scaling laws of deformation. To this end, we compare scaling properties and multifractality of simulated divergence and shear strain rate, as proposed in the Sea Ice Rheology Experiment (SIREx) studies, with those derived from the RADARSAT Geophysical Processor System (RGPS). Results show that including a plastic damage parametrization in the standard viscous–plastic model increases the spatial but decreases the temporal localization of simulated linear kinematic features (LKFs) and brings all spatial deformation rate statistics in line with observations from RGPS without the need to increase the mechanical shear strength of sea ice as recently proposed for lower-resolution viscous–plastic sea ice models. In fact, including damage with a healing timescale of th=30 d and an increased mechanical strength unveils multifractal behavior that does not fit the theory. Therefore, a plastic damage parametrization is a powerful tuning knob affecting the deformation statistics of viscous–plastic sea ice.

Disorder-Broadened Phase Boundary with Enhanced Amorphous Superconductivity in Pressurized In2Te5.

As an empirical tool in materials science and engineering, the iconic phase diagram owes its robustness and practicality to the topological characteristics rooted in the celebrated Gibbs phase law free variables (F) = components (C) - phases (P) + 2. When crossing the phase diagram boundary, the structure transition occurs abruptly, bringing about an instantaneous change in physical properties and limited controllability on the boundaries (F = 1). Here, the sharp phase boundary is expanded to an amorphous transition region (F = 2) by partially disrupting the long-range translational symmetry, leading to a sequential crystalline-amorphous-crystalline (CAC) transition in a pressurized In2Te5 single crystal. Through detailed in situ synchrotron diffraction, it is elucidated that the phase transition stems from the rotation of immobile blocks [In2Te2]2+, linked by hinge-like [Te3]2- trimers. Remarkably, within the amorphous region, the amorphous phase demonstrates a notable 25% increase of the superconducting transition temperature (Tc), while the carrier concentration remains relatively constant. Furthermore, a theoretical framework is proposed revealing that the unconventional boost in amorphous superconductivity might be attributed to an intensified electron correlation, triggered by a disorder-augmented multifractal behavior. These findings underscore the potential of disorder and prompt further exploration of unforeseen phenomena on the phase boundaries.

Investigation on the Nanopore Heterogeneity in Coals and its Influence on Methane Adsorption: A Multifractal Theory Study.

Coalbed methane (CBM) reservoirs constitute a distinct class of dense organic rocks characterized by extremely low porosity and permeability. Conducting an in-depth investigation into pore heterogeneity assumes paramount importance for the exploration and development of CBM. This study focuses on the multifractal analysis of the pores with diameters below 300 nm in six coal samples sourced from the Junggar Basin and the Qinshui Basin in China. The analysis is based on a series of experiments involving CO2 adsorption, low-temperature N2 adsorption/desorption, and CH4 isothermal adsorption. This work delves into the influence of pore heterogeneity on gas adsorption capacity by linking the structural parameters to CH4 adsorption properties. The results indicate that both the micropores, as assessed through CO2 adsorption, and mesopores to macropores, measured via N2 adsorption, exhibit multifractal behavior. In contrast to micropores, the mesopores and macropores display stronger heterogeneity and lower connectivity. Generally, uniform and well-connected nanopores are anticipated to positively contribute to gas adsorption. However, there is a positive correlation between the Langmuir volume and the heterogeneity degree of micropores. This phenomenon is ascribed to the fact that the greater surface complexity in micropores involves a larger specific surface area and a higher abundance of adsorption sites. This research contributes to a more profound and precise comprehension of the heterogeneous pore structure within CBM reservoirs, thereby establishing a theoretical foundation for the sustainable exploitation of CBM.