Classical and quantum approaches to probabilistic modeling of fire occurrence in anthracite grade coal

Despite having definite ignition points for a substance, auto ignition may be observed at different temperatures and energies. This paper presents a detailed quantitative analysis of anthracite coal ignition probability across various energy and temperatures under classical and quantum frameworks. The objective is to establish a probabilistic framework to describe ignition behavior as a function of thermal and energetic variability rather than as a fixed threshold. Energy temperature distributions are derived using specific heat temperature relations and the black body radiation equation. General equations for calculating ignition probability at different temperatures and energies measured independently or simultaneously are derived from probability theorems, distribution equations and curves. Further, the probability calculations for 500 K and 1000 kJ/kg are depicted. In the quantum approach, the validity of the energy-temperature uncertainty relationship of quantum thermodynamics is checked in the domain of measurement. Results depict that probability increases from 0.0047 at 100 °C to 0.9935 at 5000 °C, crossing 50% at 769.15 °C. Further, the probability increases from 0.0000 at 200 J/g to 0.9930 at 5000 J/g, crossing 50% at 789.86 J/g. The concept of the ignition line, along with its corresponding equation, is also established. Understanding the probabilistic framework of ignition enhances combustion efficiency, safety, and fire prevention in mining and storage by moving beyond deterministic ignition points to account for thermal and energetic variability. Moreover, the integration of quantum probability principles provides deeper insight into fire occurrence mechanisms, enabling the analysis of ignition behavior under inherently uncertain and fluctuating microthermal conditions.