## 3 Theoretical Approach
### Statistical Mechanics to Describe Energy  
At the origin of every transformation, there is energy. The energy stored in any particle (atoms or molecules) can be analyzed using Schrodinger's equation [26] and used as a vehicle to provide a physical basis for the upscaling process. The resolution of the Schrodinger's equation [25] gives us information about the quantum particles and their state of energy. Thus, the energy of a system is composed of the kinetic energy of each part of the system and the potential energy of all the interactions between all parts of the system.  
For a perfect monoatomic gas, the interactions can be considered negligible. Hence the global energy can be described as the sum of the energy of all parts. If the interactions cannot be neglected, then the forces between atoms need to be included in the description of the overall energy of the system. The potential energy of the interactions adds a significant level of complexity. Indeed, in many cases, the interactions are not well-known. If strong interactions exist, then assuming that the global energy of the system is equal to the sum of the energy of all elementary particles (assumption of equipartition) brings uncertainty and potentially significant errors about using the Boltzmann statistics in the Arrhenius equation development. This is an initial and first key assumption that has yet to be demonstrated for materials of the complexity of those involved in fires.  
This is a strong assumption, but it enables the description of the energy repartition by the following equation:  
\[U=\sum_{i=0}^{I}E_{i}n_{i} \tag{3}\]  
Here \(U\) represents the total energy of the system, \(E_{i}\) the level of energy in the state \(i\) and \(n_{i}\) the number of particles in the energetic state \(i\) (this definition is consistent with the quantum mechanics definitions, see [27]). The total number of particles \(N\) in all energetic states \(i\) is given by:  
\[N=\sum_{i}n_{i} \tag{4}\]  
Based on the previous assumptions, [PERSON] uses statistical mechanics [25; 28] to develop the path between the microscopic and macroscopic scale (see Appendix A about the Boltzmann development and the key steps with the Equation (A4) and the Equation (A15), both use the Equations (3) and (4)). To do that, [PERSON] included the configurations and weights of all molecules and all molecule states. Indeed, at any instance, there will be \(n_{0}\) molecules in the state \(0\) with energy \(E_{0}\) and so forth for states \(1\), \(2\), etc. (see Appendix A, Equation (A1)).  
To account for the micro-state, in [PERSON]'s statistics, the particles must be identic and distinguishable [29], here it is mean that the position and the velocity of each particle are known (contrary to quantum particles, for example). This is the second key assumption.  
Finally, after an extensive derivation (see Appendix A), [PERSON] determines the following equation:  
\[\frac{n_{i}}{N}=\frac{g_{i}}{Z}e^{-\frac{E_{i}}{R_{b}T}} \tag{5}\]  
Thus, Equation (5) represents the Boltzmann distribution, which describes a number \(n_{i}\) of particles in the energetic state \(E_{i}\). In this equation, \(g_{i}\) represents the degeneracy of the state \(i\), \(Z\) is the partition function (see Equation (A12) in Appendix A), \(k_{b}\) is the Boltzmann constant and \(T\) the temperature [29].  
Equation (5) describes the behavior of an ensemble of particles by a macroscopic expression, and thus the transition between microscopic and macroscopic scale is mathematically formalized.  
The Boltzmann equation and its development is the key concept that enables the understanding of the kinetic equation and any potential limitations to it. In the following sections, the main assumptions used in developing Equation (5) are further analyzed to identify other key issues of relevance. These assumptions are then evaluated in the context of the general kinetic law. In doing so, this work extends the study by [PERSON] and [PERSON] [17].