Correct. But between 5:16 and 5:20 the transition from discrete to continuous is being made. The reason this can be done is that n is always large (and Stirling's formula is a good approximation)kidsasd987 said:However its a finite discrete number. as far as I know, derivative is defined on continuous(complete) functions
The Boltzmann distribution is a statistical distribution that describes the probability of a system being in a particular state at a given temperature. It is used to describe the distribution of energy among the particles in a system, and is an important concept in statistical mechanics.
The Boltzmann distribution is derived using the principles of statistical mechanics, specifically the concept of maximizing entropy subject to constraints. This results in the Boltzmann factor, which relates the energy of a state to its probability.
The Boltzmann distribution derivation assumes that the system is in thermal equilibrium, that the particles are distinguishable, and that there is no interaction between them. It also assumes that the system is in a closed system and that the energy levels are discrete.
The Boltzmann distribution is significant in thermodynamics because it allows us to calculate the probabilities of different energy states in a system, which is essential for understanding how a system behaves at a microscopic level. It also helps us understand how energy is distributed in a system and is used in many thermodynamic equations.
The Boltzmann distribution can be applied to any system that meets the assumptions made in its derivation, such as a gas or a solid. It is also commonly used in quantum mechanics to describe the behavior of particles in a system at the atomic level.