In any case, you need to consider the density of states (per unit whatever the integration variable is). Or you might formulate it as an integral over energies. You might formulate the problem as an integral over coordinates $x$, or coordinates-and-momenta, $(x,p)$. We describe the main areas of research that laid the foundations for this principle and discuss its modern form and limitations. The chosen distribution maximizes the entropy function subjected to satisfying information constrains via the method of Lagrange multipliers. It is necessary, though, to introduce a measure, for the integration. The maximum entropy production principle (MEPP) was repeatedly and independently proposed in the mid-20th century in various fields of physics and proved to be extremely effective in various nonequilibrium problems. developed the principle of maximum entropy (POME) as a tool for choosing some specific probability distribution from the set of feasible solutions. Generalizing to the continuum case is discussed on the relevant Wikipedia page, and involves calculus of variations, with summations replaced by integrals, and Lagrange multipliers playing their usual role. Let us consider a particle that may occupy any discrete energy level $\mathcalĪs it is used in standard thermodynamics (all in units of $k_B=1$). The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy, in the context of precisely stated prior data (such as a proposition that expresses testable information). I'm stuck halfway through a derivation of the Boltzmann distribution using the principle of maximum entropy.
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