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The Maxwell-Boltzmann distribution and the Boltzmann distributions are probability distributions, i.e. functions $\rho(\vec x,\vec v)$ of velocity and position of a particle, that say what is the probability density that the velocity and position belong to the small cube around the given value of them. The Boltzmann distribution is the more general one, $\... 3 Entropy Demystified (The Second Law Reduced to Plain Common Sense) by Arieh Ben-Naim. Authored discussed not only the thermodynamics origin of entropy but also the same notion in the context of information theory developed by Claude Shannon. 3 Your equation (2) is trivially a solution of (1), because$v$and$T$are constant. This is a disappointing answer, because it leaves unanswered the question what makes the Boltzmann distribution unique. The answer is that you only wrote down the collision-less Boltzmann equation, but in the real world collisions are always present (and indeed, systems ... 1 Density is the average amount of mass per unit volume$\rho(\vec{r},t) = \frac{M}{V_r}$. Distribution function is defined as a number of particles per unit phase space volume$ f(\vec{r}, \vec{c},t) = \frac{N}{V_r V_c} $(which is space volume times velocity volume). Each particle has mass$M$, so to get the total mass density, we need to sum distribution ... 1 This has been open for a while so I will bite. The information paradox has two versions or iterations. The previous one is that information is demolished by black holes by the entropy of its event horizon and that Hawking radiation that is emitted is in a pure blackbody distribution. A blackbody distribution of radiation is maximally random. If you make a ... 1 But for an ideal gas, internal energy is only a function of temperature and so internal energy remains constant here,no change in average kinetic energy of gas particles takes place, so where does the chaos come from to increase entropy of the system. 'Chaos' is not a very well defined term in context of statistical physics. It is not necessary to use it ... 1 Consider the probability of state$n$divided by the probability of state 0: $$\frac{\text{prob of }\left \lvert E_n \right \rangle}{\text{prob of }\left \lvert E_0 \right \rangle} = \frac{\exp \left( -E_n / k T \right)}{\exp \left( - E_0 / k T \right)} = \exp \left( - (E_n - E_0) / kT \right) \, .$$ If$E_n > E_0$, then as$T \rightarrow 0$the ... 1 There is no single point where this becomes true - it is a very gradual change. The buzzwords are microscopic$\to$mesoscopic$\to\$ macroscopic. There is no special kind of mathematics involved; in the mesoscopic domain one uses a mix of quantum mechnaics and statistical mechanics. See https://en.wikipedia.org/wiki/Mesoscopic_physics