# On the relations among probability distributions in statistical mechanics [duplicate]

Suppose I have a microcanonical ensemble made of $$N$$ particles immersed in a (viscous) fluid at equilibrium temperature $$T$$. When $$N$$ gets very large, the particles can be treated as a continuum. Their current density vector is defined to be $$\mathbf{j}(\mathbf{r}, t) = \rho(\mathbf{r},t)\,\mathbf{v}(\mathbf{r}, t)$$, where $$\rho$$ is a pdf (giving the fraction of particles per unit volume at time $$t$$) and $$\mathbf{v}$$ is the flow velocity.

Question.What is the link between $$\rho(\mathbf{r},t)$$ defined above, and the Boltzmann distribution, which gives the fraction of particles at a given energy $${\displaystyle p_{i}={\frac {1}{Q}}}{e^{-{\varepsilon }_{i}/(kT)}={\frac {e^{-{\varepsilon }_{i}/(kT)}}{\sum _{j=1}^{M}{e^{-{\varepsilon }_{j}/(kT)}}}}}$$ where $$p_i$$ is the probability of state $$i$$, $$\varepsilon_i$$ the energy of state $$i$$, $$k$$ the Boltzmann constant, $$T$$ the absolute temperature of the system and $$M$$ is the number of all states accessible to the system at temperature $$T$$.

• You are likely mixing here classical and quantum cases (viscous liquid and quantum version of the Boltzmann distribution). Apart from that, fluid mechanics is indeed derivable from the Boltzmann equation (equation for a non-equilibrium distribution) - e.g., see the references in my answer in the thread linked above. Commented Dec 21, 2022 at 14:31
• I'm sorry but... the Boltzmann equation seems a little different from the Boltzmann distribution. I'm just trying to reconstruct the Einstein derivation of Brownian motion Commented Dec 21, 2022 at 15:10
• Boltzmann equation describes more general situation than Boltzmann distribution (which is an equilibrium solution of this equation.) But what you wrote in the OP is quantum Boltzmann distribution. Commented Dec 21, 2022 at 15:43
• Ok, and ... what do Navier-Stokes equations have to deal with the link between $\rho(\mathbf{r}, t)$ and $\rho(E, T)$? Commented Dec 21, 2022 at 16:01
• Fluid dynamics equations define the density $\rho(\mathbf{r},t)$ and flow velocity $\mathbf{v}(\mathbf{r},t)$ in continuum limit - this is what you ask about. Commented Dec 21, 2022 at 16:09