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Couchyam

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- TL;DR Summary
- Are the classical laws of statistical mechanics incomplete?

Einstein famously derived his relation between the diffusion constant of Brownian motion, particle mobility in a disippative medium, and temperature by considering Brownian motion in a harmonic oscillator potential. The result, $D = \mu k_BT$, is derived assuming that the mobility $\mu$ is constant, which is reasonable in the vast majority of cases where the relation is used. However, there are systems in which the mobility depends significantly on the particle configuration, such as when a Brownian particle (something like a largish molecule or pollen grain) undergoes random motion near a static wall, or even in close proximity to another Brownian particle. This would naturally lead to a spatially varying or configuration-dependent diffusion constant.

In physics, we are often taught to accept the results of statistical mechanics without question; if a system (with configuration space $C$ and configurations denoted by $q$) is observed with a (free) energy landscape $U(q)$, the probability of finding the system near state $q$ should be proportional to $\exp(-\beta U(q))$. However, a configuration dependent diffusion constant appears gives rise to a spatially varying probability density, even in the absence of an external energy landscape; heuristically, particles spend more time in regions where they meander slowly (in comparison to regions where the diffusion parameter is large.)

How is this paradox generally resolved, and what might its implications be? That is, should the Boltzmann-derived measure from statistical mechanics be modified to take into account a kinetic effect (the locally varying diffusion parameter), and if so, how, or is there a way to reconcile the (evidently kinetic) effects of a spatially varying diffusion constant on the probability density with equilibrium statistical mechanics? How can we be absolutely sure that Einstein's relation remains valid in systems where there is no intrinsic translational symmetry?

In physics, we are often taught to accept the results of statistical mechanics without question; if a system (with configuration space $C$ and configurations denoted by $q$) is observed with a (free) energy landscape $U(q)$, the probability of finding the system near state $q$ should be proportional to $\exp(-\beta U(q))$. However, a configuration dependent diffusion constant appears gives rise to a spatially varying probability density, even in the absence of an external energy landscape; heuristically, particles spend more time in regions where they meander slowly (in comparison to regions where the diffusion parameter is large.)

How is this paradox generally resolved, and what might its implications be? That is, should the Boltzmann-derived measure from statistical mechanics be modified to take into account a kinetic effect (the locally varying diffusion parameter), and if so, how, or is there a way to reconcile the (evidently kinetic) effects of a spatially varying diffusion constant on the probability density with equilibrium statistical mechanics? How can we be absolutely sure that Einstein's relation remains valid in systems where there is no intrinsic translational symmetry?