# Effects of a spatially nonuniform diffusion parameter

• A
Couchyam
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?

Mentor
I am not seeing the paradox

Couchyam
I am not seeing the paradox
An example might help illustrate the basic gist. Consider a particle undergoing diffusion along one dimension, at a temperature ##T##. Let's suppose that after a long time has passed, a (spatially varying) probability density ##\rho(x)## emerges. According to canonical statistical mechanics, the density ##\rho(x)## reflects a (nontrivial) free energy landscape ##F(x)##, in the sense that ##\rho(x) = A\exp(-\beta F(x))##, or ##F(x) = -k_BT\log(\rho(x)) + F_0## (here ##\exp(-\beta F(x))## is the canonical Boltzmann weight derived either from entropy maximization or the microcanonical ensemble in conjunction with equal a priori probabilities.)

However, the density ##\rho(x)## also could have been caused by a spatially varying diffusion operator ##D(x)##, ##D'(x)\neq 0##. This isn't the case for all density functions (as certain 'pathological' stationary distributions may require a negative diffusion constant over some interval, making them unstable) but a general solution ##\phi(x)## to the inhomogeneous Laplace equation
$$(D(x)\phi'(x))' = D'(x)\phi'(x) + D(x)\phi''(x) = 0$$
(where ##D(x) > 0##) is itself non-constant. Hence, given a probability density ##\rho(x)##, how can we know which characteristics of ##\rho(x)## are attributable to an equilibrium free energy landscape, and which are caused by a spatially varying diffusion operator (whether from hydrodynamic interactions or something else that might influence the mobility of particles?) Is there an intrinsically non-equilibrium quality of systems with position-dependent diffusion coefficients? (If so, how can this quality be reconciled with equilibrium statistical mechanics, or of the existence of a local equilibrium in Brownian motion?)

Mentor
That seems like less of a paradox and more of an ambiguity.

Couchyam
That seems like less of a paradox and more of an ambiguity.
May I inquire as to where the ambiguity might be?

Mentor
Here:
the density ρ(x) also could have been caused by a spatially varying diffusion operator D(x),
It is not that what you describe is a logical self-contradiction (a paradox), but rather that from this one observation it is impossible to know which of these two sources is the dominant factor.

Couchyam
Here:It is not that what you describe is a logical self-contradiction (a paradox), but rather that from this one observation it is impossible to know which of these two sources is the dominant factor.
Thanks for clarifying!

If there is a paradox, it might be over how the ambiguity resolves itself in the derivation of the Einstein relation ##D = \mu k_BT## (for instance, whether there is a latent assumption that a given probability distribution such as a Gaussian emerges from an equilibrium free energy landscape [of a harmonic oscillator] or if additional assumptions need to be invoked.) Another (apparent) paradox might be that, with a position-dependent mobility, the steady-state probability distribution of a Brownian process would differ from the Boltzmann distribution associated with the drift force/potential energy.

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