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Friction arises as an effective macroscopic force, unlike the four fundamental interactions in nature which are there at the microscopic level. How can we understand, the appearance of friction as a dissipative force originating from the fundamental interactions between atoms or molecules? In particular, how can we understand the velocity dependence (linear or quadratic) of the frictional force through a mathematical model of microscopic dynamics.

In other words, can the Stoke's law (the velocity dependence of viscous drag) be derived from microscopic considerations?

SRS
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2 Answers2

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Stokes law for the force of friction experienced by a sphere in a slowly moving fluid $$ F_d= 6\pi \eta a u, $$ where $u$ is the fluid velocity, $\eta$ is the viscosity and $a$ is the radius, is a rigorous consequence of the Navier-Stokes equation in the limit of small Reynolds number $R=ua/\nu$ where $\nu=\eta/\rho$ ($\rho$ is the mass density of the fluid).

Indeed, the Navier-Stokes equation also predicts higher order terms in $R$. The next term (due to Oseen) provides as correction $$ 1+ \frac{3}{8}R, $$ which gives a quadratic dependence on $u$. Higher order terms are more complicated (terms like $R^2\log(R)$), and eventually turbulence implies that the expansion ceases to be useful, and the drag has to be computed numerically.

The Navier-Stokes equation itself can be derived using more microscopic theories, like (quantum) kinetic theory. In particular, for dilute gases the Boltzmann equation determines the viscosity $\eta$ in terms of the scattering cross section between atoms. A simple estimate is $$ \eta =\frac{\sqrt{2mT}}{3\sigma}\, , $$ where $T$ is the temperature, $m$ is the mass of the atoms, and $\sigma$ is the cross section. This formula is roughly correct for air, but liquids (like water) are more complicated, and $\eta$ has to be computed numerically. This is explained in standard text books on kinetic theory, like vol X of Landau and Lifshitz (or more introductory text books on stat mech, like Kerson Huang's book).

The scattering cross section between atoms can be computed using the laws of quantum mechanics. For neutral atoms the long range potential is the van der Waals (Casimir-Polder) potential, which arises from two-photon exchange and is governed by the polarizability of the atoms.

Note that in passing from many body quantum mechanics to a macroscopic theory like kinetic theory or fluid dynamics we have to coarse grain, and as a result we go from time reversible microscopic dynamics to irreversible macroscopic dynamics. However, the important point is that the parameters of the macroscopic theory (shear viscosity, in particular) are completely fixed by microscopic dynamics.

Solid-on-solid friction is a slightly different subject, see Can the coefficient of friction be derived from fundamentals? .

Thomas
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  • Very useful answer. Can you provide a reference for the derivation of Navier-Stokes equation and the determination of $\eta$ from microscopic interactions as you suggested? @Thomas – SRS May 04 '17 at 08:12
  • In the kind of derivations you mention, does the electromagnetic nature of the microscopic forces play any role or the microscopic forces are assumed to be random/stochastic and their origin is left unspecified? I ask this question because very often I hear that friction ultimately comes from electromagnetic interactions.

    Is it possible to model the problem in some manner where a macroscopic object interacting with its environment give rise to a macroscopic frictional force on the object (without going though Navier-Stoke's equation) on integrating out the environment? @Thomas

     – SRS May 04 '17 at 08:12
  • Yes, the fundamental forces between atoms (which ultimately cause friction) are electromagnetic. An example of a (more) microscopic simulation that accounts for friction is molecular dynamics. – Thomas May 04 '17 at 13:13
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A good derivation of the Navier-Stokes equation and its dissipative term is given in the excellent statistical physics book by Linda Reichl.

In general, the lowest order corrections to conservative effective theories (here the Euler equations) approximating a microscopic theory (here, e.g., classical N-particle theory, or a Boltzmann equation) by appropriate coarse-graining are dissipative, as the dissipation accounts for the loss of energy to unmodelled high frequency modes. The additional terms arising contain integrals over 2-point correlation functions that give the mean contributions of the bilinear terms in an expansion in terms of powers of the fluctuations. The linear terms in this expansion don't contribute since their mean is zero.

This is independent of any electromagnetic stuff, but one can generalize the derivations to get dissipative equations for charged fluids, and these involve the electromagnetic field.

Arnold Neumaier
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