I Equipartition theorem and Drag

[Opus_723]

If I have a many-body Hamiltonian, and I choose a coordinate x with canonical momentum p, I can say that by the generalized equipartition theorem that

<p(dH/dx)> = -<p(dp/dt)> = 0

Since p and x are distinct phase space variables, and since by the Hamiltonian equations of motion the force (dp/dt) on coordinate x is equal to -dH/dx. So there is no correlation between a canonical momentum and its own rate of change, or the force acting on that coordinate.

On the other hand, if I imagine that my many body system is, for example, a heavy particle in a liquid or gas made of light particles, then there should be something like a drag force on the heavy particle. And a drag force is very obviously correlated with the momentum, with the force tending to be opposite the momentum. So it seems like <p(dp/dt)> should be nonzero.

Another way to see this is to simply look at the correlation in a typical Langevin equation.

So how do I reconcile the correlation between momentum and force at the more macroscopic level where we see 'drag' and the equipartition theorem telling me there should be no such correlation between a canonical momentum and its own rate of change?

 

[vanhees71]

To derive dissipation (friction) from Hamiltonian dynamics you need to use some "coarse graining" in the sense of statistical mechanics. One way is to start from the Liouville equation, which is still an exact description of the $N$-particle distribution function and use it to derive the Boltzmann equation for the one-particle distribution function. Doing so, you'll see that you get first an equation of motion for the one-particle distribution function containing the two-particle distribution function, for the two-particle distribution function you need the three-particle distribution function and so on (the so-called BBGKY hierarchy). To truncate this "tower of equations" for the distribution functions you assume that you can neglect two-particle correlations and substitute the two-particle correlation function by the product of two one-particle correlation functions, i.e., you only keep the uncorrelated part. This leads to the Boltzmann equation, and the H-theorem can be derived, which shows that you have entropy production and dissipation.

With the Boltzmann equation you can also derive the Fokker-Planck equation for the motion of a "heavy particle" in a "heat bath" of "light particles", because then in each two-body collision the momentum transfer to the heavy particle is small, and you can expand the collision term of the Boltzmann equation up to 2nd order in the momentum transfer, which reduces the Boltzmann integro-differential equation to the partial differential Fokker-Planck equation, which itself is again equivalent to a Langevin process, describing the interaction of the heavy particle with the medium by a friction term and a random-force term (friction/drag=dissipation and fluctuations=diffusion).

A very good introduction can be found in Landau and Lifshitz, Course of theoretical physics, Vol. X.

 

[Opus_723]

Okay, that makes sense, but I'm having trouble seeing how the two pictures connect. If I measure <p(dp/dt)> for one particle, it should be zero, and yet the motion of that particle could be well described by a Langevin equation that has a nonzero <p(dp/dt)> due to the drag term. Is it just that Langevin equations get lots of statistics right but fundamentally fail to reproduce the correct <p(dp/dt)> because of the approximations involved, or am I still missing something that actually reconciles the two predictions?