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The Schrödinger equation in its variants for many particle systems gives the full time evolution of the system. Likewise, the Boltzmann equation is often the starting point in classical gas dynamics.

What is the relationship, i.e. the classical limit, which connects these two first order in time equations of motions?

How does one approach this, or is there another way in which one sees the classical time evolution?

Where are these considerations relevant?

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I have seen something very similar with mean-field equations in a lecture notes by Francois Golse. He shows the connection between Vlasov equation and the Hartree-Fock equation through just quantization. But the Boltzmann equation is not a mean-field equation... –  r.g. Feb 13 '12 at 16:51
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3 Answers 3

There are two differnt levels to see this connection. Formally, you can derive a Fokker-Planck equation from the Boltzmann equation and do a Wick rotation on the time variable. This can be seen as a mathematical curiosity presently.

But there is a more relevant way to recover this and is given by a formulation of the quantum Boltzmann equation. There is a beautiful Physics Report by Bassano Vacchini and Klaus Hornberger that can be downloaded here. This equation is relevant to understand the behavior of matter waves in interference experiments involving large molecules with their decoherence effects as realized by Anton Zeilinger and Markus Arndt.

When the formal limit $\hbar\rightarrow 0$ is taken, quantum Boltzmann equation reduces to ist classical counterpart.

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Thank you for the article. What do you mean by "presently"? Btw. I'm actually a student at the Boltzmann institute in Vienna. Fun fact about Markus Arndt: It's said that he never sleeps. As a matter of fact, it's been examined that he sends emails to his different PhD students, sometimes in 3 hour intervals, at times like 2am, 5am... Last time I saw him in front of the coffee machine, asking "Do you know where my friend Sarah (PhD Student) is?" His answer: "Working I hope!". But don't get me wrong, he is a very nice guy. :) –  NikolajK Feb 14 '12 at 9:05
@NickKidman: You are welcome. By "presently" I mean that there is no consistent formulation of a Boltzmann equation in a context of stochastic processes relating it to the Schroedinger equation and so this is just a formal coincidence. Nice to know about Markus: The work he is performing is really striking. –  Jon Feb 14 '12 at 9:14
Actually, your comment about the Fokker-Planck equation got me a little confused about the hierarchy of these equations. Could you add a comment on declining order, if there is one? I'd see the abstract Schrödinger equation on the top of the chain, but a relation (not limit) with that classical equation feels strange. There are also things like the Master equation, which sometimes seems to come before the Boltzmann equation even. –  NikolajK Feb 14 '12 at 9:20
The idea is this (e.g. K. Huang, Statistical Mechanics): From Boltzmann equation you can derive the diffusion equation that has the from $$\partial_t\Theta=D\partial_{xx}\Theta.$$ Now, take $D$, the diffusion coefficient, arbitrary and rotate time as $t\rightarrow -it$. This gives you, in a strict formal sense, the Schroedinger equation. –  Jon Feb 14 '12 at 9:33
@Jon: The relation between Boltzmann equation and diffusion equation you mention is extremely formal, and very incidental. The interpretation is that the number of particles is diffusing, the BE is not an equation describing a single particle. So the rotation to the Schrodinger equation is certainly devoid of even the usual little bit of physical content of Wick rotation. if you use a 12 dimensional BE describing pairwise correlations, it would have no relation to SE, so this is really just a low-dimensional coincidence that both describe diffusion. –  Ron Maimon Feb 14 '12 at 10:48
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You are probably asking if there is a limit where the Schrodinger equation for many particles interacting with a potential reproduces the Boltzmann equation for many classical particles colliding in a potential. The answer is no, because the Boltzmann equation is irreversible in time, while the Schrodinger equation is reversible. The first order BE does not have a symmetry between forward and backward in time evolution, it's not an equation for complex amplitudes. It has an entropy which constantly increases. The Schrodinger equation is completely reversible.

It should be added that this is also true of classical particle dynamics--- as Loschmidt noted, it is impossible for the reversible classical particle dynamics to ever produce exact irreversible Boltzmann evolution. But Boltzmann understood that the equation was only approximate, valid only when multiparticle correlations could be ignored. But Boltzmann also understood from physical intuition that this was the case most of the time in real gasses. So there is a sense in which it is possible to arrive at the Boltzmann equation from a statistical description of a classical gas. But it requires a truncation of the statistical description to only the function f(x,p) which describes the expected number of particles at position x and momentum p. This truncation is lossy, and it is the reason for the emergence of irreversibility.

So the more nuanced answer is that you can find a Boltzmann equation when you can truncate the statistical description into a low dimensional projection, and get the best approximate statistical evolution in this truncation.

Your intuition was probably that the SE should reduce to a BE because both describe the behavior of a bunch of particles in a statistical way. This is incorrect, because the SE is not a statistical description. Absent a measurement, which is not described by SE anyway, the SE gives you the time evolution of the complete state. There is nothing statistical going on without measurement.

The SE is also describing waves in a humongous 3N dimensional space, so that it describes all the entanglements between all the particles. To get to the BE, you need to truncate the space to just the expected number of particles at position x with momentum p. This truncation doesn't work with probability amplitudes--- only probabilities can be truncated like this. The reason is that if you have a truncated state X which internally can be one of two possible micro-states A or B, if you have probability, you can just say "the probability of X is the sum of the probability of A and the probability of B", but you can't say "the amplitude of X is the sum of the amplitude of A and the amplitude of B", because that's just wrong for quantum evolution.

So truncated partial descriptions are only for classical probabilities. So to reproduce classical Boltzmann dynamics you need to pass to a statistical description, which means density matrices, then take the classical limit of SE for density matrices, then project this probabilistic description from the 6N dimensional phase space to the 6 dimensional Boltzmann function.

The first step is the classical limit of QM, the second step is the original derivation of the Boltzmann equation from the full description of the stochastic gas in 6N dimensional phase space. You can't relate the equations in any other meaningful way.

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I was just wondering, since both are tool with which one could describe several gas systems microscopically, there must be a connection at least of all the expectation values. Also, I'm a bit confused about the whole irreversibility thing: In view of the second law of thermodynamics, isn't the irreversibility a good thing? Don't I want the (more) fundamental description to have this kind of property? –  NikolajK Feb 15 '12 at 9:36
@Nick: yes--- in principle, if you have a gas of many atoms, and you evolve their wavefunction by SE, and you look at only the density of atoms with position x and momentum p, you should be able to reproduce a Boltzmann equation in some limit. But the limit is going to be complicated, because it is essentially two separate limits--- the classical limit (where the separation is large compared to the wavelength) plus the Boltzmann limit (where the multiparticle correlations are ignored). The second limit leads to the irreversibility, it corresponds to throwing away certain complex information. –  Ron Maimon Feb 15 '12 at 14:16
@RonMainmon: One last thing: The Boltzmann equation is used in Plasma physics. But there is some wider range interaction going on, so how does that fit together with the statement? –  NikolajK Feb 15 '12 at 15:12
@Nick Kidman: The essential approximation for the Boltzmann equation is that you can describe the statistics of the particles using the density at position x and momentum p, and that this density obeys a closed equation from 2-particle scattering. You can add a term which describes the interaction of each particle with an overall electrostatic field due to all the other particles, because you can extract the charge density from the position density distribution of the charged particles in the plasma, so it is also possible to write a closed equation. But this is classical dynamics always. –  Ron Maimon Feb 15 '12 at 15:21
Truncated partial descriptions are definitely not only for classical systems. it is known for a long time that the Boltzmann equation can be obtained without going thrrough a classical intermediate stage, and variants with more realistic quantum collision operators must be derived in that way to give results consistent with experiment. See my own answer. –  Arnold Neumaier Mar 18 '12 at 14:08
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Irreversible equations such as the Boltzmann equation can be obtained rigorously as scaling limits of reversible microscopic equations such as a multiparticle Schroedinger equation.

A good entry point for studied about your question is the survey paper by

  • H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Mod. Phys. 52 (1980) 569.

    You can follow up with reading one of the many papers citing it obtained with author:spohn kinetic in http://scholar.google.com

    Less rigorous versions of the same technique are ubiquitous in nonequilibrium statistical mechanics. I recommend two nice books: the book by

  • Grabert, Projection operator techniques (very thorough theoretically), and the book by
  • Oettinger, Beyond equilibrium thermodynamics (much more applied).

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