olegt
Posts: 1405
Joined: Dec. 2006
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Missing Shade of Blue,
I work in physics of condensed matter, studying all kinds of magnetic phenomena, as a theorist. My interest in the foundations of statistical mechanics is not exactly idle: I teach a graduate course in statistical physics every other year. So I am interested in what mathematicians think about this.
| Quote | | I do have a slight quibble with the notion that glass "becomes non-ergodic" at low temperatures. The phrase suggests that the system in its high-temperature equilibrium state is ergodic, and that it loses this ergodicity when it freezes into a glass state. But there is no reason to believe that the equilibrium state is ergodic if you reject the standard stat. mech. assumption that thermodynamic equilibrium implies ergodicity. |
I am not sure that there are sufficient grounds for rejecting the link between ergodicity and thermodynamic equilibrium. A key point to remember is this: physicists like to work in the so-called thermodynamic limit (infinite number of particles). If your system is too small, you can't even define a temperature, so there is no thermodynamics to speak of. For example, even though N>1 hard spheres in a box are an ergodic system, the velocity distribution becomes Maxwellian only in the limit of an infinite N.
| Quote | | We don't have proofs of ergodicity for virtually all the systems to which we usually apply classical stat. mech. In fact, there's good reason to think many of these systems are not ergodic. The KAM Theorem leads us to expect that a typical Hamiltonian system with finite degrees of freedom will contain islands of non-ergodic flow. |
Yes, KAM theory tells us that some invariant tori of an integrable system remain such even after the addition of a nonlinear perturbation. That is quite significant since it follows that realistic dynamical systems can be non-ergodic. However, there is an important caveat: the perturbation must be sufficiently weak. And furthermore, the bounds on the strength of the perturbation become more stringent as the number of degrees of freedom increases, so in the thermodynamic limit invariant tori shrink and a typical system becomes ergodic.
Let me quote Henk Broer's brief overview KAM theory: the legacy of Kolmogorov’s 1954 paper (PDF file):
| Quote | | On the other hand, and from a more global point of view, the measure-theoretical part of KAM theory implies that for typical Hamiltonian systems in finitely many degrees of freedom, no ergodicity holds, since the energy hypersurfaces can be decomposed in several disjoint invariant sets of positive measure. This is of particular interest for statistical physics, where the ergodic hypothesis roughly claims that the system, when confined to bounded energy hypersurfaces, is ergodic. This paradox probably is resolved as the number of particles is increasing since the obstruction to ergodicity provided by the KAM tori then seems to decrease rapidly in importance. |
Towards the end of the paper he discusses a particular case of N coupled nonlinear oscillators, where the allowed perturbation strength goes down exponentially with N.
Of course, ergodicity in systems with an infinite number of degrees of freedom is not yet well understood. But it looks to me like physicists need not be alarmed just yet: while we now know that ergodicity is broken in some realistic finite systems, there are as yet no reasons to question ergodicity of realistic systems in the thermodynamic limit.
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