# The role of the ergodic hypothesis

Statistical mechanics textbook often invoke the so-called ergodic hypothesis (assuming that every phase space trajectory comes arbitrarily close to every phase space point with the same values of all conserved variables as the initioal point of the trajectory) to derive thermodynamics from the foundations. However, textbook statistical mechanics gives only a gross simplification of the power of thermodynamics. The ergodic hypothesis is not needed to make thermodynamics valid. Indeed, the ergodic hypothesis is invalid in many cases - namely always when the system needs additional variables to be thermodynamically described.

This is the case for fluids near the critical point, for finite objects at their surfaces, for systems with interfaces, for metastable states, for molecular systems in the absence of chemical reactions (here the number of molecules of each species is conserved), etc.

But this does not invalidate thermodynamics - the latter only requires that a sufficiently large set of macroscopic variables (in the above sense) is included in the list of thermodynamic variables. Indeed, traditional thermodynamics accounts for molecules, surface tension, metastability, etc., without any change to the formalism.

Probably the ergodic hypothesis, restricted to a limited piece of a submanifold of the phase space with fixed values of the macroscopic variables (whether conserved or not) is ''roughly'' equivalent to the completeness of the set of distinguished macroscopic observables, in the sense that every other macroscopic observable can be defined in terms of the distinguished ones. But ...

1. It is the latter property (only) which can be checked experimentally: Completeness holds if and only if the properties of the system under study are indeed predicable by the thermodynamics of the distinguished observables. Experiment (or experience), together with simplicity of the description, decides in _all_ practical situations what is the set of distinguished observables.
Indeed, we refine a model whenever we discover significant deviations from the thermodynamical behavior of a previous simpler model. Thus thermodynamics takes the form of a setting for describing material properties to which any successful description has to conform by axiomatic decree.

2. The ergodic hypothesis can be proved only for extremely simple systems. In particular, these systems must conform to classical mechanics - there is no simple quantum version of ergodic dynamics. Moreover, there are many classical systems which are chaotic only in part of their phase space - they are probably not ergodic, as the number of conserved quantities depends on where in the phase space one is.

3. Thermodynamics applies also for nearly conserved quantities, where the ergodic argument becomes vague; conversely, near ergodicity (up to the model accuracy) is enough to make a thermodynamic description valid. In particular, thermodynamics applies near a critical point where there cannot be an ergodic argument since there is no extra conserved quantity but an order parameter is needed to give a correct description. (At which distance from the critical point should one ignore the order parameter? Ergodic arguments have nothing to say here.)

4. There are studies about the nonergodic behavior of supercooled liquids, e.g., Phys. Rev. A 43, 1103 - 1106 (1991).

Thus I think it is best to ignore the ergodic hypothesis as a means for explaining statistical mechanics, except in some simple model cases. It should have no deeper relevance than the hard sphere model of a monatomic gas (which has been shown to be ergodic, I believe).

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ