What is a Cross Over? Definition & Effects

[atyy]

What is a cross over? It seems to be something like a "qualitative change" without a phase boundary, in the vicinity of the end of a phase boundary. But what is the proper definition? How is it different from other sorts of continuous changes? Is it something that an experimentalist can detect?

 

[Timo]

I don't think that crossover is a standing term with a proper definition. For example, I don't even have a clue what kind of physics you are talking about ;). We did actually use the term in one of our latest publications (computational soft matter physics) in the sense of crossover between two universality classes of a phase transition as an effect of the necessarily finite system sizes in (computer) experiments.

To understand this, one must first know that in the theory phase transitions are only defined in the thermodynamic limit of an infinitely large system (*). Naturally, the infinite size of the system can cause contributions that one may expect to be small to actually become enormous. Consider for example the two-dimensional random field Ising model (RFIM), i.e. an Ising system with a random external magnetic field on each lattice site (e.g. each field independently drawn from a Gaussian distribution around zero). It is generally acknowledged (**) that for any non-zero coupling of the spins to the random extra field the paramagnetic<->ferromagnetic phase transition is destroyed for T>0. (it will still exist in 3+ dimensions, but also be altered there).

Does that mean that this happens in reality? Not really. Certainly, for a given system of finite size, one can tune down (in simulations) the coupling to the random external fields to some small but non-zero value such that one sees something like a transition to a ferromagnetic state at some non-zero temperature. Possibly more interesting, for a set of parameters where one observes the equivalent of ferromagnetism, one can increase the system size until the ferromagnetic-like state is vanished - we know from theory that the ferromagnetic-like must vanish somewhere in the limit of the system size approaching infinity, after all.

Back to your original question: I'm a simulator, and computer simulations in statistical physics notoriously suffer from the problem that the simulated systems are much, much, much smaller than the respective real-world systems of interest. To get around this problem one does simulate systems of various sizes, looks at systematic dependencies of the system size, and tries to extrapolate to the thermodynamic limit. What we did encounter is that if the system sizes in a simulation were chosen too small, then despite of the presence of a (small) coupling to a random field (***) we saw the behavior of the Ising model, even after extrapolation to infinitely-large systems. Only after simulating larger systems the transition-disrupting effect of the random field became apparent and typical properties of the Ising model vanished. We called that a crossover from Ising behavior to the behavior of the RFIM.

So could one see such a crossover in an experiment? In principle yes. For any strength of coupling to the random field (or equivalently: density of sites with a random field) there should be some size regime where the crossover occurs. If you happen to find a system where this regime is in experimentally-accessible scales then one might see this. In practice, an experimentalist will probably put some effort into being away from such a crossover, e.g. by using samples of high purity when trying to experiment on Ising-like systems.

Bottom line: I am not aware of a proper definition of crossover. It does sound like a great excuse for "our data don't look very convincing", tough. I have used it to refer to an effect of system size, i.e. as an effect that exists because one is not in the thermodynamic limit (where proper phase transition behavior is defined). It is experimentally detectable in principle, and possibly even relevant for some real and therefore necessarily finite systems (but that's just my personal point of view).

footnotes:
(*) This has the interesting effect that strictly speaking phase transitions do not exist in nature
(**) i.e. proven, as far as I know, I just don't have a reference at hand right now.
(***) We actually did not simulate a 2D Ising system but a 3D liquid one; I've simplified a bit for the sake of readability.

 

[atyy]

Thanks!

 

[PAllen]

To add to 'lack of definition', I once worked in the audio speaker field. There a 'crossover' is a circuit for allocating signal between woofer and midrange, midrange and tweeter, etc., with the goal being to simulate a magic driver that accurately responds to all frequencies.

 

[EmpaDoc]

Crossovers don't have to be in the vicinity of phase transition. Consider the http://www.aip.org/pnu/2004/split/671-1.html" , which sees pairs of fermionic atoms change from Cooper pairs to molecules as the interaction between the fermions is tuned (as they can do, quite amazingly, with cold atoms). It will occur also in the thermodynamic limit (though experiments are done in atomic clouds with typically millions of atoms - best described as mesoscopic), and there is no parameter to change in order to turn the crossover into a phase transition.

So what is a crossover? I don't think it can be well defined, because the notion of "qualitative change" is so elusive; it's more or less a question of taste to decide whether a change is qualitative or not! A phase transition can be rigorously defined -- a discontinuous change in a macroscopic parameter -- and a crossover is just what we call it when there is not a phase transition, but we still mean that there has been a qualitative change.