Dimension, fluctuations, and phase transitions

[VortexLattice]

Hey all,

I'm reading Chaikin's Principles of Condensed matter, and he's talking about the effect fluctuations have in various systems. He says:

Below three dimensions, fluctuations become so violent that they can destroy the ordered state and finite temperature phase transitions. In one dimension, fluctuations destroy all long-range order and phase transitions. This is essentially a problem of connectivity. The only way one end of a one-dimensional system knows what is going on at the other end is via information transmitted directly along the chain. For an infinitely long system, any fluctuation cuts the flow of information and hence the order. Since there are always fluctuations at any finite temperature, a one-dimensional system cannot be ordered except at zero temperature. In two-dimensional systems, there are many paths that can connect one part of the system to others.

So I get why order is destroyed in 1D, and not in 2D. But I don't see why they destroy the phase transitions. Can anyone tell me?

Thanks!

 

[daveyrocket]

Well if there is no order, how can you have a phase transition?