- #1
JorisL
- 489
- 189
Hey,
In a course were we are treating phase transitions from a mathematically exact point of view, Cylinder sets were introduced. I'll first outline the context some more.
So we are considering systems on a lattice with a finite state space for each lattice point, for simplicity. E.g. an Ising spin model on a d-dimensional lattice ##\mathcal{L} = \mathbb{Z}^d##.
The spins can take values in ##S = \{-1,+1\}## and the entire configuration is an element of ##\Omega = S^\mathcal{L}##.
From Tychonoff's theorem we get that the configuration space over the entire lattice is compact (product of compact spaces).
Now denote ##\Lambda \subset \mathcal{L}## as a finite subset (for example the d-dimensional hypercubes ##\Lambda_n = [-n,n]^d##).
Furthermore we use ##\sigma , \nu , \xi## to denote configurations on the lattice i.e. they are exactly the elements of ##\Omega##.
Finally we have this notion we can use to have an idea of distance (not very exact I think),
\xi \equiv \sigma \,\text{on}\, \Lambda \Leftrightarrow \forall x\in\Lambda ,\, \xi(x) = \sigma(x)
We defined the cylinder sets as
\mathcal{N}_\Lambda(\sigma) = \left\{ \xi\in\Omega : \xi \equiv \sigma \,\text{on}\, \Lambda\right\},\,\sigma\in\Omega
I hope this is somewhat clear so far.
Mathematically it kind of makes sense to me, what I don't have is an intuitive idea of these cylinder sets.
How useful are these without the notion of distance (at least we haven't defined this so I guess it is not needed)?
Why are they called cylinder sets? Were they first constructed for a space where they actually are cylinders (or are isomorphic to cylinders)?
Can they be represented in a diagram (with appropriate projections/maps between space) like in the case of a manifold with its tangent spaces/bundle and cotangent spaces? I ask this because I attempted something like that and got stuck with making this into a clear diagram (other spaces are added afterwards like a Banach space of observables).
Thanks
Joris
In a course were we are treating phase transitions from a mathematically exact point of view, Cylinder sets were introduced. I'll first outline the context some more.
So we are considering systems on a lattice with a finite state space for each lattice point, for simplicity. E.g. an Ising spin model on a d-dimensional lattice ##\mathcal{L} = \mathbb{Z}^d##.
The spins can take values in ##S = \{-1,+1\}## and the entire configuration is an element of ##\Omega = S^\mathcal{L}##.
From Tychonoff's theorem we get that the configuration space over the entire lattice is compact (product of compact spaces).
Now denote ##\Lambda \subset \mathcal{L}## as a finite subset (for example the d-dimensional hypercubes ##\Lambda_n = [-n,n]^d##).
Furthermore we use ##\sigma , \nu , \xi## to denote configurations on the lattice i.e. they are exactly the elements of ##\Omega##.
Finally we have this notion we can use to have an idea of distance (not very exact I think),
\xi \equiv \sigma \,\text{on}\, \Lambda \Leftrightarrow \forall x\in\Lambda ,\, \xi(x) = \sigma(x)
We defined the cylinder sets as
\mathcal{N}_\Lambda(\sigma) = \left\{ \xi\in\Omega : \xi \equiv \sigma \,\text{on}\, \Lambda\right\},\,\sigma\in\Omega
I hope this is somewhat clear so far.
Mathematically it kind of makes sense to me, what I don't have is an intuitive idea of these cylinder sets.
How useful are these without the notion of distance (at least we haven't defined this so I guess it is not needed)?
Why are they called cylinder sets? Were they first constructed for a space where they actually are cylinders (or are isomorphic to cylinders)?
Can they be represented in a diagram (with appropriate projections/maps between space) like in the case of a manifold with its tangent spaces/bundle and cotangent spaces? I ask this because I attempted something like that and got stuck with making this into a clear diagram (other spaces are added afterwards like a Banach space of observables).
Thanks
Joris