# What is the probability distribution for a subsystem in canonical ensemble?

Suppose we have a 3d Ising Model(NN interaction) in simple cubic lattice, if we define a subsystem of it to be a 2d plane of spins(for example all sites with z = L/2, L being the linear system size) and also define this subsystem energy just like a 2d Ising model(only 4 neighbor).

What is the probability distribution of this subsystem states?

For the 3d system itself this probability distribution is proportional to exp(-E/T) but I do not think for the 2d subsystem this holds.

• 3d Ising model is a hard problem. I think it is possible to find an explicit probability distribution of a subsystem states for the simple 1d Insing model and for the not so simple 2d Ising model. – Gec Nov 22 '18 at 17:41

The total density operator for the 3D system is: $$\rho = \frac{1}{\mathcal Z} e^{-\beta H}$$ where $$H \equiv - J \sum_{\langle i,j\rangle} \sigma_i^z \sigma_j^z$$ is the Ising Hamiltonian with nearest neighbor interactions. Now if you want to describe the density operator of the 2D subsystem (and hence the probability distribution), which I'll call $$A$$, you need to take a partial trace over all other degrees of freedom ($$\bar A$$), i.e. all spins outside that 2D plane. So the density operator of the subsystem is: $$\rho_A \equiv \mathrm{Tr}_{\bar A} (\rho) = \frac{1}{\mathcal Z} \mathrm {Tr}_{\bar A}(e^{\beta J \sum_{\langle i,j\rangle} \sigma_i^z \sigma_j^z})$$ I don't think there is an analytic way to exactly evaluate such an expression; so you would probably need to resort to numerical methods for it.