ising model

The ising model was originally conceived as a simple explanation for ferromagnetism, and analyses the interaction between particles which can be in one of two states / “spins”: UP (+1) or DOWN (-1).
These particles are placed on a lattice.

The hamiltonian $H$ represents the energy of the system, $J$ is the interaction strength / coupling constant between neighboring spins, and $s_i,s_j\in \{-1,1\}$ are the spins at sites $i$ and $j$.

$$H=-J{s}_1{s}_2=\pm J$$

When the spins are parallel, their energy will be $-J$, when they are opposite, the energy of the system will be $+J$.
Two spins together will spend most of the time in the aligned state, the lowest energy state.

Things get interesting when the system is bigger:

$N$ … number of particles. White=UP, Black=DOWN.
As the temperature decreases, the spins tend to become more aligned, resulting in larger domains of uniformly oriented spins. In this configuration, larger systems exhibit a phase transition from a disordered state (at high temperature) to an ordered state (at low temperature), where most spins are aligned.
In this configuration, where we have all to all connectivity, the more nodes the more stable the system becomes. But not all systems exhibit this behavior; the nature of the phase transition depends on the dimensionality of the lattice and the temperature as we will see.

The energy of the system refers to the total Hamiltonian, considering all pairs of neighboring spins.

In a more general form for a lattice with many spins, the Hamiltonian is expressed as:

$$H=-J\sum _{⟨i,j⟩}{s}_i{s}_j$$

where $⟨i,j⟩$ denotes a sum over all pairs of neighboring spins.

At thermal equilibrium the system does not minimize energy, but the free energy:

$$F(T)=E-TS$$

The change in free energy $F$ is equal to the change in energy $E$ minus the temperature $T$ times the change in entropy $S$.

$$\Delta F=\Delta E-T\Delta S$$

Systems tend to evolve towards states that minimize the free energy.

In a 1-dimensional topology (where the spins are chained like a linked list), there are $N-1$ possible “domain walls”, where two opposing spins face each other.

If a single domain wall exists, the Hamiltonian of the system (given by $H=-J{\sum }_i{s}_i{s}_{i+1}$) results in an energy contribution of $2J$ from the domain wall: Introducing a domain wall costs energy.
Without any domain walls (i.e., all spins are aligned), the energy contribution from interactions is zero.
Now, considering the free energy, the expression $\Delta F=2J-T\ln (N-1)$ represents the change in free energy when a domain wall is present. The term $2J$ accounts for the energy cost of the domain wall, while $T\ln (N-1)$ represents the entropy contribution.
For large $N$ (i.e., $N\to \infty$), the entropy term becomes dominant at any temperature $T$, which implies that the free energy will be negative ($\Delta F<0$), no matter how strong the coupling strength, or how low the temperature.
(At least I think…) Since the free energy is decreasing regardless of whether there are there is a domain barrier added, they can appear spontaneously and break the global ordering of the spins. This means that the 1D Ising model does not have a coherent phase, i.e. the system will not achieve global ordering / equilibrium.

For a 2D-system (grid with nearest neighbour interaction), $\Delta E=J\mathcal{P}$, where $\mathcal{P}$ is the perimiter of the domain barrier.
The change in entropy is the number of perimiters there of length $\mathcal{P}$ there can be.
This turns out to be $\text{Number of configurations}(\mathcal{P})\le 3^{\mathcal{P}}$ (plus some constant), so the change in entropy is at most $\mathcal{P}\log (3)$.

Rearranging, we arrive at the following expression for the change in free energy:

$$\Delta F\ge \mathcal{P}(2J-T\log 3)$$

For a sufficiently low temperature ($T\le \frac{2J}{\log 3}$), the free energy is guaranteed to increase.
→ The creation of a domain barrier is thermodynamically not favourable → The system will look homogenous.
Even if the connection strength is weak, you can still have an ordered phase at a sufficiently low temperature.

The ability to self-organize into an ordered configuration only depends on the topology of the system, and not the interaciton strength.

References

Requirements for self organization