underlying set of states is given by a full shift, then the sum may be trivially evaluated as If neighboring spins are only allowed in certain specific configurations, then the state space is given by a subshift of finite type. The partition function may then be written as where card is the cardinality or count of a set, and Fix is the set of fixed points of the iterated shift function: The q × q matrix A is the adjacency matrix specifying which neighboring spin values are allowed. Interacting model The simplest case of the interacting model is the Ising model, where the spin can only take on one of two values, sn ∈ {−1, 1} and only nearest neighbor spins interact. The interaction potential is given by This potential can be captured in a 2 × 2 matrix with matrix elements with the index σ, σ′ ∈ {−1, 1}. The partition function is then given by The general solution for an arbitrary number of spins, and an arbitrary finite-range interaction, is given by the same general form. In this case, the precise expression for the matrix M is a bit more complex. The goal of solving a model such as the Potts model is to give an exact closed-form expression for the partition function and an expression for the Gibbs states or equilibrium states in the limit of n → ∞, the thermodynamic limit. Applications Signal and image processing The Potts model has applications in signal reconstruction. Assume that we are given noisy observation of a piecewise constant signal g in Rn. To recover g from the noisy observation vector f in Rn, one seeks a minimizer of the corresponding inverse problem, the Lp-Potts functional Pγ(u) which is defined by The jump penalty forces piecewise constant solutions and the data term couples the minimizing candidate u to the data f. The parameter γ > 0 controls the tradeoff between regularity and data fidelity. There are fast algorithms for the exact minimization of the L1 and the L2-Potts functional. In image processing, the Potts functional is related to the segmentation problem. However, in two dimensions the problem is NP-hard. See also Random cluster model Critical three-state Potts model Chiral Potts model Square-lattice Ising model Minimal models Z N model References External links Spin models Exactly solvable models Statistical mechanics Lattice models