The Sznajd model or United we stand, divided we fall (USDF) model is a sociophysics model introduced in 2000^[1] to gain fundamental understanding about opinion dynamics. The Sznajd model implements a phenomenon called social validation and thus extends the Ising spin model. In simple words, the model states:

• Social validation: If two people share the same opinion, their neighbors will start to agree with them.
• Discord destroys: If a block of adjacent persons disagree, their neighbors start to argue with them.

Statistical mechanics
Thermodynamics Kinetic theory
Particle statistics Spin–statistics theorem Indistinguishable particles Maxwell–Boltzmann Bose–Einstein Fermi–Dirac Parastatistics Anyonic statistics Braid statistics
Thermodynamic ensembles NVE Microcanonical NVT Canonical µVT Grand canonical NPH Isoenthalpic–isobaric NPT Isothermal–isobaric
Models Debye Einstein Ising Potts
Potentials Internal energy Enthalpy Helmholtz free energy Gibbs free energy Grand potential / Landau free energy
Scientists Maxwell Boltzmann Bose Gibbs Einstein Ehrenfest von Neumann Tolman Debye Fermi
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Mathematical formulation

For simplicity, one assumes that each individual ${\displaystyle i}$ has an opinion S_i which might be Boolean (${\displaystyle S_{i}=-1}$ for no, ${\displaystyle S_{i}=1}$ for yes) in its simplest formulation, which means that each individual either agrees or disagrees to a given question.

In the original 1D-formulation, each individual has exactly two neighbors just like beads on a bracelet. At each time step a pair of individual ${\displaystyle S_{i))$ and ${\displaystyle S_{i+1))$ is chosen at random to change their nearest neighbors' opinion (or: Ising spins) ${\displaystyle S_{i-1))$ and ${\displaystyle S_{i+2))$ according to two dynamical rules:

1. If ${\displaystyle S_{i}=S_{i+1))$ then ${\displaystyle S_{i-1}=S_{i))$ and ${\displaystyle S_{i+2}=S_{i))$. This models social validation, if two people share the same opinion, their neighbors will change their opinion.
2. If ${\displaystyle S_{i}=-S_{i+1))$ then ${\displaystyle S_{i-1}=S_{i+1))$ and ${\displaystyle S_{i+2}=S_{i))$. Intuitively: If the given pair of people disagrees, both adopt the opinion of their other neighbor.

Findings for the original formulations

In a closed (1 dimensional) community, two steady states are always reached, namely complete consensus (which is called ferromagnetic state in physics) or stalemate (the antiferromagnetic state). Furthermore, Monte Carlo simulations showed that these simple rules lead to complicated dynamics, in particular to a power law in the decision time distribution with an exponent of -1.5.^[2]

Modifications

The final (antiferromagnetic) state of alternating all-on and all-off is unrealistic to represent the behavior of a community. It would mean that the complete population uniformly changes their opinion from one time step to the next. For this reason an alternative dynamical rule was proposed. One possibility is that two spins ${\displaystyle S_{i))$ and ${\displaystyle S_{i+1))$ change their nearest neighbors according to the two following rules:^[3]

1. Social validation remains unchanged: If ${\displaystyle S_{i}=S_{i+1))$ then ${\displaystyle S_{i-1}=S_{i))$ and ${\displaystyle S_{i+2}=S_{i))$.
2. If ${\displaystyle S_{i}=-S_{i+1))$ then ${\displaystyle S_{i-1}=S_{i))$ and ${\displaystyle S_{i+2}=S_{i+1))$

Relevance

In recent years, statistical physics has been accepted as modeling framework for phenomena outside the traditional physics. Fields as econophysics or sociophysics formed, and many quantitative analysts in finance are physicists. The Ising model in statistical physics has been a very important step in the history of studying collective (critical) phenomena. The Sznajd model is a simple but yet important variation of prototypical Ising system.^[4]

In 2007, Katarzyna Sznajd-Weron has been recognized by the Young Scientist Award for Socio- and Econophysics of the Deutsche Physikalische Gesellschaft (German Physical Society) for an outstanding original contribution using physical methods to develop a better understanding of socio-economic problems.^[5]

Applications

The Sznajd model belongs to the class of binary-state dynamics on a networks also referred to as Boolean networks. This class of systems includes the Ising model, the voter model and the q-voter model, the Bass diffusion model, threshold models and others.^[6] The Sznajd model can be applied to various fields:

• The finance interpretation considers the spin-state ${\displaystyle S_{i}=1}$ as a bullish trader placing orders, whereas a ${\displaystyle S_{i}=0}$ would correspond to a trader who is bearish and places sell orders.