Ising's Roots and the Transfer-Matrix Eigenvalues.

Today, the Ising model is an archetype describing collective ordering processes. As such, it is widely known in physics and far beyond. Less known is the fact that the thesis defended by Ernst Ising 100 years ago (in 1924) contained not only the solution of what we call now the 'classical 1D Ising model' but also other problems. Some of these problems, as well as the method of their solution, are the subject of this note. In particular, we discuss the combinatorial method Ernst Ising used to calculate the partition function for a chain of elementary magnets. In the thermodynamic limit, this method leads to the result that the partition function is given by the roots of a certain polynomial. We explicitly show that 'Ising's roots' that arise within the combinatorial treatment are also recovered by the eigenvalues of the transfer matrix, a concept that was introduced much later. Moreover, we discuss the generalization of the two-state model to a three-state one presented in Ising's thesis, which is not included in his famous paper of 1925 (E. Ising, Z. Physik 31 (1925) 253). The latter model can be considered as a forerunner of the now-abundant models with many-component order parameters.

The diaspora model for human migration.

Migration's impact spans various social dimensions, including demography, sustainability, politics, economy, and gender disparities. Yet, the decision-making process behind migrants choosing their destination remains elusive. Existing models primarily rely on population size and travel distance to explain the spatial patterns of migration flows, overlooking significant population heterogeneities. Paradoxically, migrants often travel long distances and to smaller destinations if their diaspora is present in those locations. To address this gap, we propose the diaspora model of migration, incorporating intensity (the number of people moving to a country), and assortativity (the destination within the country). Our model considers only the existing diaspora sizes in the destination country, influencing the probability of migrants selecting a specific residence. Despite its simplicity, our model accurately reproduces the observed stable flow and distribution of migration in Austria (postal code level) and US metropolitan areas, yielding precise estimates of migrant inflow at various geographic scales. Given the increase in international migrations, this study enlightens our understanding of migration flow heterogeneities, helping design more inclusive, integrated cities.

Individual bias and fluctuations in collective decision making: from algorithms to Hamiltonians.

In this paper, we reconsider the spin model suggested recently to understand some features of collective decision making among higher organisms (Hartnettet al2016Phys. Rev. Lett.116038701). Within the model, the state of an agentiis described by the pair of variables corresponding to its opinionSi=±1and a biasωitoward any of the opposing values ofSi. Collective decision making is interpreted as an approach to the equilibrium state within the nonlinear voter model subject to a social pressure and a probabilistic algorithm. Here, we push such a physical analogy further and give the statistical physics interpretation of the model, describing it in terms of the Hamiltonian of interaction and looking for the equilibrium state via explicit calculation of its partition function. We show that, depending on the assumptions about the nature of social interactions, two different Hamiltonians can be formulated, which can be solved using different methods. In such an interpretation the temperature serves as a measure of fluctuations, not considered before in the original model. We find exact solutions for the thermodynamics of the model on the complete graph. The general analytical predictions are confirmed using individual-based simulations. The simulations also allow us to study the impact of system size and initial conditions on the collective decision making in finite-sized systems, in particular, with respect to convergence to metastable states.

Generalized Ising Model on a Scale-Free Network: An Interplay of Power Laws.

We consider a recently introduced generalization of the Ising model in which individual spin strength can vary. The model is intended for analysis of ordering in systems comprising agents which, although matching in their binarity (i.e., maintaining the iconic Ising features of '+' or '-', 'up' or 'down', 'yes' or 'no'), differ in their strength. To investigate the interplay between variable properties of nodes and interactions between them, we study the model on a complex network where both the spin strength and degree distributions are governed by power laws. We show that in the annealed network approximation, thermodynamic functions of the model are self-averaging and we obtain an exact solution for the partition function. This allows us derive the leading temperature and field dependencies of thermodynamic functions, their critical behavior, and logarithmic corrections at the interface of different phases. We find the delicate interplay of the two power laws leads to new universality classes.

Self-averaging in the random two-dimensional Ising ferromagnet.

We study sample-to-sample fluctuations in a critical two-dimensional Ising model with quenched random ferromagnetic couplings. Using replica calculations in the renormalization group framework we derive explicit expressions for the probability distribution function of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energies is Gaussian, and the typical sample-to-sample fluctuations as well as the average value scale with the system size L like ∼Llnln(L). In contrast, the specific heat is shown to be self-averaging with a distribution function that tends to a δ peak in the thermodynamic limit L→∞. While previously a lack of self-averaging was found for the free energy, we here obtain results for quantities that are directly measurable in simulations, and implications for measurements in the actual lattice system are discussed.

Universal free-energy distribution in the critical point of a random Ising ferromagnet.

We discuss the non-self-averaging phenomena in the critical point of weakly disordered Ising ferromagnet. In terms of the renormalized replica Ginzburg-Landau Hamiltonian in dimensions D<4, we derive an explicit expression for the probability distribution function (PDF) of the critical free-energy fluctuations. In particular, using known fixed-point values for the renormalized coupling parameters, we obtain the universal curve for such PDF in the dimension D=3. It is demonstrated that this function is strongly asymmetric: its left tail is much slower than the right one.