##
**Alessio Squarcini (Max-Planck-Institute for Intelligent Systems, Stuttgart)**

### Long-range medium-mediated interactions: two exactly solvable models

(Egalement par zoom

https://us06web.zoom.us/j/88613589674?pwd=djRyNVN0dUVJMXhMOG1rem9OVG5WUT09

Meeting ID: 886 1358 9674

Passcode: 253305)

In this talk I will discuss two situations in which long-range effects such as ordering and forces emerge in certain systems of classical statistical mechanics and how such models are handled through exact techniques.

The first part of my talk revolves around the following question: can order extend over distances larger than the bulk correlation length? I will show how a network of Ising boxes connected by channels is able to exhibit an extraordinarily long-range ferromagnetic order over distances which grow exponentially with the cross sections of the channels. The emergence of such a new length scale follows from an exact calculation based on the diagonalization of the transfer matrix for the square lattice Ising model. The analytical study is flanked by extensive Monte Carlo simulations [1].

The second phenomenon I will discuss is the critical (or thermodynamical) Casimir effect [2]. Chemically inhomogeneous colloidal particles immersed in a critical binary mixture are subjected to a fluctuation-induced-force known as the critical Casimir effect. By modeling a binary mixture at its demixing critical point by means of the critical Ising model in two dimensions, and exploiting its scaling limit description in terms of a Conformal Field Theory, I determine the exact density profiles and correlation functions around various particles whose boundaries are formed by patches with different chemical structure and preference of the binary mixture components. The formalism encompasses several interesting configurations, including Janus particles, colloidal quadrupoles and needles with inhomogeneous patches of symmetry breaking boundary conditions. Within the framework of the ‘’Small Particle Operator Expansion’’ I determine the exact asymptotic behavior of the interaction free energy between these colloids, and colloids confined by a wedge-shaped wall. The theoretical predictions are confirmed by numerical results available in the literature.

References

[1] D. B. Abraham, A. Maciolek, AS, and O. Vasilyev, Action at a distance in classical uniaxial ferromagnetic arrays, Phys. Rev. E 96, 042154 (2017).

[2] AS, A. Maciolek, E. Eisenriegler, and S. Dietrich, Critical Casimir interaction between colloidal Janus-type particles in two spatial dimensions, J. Stat. Mech. 043208 (2020).