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Abstract |
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). |
