Thermodynamic limit | EPFL Graph Search

In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number N of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles. The thermodynamic limit is defined as the limit of a system with a large volume, with the particle density held fixed. : N \to \infty,, V \to \infty,, \frac N V =\text{constant} In this limit, macroscopic thermodynamics is valid. There, thermal fluctuations in global quantities are negligible, and all thermodynamic quantities, such as pressure and energy, are simply functions of the thermodynamic variables, such as temperature and density. For example, for a large volume of gas, the fluctuations of the total internal energy are negligible and can be ignored, and the average internal energy can be predicted from knowledge of the pressure and temperature of the gas. Note that not all types of thermal fluctuations disappear

Level crossings induced by a longitudinal coupling in the transverse field Ising chain

Frédéric Mila, Grégoire Vionnet

We study the effect of antiferromagnetic longitudinal coupling on the one-dimensional transverse field Ising model with nearest-neighbor couplings. In the topological phase where, in the thermodynamic limit, the ground state is twofold degenerate, we show that, for a finite system of N sites, the longitudinal coupling induces N level crossings between the two lowest-lying states as a function of the field. We also provide strong arguments suggesting that these N level crossings all appear simultaneously as soon as the longitudinal coupling is switched on. This conclusion is based on perturbation theory and a mapping of the problem onto the open Kitaev chain, for which we write down the complete solution in terms of Majorana fermions.

2017

Inferring the flow properties of epithelial tissues from their geometry

Valentin Camille Druelle, Marko Popovic, Matthieu Wyart

Amorphous materials exhibit complex material properties with strongly nonlinear behaviors. Below a yield stress they behave as plastic solids, while they start to yield above a critical stress sigma(c). A key quantity controlling plasticity which is, however, hard to measure is the density P(x) of weak spots, where x is the additional stress required for local plastic failure. In the thermodynamic limit P(x) similar to x(theta) is singular at x = 0 in the solid phase below the yield stress sigma(c). This singularity is related to the presence of system spanning avalanches of plastic events. Here we address the question if the density of weak spots and the flow properties of a material can be determined from the geometry of an amorphous structure alone. We show that a vertex model for cell packings in tissues exhibits the phenomenology of plastic amorphous systems. As the yield stress is approached from above, the strain rate vanishes and the avalanches size S and their duration tau diverge. We then show that in general, in materials where the energy functional depends on topology, the value x is proportional to the length L of a bond that vanishes in a plastic event. For this class of models P(x) is therefore readily measurable from geometry alone. Applying this approach to a quantification of the cell packing geometry in the developing wing epithelium of the fruit fly, we find that in this tissue P(L) exhibits a power law with exponents similar to those found numerically for a vertex model in its solid phase. This suggests that this tissue exhibits plasticity and non-linear material properties that emerge from collective cell behaviors and that these material properties govern developmental processes. Our approach based on the relation between topology and energetics suggests a new route to outstanding questions associated with the yielding transition.

IOP PUBLISHING LTD2021

Inferring the flow properties of epithelial tissues from their geometry

Valentin Camille Druelle, Marko Popovic, Matthieu Wyart

IOP PUBLISHING LTD2021