# Qualitative argument to determine energy of defects

In a book of "LES HOUCHES - Critical Phenomena, Random systems, Gauge theories" the author Frolich says that:

2D

In two dimensions, the mean energy of an isolated point defect in a square area of diameter $L$ is proportional to $\log(L)$. The total number of possible positions is proportional to $L^2$, i.e. the entropy grows logarithmically in $L$. Hence the free energy behaves like:

$F=E-TS \sim \textrm{const.} \log(L)-kT \textrm{const.}^\prime \log(L)$

Thus, for $T$ large enough, a dilute system of bound point defects becomes unstable in the thermodynamic limit, i.e. defects unbind and form a plasma

3D

In three dimensions, dislocations are line defects with a self-energy roughly proportional to their lenght, $L$. In a cubic area of diameter $\sim \textrm{const.} L$ the number of possible configurations of a single dislocation loop of lenght $L$ is clearly proportional to $\exp(\textrm{const.} L)$, so the entropy grows linearly in $L$. The free energy thus behaves like

$F \sim \textrm{const.} L - kT \textrm{const.}^\prime L$

### I have not understood the following points:

1. What is the qualitative argument that says that the mean energy of a point defect in a square is $∼\log(L)$?

2. The same question of 1. for the 3D case.

3. How he derives the proportionality of $\exp(\textrm{const.} L)$ ?

4. The same question of 3. for the 2D case

5. How can he says that for $T$ large the system of point defects is unstable in the thermodynamic limit?

6. In 2D he speaks of "mean energy" in 3D of "self-energy". Why does he make this difference?

Thanks!