Error in Schakel's 'Boulevard of broken symmetries'?

[tesselation]

I'm working through Schakel's 'Boulevard of broken symmetries' and I don't get an equation connect to the Coulomb gas problem.

The equation in question is (5.77) on page 161.
Luckily, you may look it up in http://books.google.com/books?id=OA...EwAQ#v=onepage&q=sine gordon coulomb&f=false".

I guess I'm missing something obvious but it seems like (5.77) is wrong; e.g. if one calculates both sides for N=2.

Any ideas?

 

[NateD]

The equation in question is (5.77) on page 161 of Schakel's 'Boulevard of Broken Symmetries.' The equation is:<math>\frac{\partial^2 \phi_n}{\partial x^2} - \sum_{m \neq n} \frac{e^{2 \phi_m}}{(x_n - x_m)^2} = \frac{\beta}{4\pi} \,\sin(2\pi \phi_n).</math>This equation describes the behavior of a Coulomb gas, which is a system of particles interacting via a long-range Coulombic potential. In this equation, the left-hand side is the Laplacian of the scalar field <math>\phi_n</math>, which is the field representing the electric potential associated with the particles in the Coulomb gas. The sum on the right-hand side is the total Coulombic potential generated by all the particles in the system. The final term on the right-hand side is the sine-Gordon potential, which is the nonlinear, short-range potential that represents the effects of finite particle size and the particle-particle interactions.The equation (5.77) is correct; however, one must be careful when calculating both sides for specific values of N. In particular, one must take into account the boundary conditions, which determine how the electric potential varies as one moves around the system. Without these boundary conditions, the equation is not valid.