Relations Among Two Methods for Computing the Partition Function of ...
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Relations Among Two Methods for Computing the Partition Function of ...
Mar 20, 2015 - arXiv:1503.06248v1 [cond-mat.stat-mech] 20 Mar 2015. Relations Among Two Methods for Computing the Partition Function.
arXiv:1503.06248v1 [cond-mat.stat-mech] 20 Mar 2015
Relations Among Two Methods for Computing the Partition Function of the Two-Dimensional One-Component Plasma Johnny Alejandro Mora Grimaldo Gabriel Téllez Departamento de Física, Universidad de los Andes, Bogotá, Colombia Abstract The two-dimensional one-component plasma —2dOCP— is a system composed by n mobile particles with charge q over a neutralizing background in a two-dimensional surface. The Boltzmann factor of this system, at temperature T , takes the form of a Vandermonde determinant to the power Γ = q 2 /(2πεk B T ), where Γ is the coupling constant of this Coulomb system. The partition function of the model has been computed exactly for the even values of the coupling constant Γ, and a finite number of particles n, by two means: 1) by recognizing that the Boltzmann factor is the square of a Jack polynomial and expanding it in an appropriate monomial base, and 2) by mapping the system onto a 1-dimensional chain of interacting fermions. In this work the connection among the two methods is derived, and some properties of the expansion coefficients for the power of the Vandermonde determinant are explored. Keywords Coulomb Gas · One-component Plasma · Jack Polynomials · Fermion Chain · Grassmann Algebra
1 Introduction The Two-Dimensional One-Component Plasma —2dOCP— is a model in statistical physics in which n mobile particles with charge q are distributed in a two-dimensional space, over a background of neutralizing charge. The statistical treatment of this model begins with the solution of the two dimensional Poisson’s equation, from which the Hamiltonian of the system is constructed. In a two-dimensional universe, the electrostatic interaction among two point-like charges q, located at positions ri and r j is given by the potential: ¯ ¯ ¯ri − r j ¯ q2 ln . (1) U (ri , r j ) = − 2πε L
Working on cgs units, we have 2πε = 1. L is an irrelevant arbitrary length that will be set to 1 in the following. If the charges are distributed over a uniformly charged disk of radius R, with background charge density −nq/(πR 2 ), the potential energy of the system —built up from the background-background interaction, the background-particle interaction and the particle-particle interaction— is [1, 2] U=
n X ¯ ¯ nq 2 X r i2 − q 2 ln ¯ri − r j ¯ +U0D , 2 2R i =1 i
