Expanded Vandermonde Powers and Sum Rules for the Two-Dimensional One-Component Plasma

Abstract

The two-dimensional one-component plasma (2dOCP) is a system of N mobile particles of the same charge q on a surface with a neutralizing background. The Boltzmann factor of the 2dOCP at temperature T can be expressed as a Vandermonde determinant to the power Γ=q ²/(k _B T). Recent advances in the theory of symmetric and anti-symmetric Jack polynomials provide an efficient way to expand this power of the Vandermonde in their monomial basis, allowing the computation of several thermodynamic and structural properties of the 2dOCP for N values up to 14 and Γ equal to 4, 6 and 8. In this work we explore two applications of this formalism, to the study of the pair correlation function of the 2dOCP on the sphere, and the distribution of radial statistics of the 2dOCP in the plane. Also provided is a finite N approximation to the pair correlation on the sphere, and a sum rule for the constant term in the large N expansion of the moments of the density in the plane.

Appendix

In this appendix we present some properties of the functions

$$ {\mathcal{I}}(k_1,k_2)= \int\!\!\!\int_{0\leq t_2 < t_1} e^{-t_1-t_2} t_1^{k_1} t_2^{k_2} \,dt_1\,dt_2 , $$

(A.1)

and

$$ {\mathcal{J}}(k_1,k_2)=\int_{0}^{\infty} \int_{0}^{\infty} e^{-t_1-t_2} t_1^{k_1} t_2^{k_2} \log\bigl( \max(t_1 ,t_2)\bigr) \,dt_1\,dt_2 , $$

(A.2)

that appear in the expansion of the density around Γ=2. First, it should be noticed that

$$ {\mathcal{I}}(k_1,k_2)+ { \mathcal{I}}(k_2,k_1)= k_1! k_2! . $$

(A.3)

Doing an integration by parts, one obtains the recurrence relation

$$ {\mathcal{I}}(k_1+1,k_2)-(k_1+1){\mathcal {I}}(k_1,k_2)=2^{-k_1-k_2-2}(k_1+k_2+1)! , $$

(A.4)

and reiterating

(A.5)

Similarly, for ${\mathcal{J}}$ we have

$$ {\mathcal{J}}(k_1+n,k_2)- \frac{(k_1+n)!}{k_1!}\,{\mathcal{J}}(k_1,k_2)= \sum _{\ell=0}^{n-1} {\mathcal{I}}(k_1+ \ell,k_2)\frac {(k_1+n)!}{(k_1+\ell+1)!} . $$

(A.6)

From (A.5), one can obtain an alternative expression for ${\mathcal{I}}$ as a sum

$$ {\mathcal{I}}(k_1,k_2)=\sum _{\ell=0}^{k_1} 2^{-k_2-\ell-1} \frac {(k_2+\ell )!k_1!}{\ell!}. $$

(A.7)

The asymptotic expansion of ${\mathcal{I}}(k_{1},k_{2})$ for large arguments, k ₁ and k ₂ of order N→∞, can be obtained by the steepest descent method. The maximum of the integrand in (A.1) is for t ₁=k ₁ and t ₂=k ₂. Therefore, the behavior of ${\mathcal {I}}(k_{1},k_{2})$ will depend on whether this maximum is in the domain of integration 0≤t ₂<t ₁ or not, i.e., if k ₂<k ₁ or not. If 1≪k ₂<k ₁, and $|k_{1}-k_{2}|/\sqrt{N}\gg1$, then the maximum of the integrand in (A.1) is deep inside the domain of integration, and a simple application of the steepest descent shows that ${\mathcal {I}}(k_{1},k_{2})\sim k_{1}! k_{2}!$. However for the calculations of Sect. 5.3, the behavior of ${\mathcal{I}}(k_{1},k_{2})$ when k ₂>k ₁ is needed. In this situation, the maximum of the integrand is outside the domain of integration. Nevertheless, ${\mathcal{I}}(k_{1},k_{2})$ will give a significant contribution when the maximum of the integrand is “close” to the border, more precisely when |k ₂−k ₁| is of order $O(\sqrt{N}\,)$. The dominant contribution to the integral (A.1) will be given by the region consisting of a strip attached and parallel to the line t ₁=t ₂ of width of order $\sqrt{N}$. To be more specific, let us do the change of integration variables v ₊=t ₁+t ₂ and v ₋=t ₁−t ₂ in (A.1) and write

$$ {\mathcal{I}}(k_1,k_2)=\int\!\!\!\int_{\mathcal{D}} dv_{+}dv_{-} e^{-g(v_+,v_{-})} \frac {dv_{+}dv_{-}}{2^{k_1+k_2+1}} , $$

(A.8)

with

$$ g(v_{+},v_{-})=v_{+}-k_1 \log(v_{+}+v_{-})-k_2\log(v_{+}-v_{-}) , $$

(A.9)

where $\mathcal{D}=\{(t_{1},t_{2}), 0\leq t_{2} < t_{1}\}$. g has its maximum for $v_{+}=v_{+}^{*}=k_{1}+k_{2}$ and $v_{-}=v_{-}^{*}=k_{1}-k_{2}$ (i.e. t ₁=k ₁ and t ₂=k ₂). Expanding g to the second order around its maximum we obtain

(A.10)

Although the domain of integration $\mathcal{D}$ cannot be simply expressed in terms of the variables v ₊ and v ₋, due to fast Gaussian decay of the integrand in (A.10), the dominant contribution is indeed obtained by the strip along the line t ₁=t ₂ mentioned above. With this in mind, it is clear that one can extend the domain of integration for v ₊ to ]−∞,+∞[ and for v ₋ to [0,+∞[, up to exponentially small corrections. Then, performing first the integral over v ₊, we obtain

$$ {\mathcal{I}}(k_1,k_2)\sim e^{-k_1-k_2} k_1^{k_1} k_2^{k_2} \sqrt { \frac{2\pi k_1k_2}{k_1+k_2}} \int_{0}^{+\infty} e^{-\frac{(v_{-}-v_{-}^{*})^2}{2(k_1+k_2)}} \,dv_{-}. $$

(A.11)

Performing now the integral over v ₋ gives

$$ {\mathcal{I}}(k_1,k_2) \sim e^{-k_1-k_2} k_1^{k_1} k_2^{k_2} \pi\sqrt {k_1k_2}\, {\mathrm{erfc}} \biggl( \frac{k_2-k_1}{\sqrt{2(k_1+k_2)}} \biggr) , $$

(A.12)

where ${\mathrm{erfc}}(x)=1-(2/\sqrt{\pi}\,)\int_{0}^{x} e^{-t^{2}}\, dt$, is the complementary error function. Recalling Stirling’s formula for the factorial, (A.12) can be written as

$$ \frac{{\mathcal{I}}(k_1,k_2)}{ k_1! k_2! } \sim \frac{1}{2}\,{\mathrm{erfc}} \biggl(\frac{k_2-k_1}{\sqrt {2(k_1+k_2)}} \biggr) . $$

(A.13)