Exact Energy Computation of the One Component Plasma on a Sphere for Even Values of the Coupling Parameter

Abstract

The two dimensional one component plasma 2dOCP is a classical system consisting of N identical particles with the same charge q confined in a two dimensional surface with a neutralizing background. The Boltzmann factor at temperature T may be expressed as a Vandermonde determinant to the power $\Gamma =q^2/(k_B T)$. Several statistical properties of the 2dOCP have been studied by expanding the Boltzmann factor in the monomial basis for even values of $\Gamma$. In this work, we use this formalism to compute the energy of the 2dOCP on a sphere. Using the same approach the entropy is computed. The entropy as well as the free energy in the thermodynamic limit have a universal finite-size correction term $\frac{\chi }{12}\log N$, where $\chi =2$ is the Euler characteristic of the sphere. A non-recursive formula for coefficients of monomial functions expansion is used for exploring the energy as well as structural properties for sufficiently large values of $\Gamma$ to appreciate the crystallization features for $N=2,3,\ldots ,9$ particles. Finally, we make a brief comparison between the exact and numerical energies obtained with the Metropolis method for even values of $\Gamma$.

Appendices

Appendix 1: Partitions Computation

All partitions $\mu$ may be found from the first one usually called the root partition $\mu _{i}^{1} = (N-i)\Gamma /2$ where we have used the following notation for the elements $\mu _{i}^{\alpha }$ of the partition $\mu$ with $(i=1,\ldots ,N)$, $(\alpha =1,\ldots ,\mathcal {N}_p)$ and $\mathcal {N}_p(N,\Gamma )$ the total number of partitions. For $N=3$ and $\Gamma =4$ the partitions are shown in Table 1.

Table 1 Partitions for $N=3$ and $\Gamma =4$

The partition elements are obtained by adding or subtracting integers to the previous partition elements holding sum $\sum _{i=1}^N{\mu _i^\alpha }$ as a constant. For instance, the second partition (4, 1, 1) may be obtained from (4, 2, 0) by a subtraction of 1 from $\mu _2^1$ and adding 1 to $\mu _3^1$. Similarly, the third partition (3, 3, 0) is obtained from the first partition by subtracting 1 from $\mu _1^1$ and adding 1 to $\mu _2^1$. These type of operations are usually referred as squeezing. Following the same rules, the 4th partition may be obtained from the third and the 5th from the 4th. Therefore, the partition elements for $N=3$ are of the form $\mu _1^\alpha = (N-1)\Gamma /2-j_1(\alpha )$, $\mu _2^\alpha = (N-1)\Gamma /2+j_1(\alpha )-j_2(\alpha )$ and $\mu _3^\alpha = (N-1)\Gamma /2+j_2(\alpha )$ where $j_i(\alpha )$ are positive integers which represent the integers transferred from $\mu _i^\alpha$ to $\mu _{i+1}^\alpha$. In general, the partitions may be obtained from the following function

$$\begin{aligned} f_i(N,\Gamma ,\alpha ) = \left\{ \begin{array}{ll} (N-i)\Gamma /2 - j_i &{} \quad \text{ if } \,\, i = 1\\ (N-i)\Gamma /2 +j_{i-1} - j_i &{} \quad \text{ if } \,\, i \in [2, N) \\ (N-i)\Gamma /2 + j_{i-1} &{} \quad \text{ if } \,\, i = N \end{array} \right. \end{aligned}$$

(35)

where $\mu _i^\alpha = f_i(N,\Gamma ,\alpha )$ only if one of the following conditions is satisfied

$$\begin{aligned} \mu _1^\alpha \ge \mu _2^\alpha \ge \cdots \ge \mu _1^\alpha \,\, \text{ if } \,\, \Gamma /2 \text{ is } \text{ even } \end{aligned}$$

(36)

or

$$\begin{aligned} \mu _1^\alpha> \mu _2^\alpha> \cdots > \mu _1^\alpha \,\, \text{ if } \,\, \Gamma /2 \text{ is } \text{ odd } \end{aligned}$$

(37)

The following lines of code written in Wolfram Mathematica 9.0 shows a way to compute the partitions for $N=3$ particles.

Appendix 2: Coefficients Computation

1.1 The Multinomial Theorem Approach

The coefficients may be computed from the following formula

$$\begin{aligned} C_{\mu }^{(N)}(\Gamma /2) = \frac{1}{(2\pi )^N} \int _0^{2\pi } d\phi _1 e^{-{{\varvec{i}}} \mu _1\theta _1 \cdots \int _0^{2\pi }d\phi _N} e^{-{{\varvec{i}}} \mu _N\theta _N} \prod _{1 \le j<k \le N} \left( e^{{{{\varvec{i}}}}\theta _k}-e^{{{{\varvec{i}}}}\theta _j}\right) ^{\Gamma /2} \end{aligned}$$

(38)

with $\Gamma /2$ and even number. The product $\prod _{1 \le j<k \le N} \left( e^{{{{\varvec{i}}}}\theta _k}-e^{{{{\varvec{i}}}}\theta _j}\right)$ into the integral is a Vandermonde determinant

$$\begin{aligned} \Pi :=\prod _{1 \le j<k \le N} \left( e^{{{{\varvec{i}}}}\theta _k}-e^{{{{\varvec{i}}}}\theta _j}\right) = \sum _{\sigma \in S_N} \text{ sgn }(\sigma )\prod _{j=1}^N \left( e^{{{{\varvec{i}}}}\theta _j}\right) ^{\sigma (j)-1} = \sum _{p=1}^{N!} \chi _{\sigma _p} \end{aligned}$$

this is a sum of N! terms of the form $\chi _p := \text{ sgn }(\sigma ^p)\exp [{{{\varvec{i}}}}\sum _{i=1}^N (\sigma ^p_i-1)\theta _i]$ with $S_N = \{\sigma ^1,\ldots ,\sigma ^{N!}\}$ and $\sigma ^p= \{\sigma ^p_{1},\ldots ,\sigma ^p_{N}\}$ is the p-th permutation of N elements. It is possible to use the multinomial theorem

$$\begin{aligned} \left( \sum _{p=1}^{M} \chi _p \right) ^n = \sum _{i_1=0}^n \sum _{i_2=0}^{i_1} \cdots \sum _{i_{M-1} =0}^{i_{M-2}} \left( {\begin{array}{c}n\\ i_1\end{array}}\right) \left( {\begin{array}{c}i_1\\ i_2\end{array}}\right) \cdots \left( {\begin{array}{c}i_{M-2}\\ i_{M-1}\end{array}}\right) \chi _1^{n-i_1} \chi _2^{i_1-i_2} \cdots \chi _M^{i_{M-1}} \end{aligned}$$

in order to evaluate the integral Eq. (38) where n and M are positive integers. If $M=N!$ and $n=\Gamma /2$ then

$$\begin{aligned} \Pi ^{\Gamma /2} = \sum _{i_1=0}^{\Gamma /2} \sum _{i_2=0}^{i_1} \cdots \sum _{i_{N!-1} =0}^{i_{N!-2}} \left( {\begin{array}{c}n\\ i_1\end{array}}\right) \left( {\begin{array}{c}i_1\\ i_2\end{array}}\right) \cdots \left( {\begin{array}{c}i_{N!-2}\\ i_{N!-1}\end{array}}\right) \prod _{p=2}^{N!}\text{ sgn }(\sigma ^p)^{i_{p-1}-i_{p}}\exp \left( {{{\varvec{i}}}} \sum _{j=1}^N K_j(\vec {i};{\sigma _j})\theta _j\right) \end{aligned}$$

(39)

where we have defined

$$\begin{aligned} K_m(\vec {i};{\sigma _m}) = K_m(i_1,\ldots ,i_{N!-1}; \sigma _m ) := \sum _{j=1}^{N!}\left( i_{j-1}-i_j\right) \left( \sigma ^j_m-1\right) \end{aligned}$$

(40)

with $i_0:=\Gamma /2$, $i_{N!} := 0$, $\sigma ^p_m$ is the mth term of the permutation $\sigma ^p$ and $\sigma _m=\left\{ \sigma ^1_m,\ldots ,\sigma ^{N!}_m\right\}$. For instance, the notation used for permutations of $N=3$ particles is

$$\begin{aligned} \left( \sigma ^p_m\right) = \left( \begin{matrix} 1 &{} \quad 2 &{}\quad 3\\ 1 &{}\quad 3 &{}\quad 2\\ 2 &{}\quad 1 &{}\quad 3\\ 2 &{}\quad 3 &{}\quad 1\\ 3 &{}\quad 1 &{}\quad 2\\ 3 &{}\quad 2 &{}\quad 1 \end{matrix}\right) = \left( \begin{matrix} \sigma ^1_1 &{}\quad \sigma ^1_2 &{}\quad \sigma ^1_3 \\ \sigma ^2_1 &{}\quad \sigma ^2_2 &{}\quad \sigma ^2_3 \\ \sigma ^3_1 &{}\quad \sigma ^3_2 &{}\quad \sigma ^3_3 \\ \sigma ^4_1 &{}\quad \sigma ^4_2 &{}\quad \sigma ^4_3 \\ \sigma ^5_1 &{}\quad \sigma ^5_2 &{}\quad \sigma ^5_3 \\ \sigma ^6_1 &{}\quad \sigma ^6_2 &{}\quad \sigma ^6_3 \\ \end{matrix}\right) \end{aligned}$$

(41)

It is important to note that $K_m(\vec {i};{\sigma _m})$ is a non-negative integer because $i_j\le i_{j-1}$ for $j=1,\ldots ,N!$ and $m=1,\ldots ,N$ as happens with the partition elements $\mu = \left\{ \mu _m\right\} = (\mu _1,\ldots ,\mu _N)$. Replacing Eq. (39) in Eq. (38) we obtain

$$\begin{aligned} C_\mu ^{(N)}(\Gamma /2)&= \frac{1}{(2\pi )^N} \sum _{i_1=0}^{\Gamma /2} \sum _{i_2=0}^{i_1} \cdots \sum _{i_{N!-1} =0}^{i_{N!-2}} \left( {\begin{array}{c}n\\ i_1\end{array}}\right) \left( {\begin{array}{c}i_1\\ i_2\end{array}}\right) \cdots \left( {\begin{array}{c}i_{N!-2}\\ i_{N!-1}\end{array}}\right) \\&\quad \times \,\prod _{p=2}^{N!}\text{ sgn }(\sigma ^p)^{i_{p-1}-i_{p}} \prod _{j=1}^{N}\int _0^{2\pi }d\phi _j \exp \left\{ -{{{\varvec{i}}}} (K_j-\mu _j)\theta _j\right\} . \end{aligned}$$

The integration problem is solved by using $\int _0^{2\pi }d\phi _j \exp \left\{ -{{{\varvec{i}}}} (K_j-\mu _j)\theta _j\right\} = 2\pi \delta _{K_j,\mu _j}$. If the first permutation is the identity $\sigma ^1_m = m$, then $\text{ sgn }(\sigma ^1) = 1$ and it is possible to write the coefficients more compactly

$$\begin{aligned} C_\mu ^{(N)}(\Gamma /2) = \prod _{j=1}^{N!-1} \sum _{i_j=0}^{i_{j-1}}\left\{ \prod _{p=1}^{N!}\text{ sgn }(\sigma ^p)^{i_{p-1}-i_{p}} \left( {\begin{array}{c}i_{p-1}\\ i_p\end{array}}\right) \prod _{l=1}^{N} \delta _{K_l,\mu _l}\right\} . \end{aligned}$$

(42)

Now, the sum $\sum _{\vec {i}} := \prod _{j=1}^{N!-1} \sum _{i_j=0}^{i_{j-1}}$ of Eq. (42) generates a set vectors of the form $\vec {i}=\left( i_1,\ldots ,i_{N!-1}\right)$. At the same time, each of these indices vector $\vec {i}$ will generate a set of non-negative integer elements $K = \left\{ K_m(i_1,\ldots ,i_{N!-1};\sigma _m) : m = 1\ldots ,N\right\}$. Finally, the Kronecker delta product $\prod _{l=1}^{N} \delta _{K_l,\mu _l}$ collects only the K sets which are partitions $K=\mu$. The number of $\vec {i}$ vectors generated with $\sum _{\vec {i}}$ is

$$\begin{aligned} M_{N,\Gamma } = \sum _{\vec {i}} 1 = \frac{1}{(N-1)!}\prod _{j=1}^{N!-1}(1+\Gamma /2) \end{aligned}$$

and these vectors belong to the following set

$$\begin{aligned} J(N,\Gamma )&:=\left\{ \vec {i}=(i_1,\ldots ,i_{N!-1}) : i_j \in \{0\}\cup \mathbb {Z}^+ \text{ with } j=1,\ldots , N!-1 \wedge \Gamma /2\right. \\&\qquad \left. \ge \, i_1 \ge i_2 \ge \cdots \ge i_{N!-1} \ge 0 \right\} . \end{aligned}$$

Hence,

$$\begin{aligned} C_\mu ^{(N)}(\Gamma /2) = \sum _{\vec {i} \in J} c(\vec {i})\delta _{\mu ,K(\vec {i},\sigma )} \text{ with } c(\vec {i}) = \prod _{\sigma =1}^{N!}\text{ sgn }(\sigma ^p)^{i_{p-1}-i_{p}} \left( {\begin{array}{c}i_{p-1}\\ i_p\end{array}}\right) . \end{aligned}$$

If $I \subset J$ is the set of vectors $\vec {i}$ which generates a partition $K=\mu$, then each partition has a set $I_\mu \subset I$ of vectors $\vec {i}$ defined by

$$\begin{aligned} I_\mu&:=\left\{ \vec {i} : \sum _{j=1}^{N!-1}\left( i_{j-1}-i_j\right) \left( \sigma _{jm}-1\right) + i_{N!-1}\left( \sigma _{N!m}-1\right) -\mu _m=0 \right. \nonumber \\&\left. \qquad \quad \forall m=1,\ldots ,N \wedge \frac{\Gamma }{2} \ge i_1 \ge i_2 \ge \cdots \ge i_{N!-1} \ge 0 \right\} . \end{aligned}$$

(43)

such that

$$\begin{aligned} I(N,\Gamma )=\bigcup _{\mu =1}^{N_p(N,\Gamma )} I_\mu \end{aligned}$$

because a set of indices $i_1,\ldots ,i_{N!-1}$ in I will generate a single partition and there are not two repeated partitions. As a result, the coefficients for a given number of particles N and value of gamma parameter $\Gamma$ may be computed with

$$\begin{aligned} C_\mu ^{(N)}(\Gamma /2) = \sum _{ (i_1,\ldots ,i_{N!-1}) \in I_\mu } \prod _{p=1}^{N!}\text{ sgn }(\sigma ^p)^{i_{p-1}-i_{p}} \left( {\begin{array}{c}i_p\\ i_{p-1}\end{array}}\right) . \end{aligned}$$

(44)

The computation of coefficients with Eq. (44) requires to find the set $I_{\mu }$ previously defined in Eq. (43). In other words, it is necessary to solve a set of N-equations of the form $K_m(\vec {i},\sigma _m)-\mu _m=0$ with $m=1,\ldots ,N$ under N!-conditions of the form $i_{j-1}-i_j \ge 0$ with $j=1,\dots ,N!$, $i_0=\Gamma /2$ and $i_{N!}=0$ where $\vec {i}=(i_1,\ldots ,i_{N!-1})$ are unknowns. Therefore, the vector solution is not unique $\text{ dim }(I_\mu ) \ge 1$ and it is possible to find more than one vector $\vec {i}$ solution associated to a single partition $\mu$ which makes harder to find the set $I_{\mu }$. The next code in addition with the one written in the previous section for partitions computation is an example of the coefficients computation for $N=3$ using Eq. (44):

Here the set $I_\mu$ is found with the function Solve of Mathematica and the output is summarized in Table 2. Taking into account that we usually have to compute the coefficients of partitions with $\mu _N=0$, then the technique described here may be used for the computation of the coefficients for values of $\Gamma$ far from 2. In fact, we have used Eq. (44) to find $C^{(N)}_\mu (\Gamma /2)$ for $N=2,3,4$ and 5 with values of $\Gamma =4,8,\ldots ,100$. In some sense the program implementation of Eq. (44) may be simple, since the only difficulty is to built $I_\mu$. However, the construction of $I_\mu$ for large values of N is hard even numerically because it will require to handle a set of N! inequalities with N equations. Although, this feature makes impractical the use of this technique even for $N > 5$, Eq. (44) provides a straightforward way to find analytically $C^{(N)}_\mu (\Gamma /2)$ for $N=2$ and $N=3$ for any even value of $\Gamma$ when $\mu _N=0$.

Table 2 Coefficients for $N=3$ and $\Gamma =8$

1.2 Coefficients for $N=2$

The partitions are given by

$$\begin{aligned} \mu _1^{\alpha } = \frac{\Gamma }{4} + \alpha - 1 \quad \text{ and }\quad \mu _2^{\alpha } = \frac{\Gamma }{4} - \alpha + 1 \end{aligned}$$

with $\alpha =1,2,\ldots , \mathcal {N}_p$ and $\mathcal {N}_p=\Gamma /4+1$ the total number of partitions. For $N=2$ the vector $\vec {i}$ has only one component $i_1$ and the $I_\mu$ set given by Eq. (43) takes the form

$$\begin{aligned} I_\mu:= & {} \left\{ i_1 : K_1(i_1,\sigma _1)=\mu _1 \wedge K_2(i_1,\sigma _2)=\mu _2 \text{ with } \frac{\Gamma }{2} \ge i_1 \ge 0 \wedge \mu _1 \ge \mu _2 \ge 0 \right\} \text{ with } \\&(\sigma ^p_m)= \left( \begin{matrix} 1 &{}2\\ 2 &{}1 \end{matrix}\right) . \end{aligned}$$

The components of the K-vector are $K_1(i_1,\sigma _1)=i_1=\mu _1$ and $K_2(i_1,\sigma _2) =\Gamma /2 - i_1=\mu _2$. From both conditions it is obtained the solution $i_1 = \Gamma /4 + (\alpha -1) = \mu _1$. Therefore, $\text{ dim }(I_\mu )=1$ and the sum $\sum _{ (i_1,\ldots ,i_{N!-1}) \in I_\mu }$ of Eq. (44) has only one term

$$\begin{aligned} C_\mu ^{(N)}(\Gamma /2) = \prod _{p=1}^{2}\text{ sgn }(\sigma ^p)^{i_{p-1}-i_{p}} \left( {\begin{array}{c}i_p\\ i_{p-1}\end{array}}\right) \end{aligned}$$

with $i_0=\Gamma /2$, $i_1=\mu _1$ and $i_2 = 0$. The final result is

$$\begin{aligned} C_\mu ^{(2)}(\Gamma /2) = (-1)^{\mu _1} \left( {\begin{array}{c}\Gamma /2\\ \mu _1\end{array}}\right) . \end{aligned}$$

(45)

This is the same formula obtained by combination of the binomial theorem and Eq. (38).

1.3 Coefficients for $N=3$ with $\mu _N=0$

The partitions and coefficients of N are connected with the previous ones of $N-1$ according to the properties $\mu _j^{\alpha (N-1,\Gamma )} + \Gamma /2 = \mu _j^{\alpha (N,\Gamma )}$ and $C_\mu ^{(N)}(\Gamma /2)=C_\mu ^{(N-1)}(\Gamma /2)$ when $\mu _N = 0$. For this reason, the coefficients computation problem for $N=3$ particles with $\mu _3=0$ is similar to the problem for $N=2$ of the previous section. In this case the partitions are given by

$$\begin{aligned} \mu _1^{\alpha } = \Gamma + 1 - \alpha , \quad \mu _2^{\alpha } = \Gamma /2 + \alpha - 1 \text{ and } \mu _3^{\alpha } = 0. \end{aligned}$$

(46)

The number of partitions is found from the condition $\mu _1^{\mathcal {N}_p}=\mu _1^{\mathcal {N}_p}$ when $\Gamma /2$ is an even number and $\mu _1^{\mathcal {N}_p}+1=\mu _1^{\mathcal {N}_p}$ when $\Gamma /2$ is an odd number. The result is $\mathcal {N}_p=\text{ Int }\left( \Gamma /4+1\right)$ and the set $I_\mu$ is

$$\begin{aligned} I_\mu :=\left\{ \vec {i} : K_1(\vec {i},\sigma _1)=\mu _1 \wedge K_2(\vec {i},\sigma _2)=\mu _2 \wedge K_3(i_1,\sigma _3)=0 \text{ with } \frac{\Gamma }{2} \ge i_1 \ge i_2 \ge \ldots \ge i_5 \ge 0 \right\} \end{aligned}$$

where $\vec {i}=\left( i_1,\ldots ,i_5\right)$. The components of the K-vector are

$$\begin{aligned} K_1(\vec {i},\sigma _1) = i_2 + i_4 , \end{aligned}$$

$$\begin{aligned} K_2(\vec {i},\sigma _2) = \Gamma /2 + i_1 - 2i_2 + 2i_3 - 2i_4 + i_5 , \end{aligned}$$

$$\begin{aligned} K_3(\vec {i},\sigma _3) = \Gamma - i_1 + i_2 - 2i_3 + i_4 - i_5 , \end{aligned}$$

and the permutation matrix is given by Eq. (41). The solution is the following

$$\begin{aligned} i_1 = i_2 = i_3 = \frac{\Gamma }{2} \quad \text{ and }\quad i_4 = i_5 = \frac{\Gamma }{2} + 1 - \alpha . \end{aligned}$$

(47)

Therefore $I_\mu$ has only one vector $\vec {i}$ when $\mu _3=0$ as is shown in Table 2 for the particular case of $\Gamma =8$ and Eq. (44) takes the form

$$\begin{aligned} C_\mu ^{(N)}(\Gamma /2) = \prod _{p=1}^{6}\text{ sgn }(\sigma ^p)^{i_{p-1}-i_{p}} \left( {\begin{array}{c}i_p\\ i_{p-1}\end{array}}\right) \end{aligned}$$

with $i_0=\Gamma /2$ and $i_6 = 0$. The final result is

$$\begin{aligned} C_\mu ^{(3)}(\Gamma /2) = (-1)^{\mu _1} \left( {\begin{array}{c}\Gamma /2\\ \mu _2-\Gamma /2\end{array}}\right) \text{ with } \mu _3 = 0. \end{aligned}$$

Appendix 3: Exact Coefficients Computation via Finite Difference Method (FDM)

We know that Vandermonde determinant $\det (z_j^{i-1})_{(i,j=1,2,\ldots ,N)}$ to the power $\Gamma /2$ with $\Gamma$ a positive even number may be written in terms of the expansion

$$\begin{aligned} \Delta _N(z_1,\ldots ,z_N)^{\Gamma /2} = \prod _{1 \le i<j\le N} \left( z_i-z_j\right) ^{\Gamma /2} = \sum _{\mu }C_{\mu }^{(N)}(\Gamma /2)m_\mu (z_1,\ldots ,z_N) \end{aligned}$$

where $\mu :=(\mu _1,\ldots ,\mu _N)$ is a partition of $N(N-1)\Gamma /4$ with the condition $(N-1)\Gamma /2\ge \mu _1\ge \mu _2\cdots \ge \mu _N\ge 0$ where $m_\mu (z_1,\ldots ,z_N)$ are the monomial symmetric functions

$$\begin{aligned} m_{\mu }(z_1,\ldots ,z_N) = \frac{1}{\prod _i m_i !} \sum _{\sigma \in S_N} z_{\sigma _1}^{\mu _1}\cdots z_{\sigma _N}^{\mu _N} \end{aligned}$$

when $\Gamma /2$ is even or antisymmetric functions when $\Gamma /2$ is odd. Since $\Delta _N(z_1,\ldots ,z_N)^{\Gamma /2}$ is a polynomial with a finite number of terms whose exponents grouped in partitions $\mu$ determine completely each coefficient $C_{\mu }^{(N)}(\Gamma /2)$ for a given N and $\Gamma$ then

$$\begin{aligned} \frac{\partial ^{\mu _1 + \cdots + \mu _N } }{\partial z_1^{\mu _1} \cdots \delta z_1^{\mu _N}} \Delta _N(z_1,\ldots ,z_N)^{\Gamma /2} = C_{\mu }^{(N)}(\Gamma /2) \frac{1}{\prod _i m_i !} \sum _{\sigma \in S_N} \frac{\partial ^{\mu _1 + \cdots + \mu _N } }{\partial z_1^{\mu _1} \cdots \delta z_1^{\mu _N}} \left( z_{\sigma _1}^{\mu _1}\cdots z_{\sigma _N}^{\mu _N} \right) \end{aligned}$$

where

$$\begin{aligned} \sum _{\sigma \in S_N} \frac{\partial ^{\mu _1 + \cdots + \mu _N } }{\partial z_1^{\mu _1} \cdots \delta z_1^{\mu _N}} \left( z_{\sigma _1}^{\mu _1}\cdots z_{\sigma _N}^{\mu _N} \right) = \prod _i m_i ! \left( \mu _1! \cdots \mu _N! \right) \end{aligned}$$

hence

$$\begin{aligned} C_{\mu }^{(N)}(\Gamma /2) = \frac{1}{\mu _1!\ldots \mu _N!} \frac{\partial ^{\mu _1 + \cdots + \mu _N } }{\partial z_1^{\mu _1} \cdots \delta z_1^{\mu _N}} \left[ \Delta _N(z_1,\cdots ,z_N)^{\Gamma /2} \right] \end{aligned}$$

(48)

Finite difference method enable us to compute derivatives of functions approximately starting from the usual limit definition of derivatives

$$\begin{aligned} \frac{d }{d x} f(x) = \lim _{h\rightarrow 0} \frac{f(x+h)-f(x)}{h} = \frac{\Delta _h^1 f(x)}{h} + O(h) \end{aligned}$$

with $\Delta _h^1 f(x) = f(x+h)-f(x)$ for forward difference. The second derivative is

$$\begin{aligned} \frac{d }{d x} f(x) = \frac{\Delta _h^2 f(x)}{h^2} + O(h) \end{aligned}$$

where $\Delta _h^2 f(x) = \Delta _h^1( \Delta _h^1 f(x) ) = \Delta _h^1 f(x+h) - \Delta _h^1 f(x) = f(x + 2h) - 2f(x+h) + f(x)$. This procedure may be generalized in order to approximate the n-order derivative

$$\begin{aligned} \frac{d^n }{d x^n} f(x) = \frac{1}{h^n} \sum _{i=0}^{n} (-1)^i \left( {\begin{array}{c}n\\ i\end{array}}\right) f(x+(n-i)h) + O(h) \end{aligned}$$

(49)

For a general function f(x) Eq. (49) give us an approximation, except in the particular case when f(x) is a polynomial of order n where Eq. (49) coincides with the exact result by virtue of

$$\begin{aligned} \sum _{i=0}^{n} (-1)^i \left( {\begin{array}{c}n\\ i\end{array}}\right) (x+(n-i)h)^m = \left\{ \begin{array}{ll} n!h^n &{} \text{ if } \quad m=n \\ 0 &{} \text{ if } \quad 0 \le m < n \\ \text{ a } \text{ function } \text{ of } x \text{ and } h \text{ if } m > n \end{array} \right. \end{aligned}$$

(50)

The cases for which $0 \le m \le n$ are independent of the value of x since they are cancelled in the expansion. As a result, we may write

$$\begin{aligned} \frac{d^n }{d x^n} x^n = n! = \frac{1}{h^n} \sum _{i=0}^{n} (-1)^i \left( {\begin{array}{c}n\\ i\end{array}}\right) (x+(n-i)h)^n \end{aligned}$$

Since n! is a constant we may choose freely the value of x. If we set $x=0$, then

$$\begin{aligned} \frac{d^n }{d x^n} x^n = \sum _{i=0}^{n} (-1)^i \left( {\begin{array}{c}n\\ i\end{array}}\right) (n-i)^n \end{aligned}$$

Therefore, if $f(x) = \sum _{i=1}^n c_i x^i$ then

$$\begin{aligned} \frac{d^n }{d x^n} f(x) = \sum _{i=0}^{n} (-1)^i \left( {\begin{array}{c}n\\ i\end{array}}\right) f(n-i)^n = n! c_n \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^n }{\partial x^n}\frac{\partial ^m }{\partial y^m} F(x,y) = \sum _{i=0}^{n} (-1)^i \left( {\begin{array}{c}n\\ i\end{array}}\right) f(n-i)^n \sum _{j=0}^{m} (-1)^j \left( {\begin{array}{c}m\\ j\end{array}}\right) g(m-j)^m = n! m! c_n p_m \end{aligned}$$

where $F(x,y):=f(x)g(y)$ with $g(y) = \sum _{i=1}^m p_i y^i$ another polynomial of order m. More generally we may write

$$\begin{aligned} \frac{\partial ^N }{\partial x_1^{n_1} \cdots \partial x_N^{n_N}} F(x_1,\ldots ,x_N)= & {} \sum _{i_1=0}^{n_1}\cdots \sum _{i_N=0}^{n_N} (-1)^{i_1 + \cdots + i_N} \left( {\begin{array}{c}n_1\\ i_1\end{array}}\right) \cdots \left( {\begin{array}{c}n_N\\ i_N\end{array}}\right) \nonumber \\&f( n_1-i_1, \ldots , n_N-i_N ) \end{aligned}$$

(51)

if $F(x_1,\ldots ,x_N) = \prod _{i=1}^N f_i(x_i)$ with $f_i(x_i)$ a polynomial function of order $n_i$. If any function $f_i(x_i)$ of $F(x_1,\ldots ,x_N)$ would have and order $n'_i$ lower than $n_i$ then Eq. (51) would be simply zero because of Eq. (50) and would give you a wrong derivative if $n'_i > n_i$. Now consider the case

$$\begin{aligned} F(x_1,\ldots ,x_N) = \sum _{i_1=0}^{n'_1}\cdots \sum _{i_N=0}^{n'_N} c_{i_1 \ldots i_N} x^{i_1}\ldots x^{i_N} \end{aligned}$$

where $i_1 + \cdots + i_N = constant$. We may obtain the coefficient $c_{i_1 \ldots i_N}$ applying $\frac{\partial ^N }{\partial x_1^{i_1} \cdots \partial x_N^{i_N}}$ according to (51) even when any derivative $\frac{\partial }{\partial x_j^{i_j}}$ of another term say $c_{i'_1 \ldots i'_N} x^{i'_1}\ldots x^{i'_N}$ give us a wrong result if $i_j < i'_j$ because the restriction $i_1 + \cdots + i_N = constant$ ensures the existence of at least one derivative say $\frac{\partial }{\partial x_k^{i_k}}$ with $i_j > i'_j$ which transform the whole term in zero. This is just the case of Eq. (48) because the partition elements have the constrain $\mu _1 + \cdots + \mu _N = N(N-1)\Gamma /4$ hence the coefficients for even values of $\Gamma /2$ take the form

$$\begin{aligned} C_{\mu }^{(N)}(\Gamma /2)= & {} \frac{1}{\mu _1!\ldots \mu _N!}\sum _{i_1=0}^{\mu _1} \cdots \sum _{i_N=0}^{\mu _N} (-1)^{i_1 + \cdots + i_N} \left( {\begin{array}{c}\mu _1\\ i_1\end{array}}\right) \cdots \left( {\begin{array}{c}\mu _N\\ i_N\end{array}}\right) \nonumber \\&\left[ \Delta _N(\mu _1-i_1,\ldots ,\mu _N-i_N)^{\Gamma /2}\right] . \end{aligned}$$

(52)

In principle, the coefficients computation with Eq. (52) does not offer remarkable implementation difficulties. Nevertheless, it is important to note that $C_{\mu }^{(N)}(\Gamma /2)$ may have a large value as $\Gamma$ or N increase. For instance, the coefficient for $N=5$ particles with $\mu = (100,100,100,0)$ at $\Gamma =100$ is

$$\begin{aligned} C_{(100,100,100,0)}^{(N=5)}(40)=2042816020019820636556288572807323741663688000. \end{aligned}$$

This value may easily overflow the maximum integer value permitted by the computer. Usually, this maximum value varies with the program used to implement the coefficients computation formula as well as the architecture of the machine. Fortunately, in order to solve this problem it is possible to use multiple precision arithmetic libraries as GMP [31] included in some of our computations.

Appendix 4: Excess Energy as a Function of N

In this section, we report the excess energy $U_{exc}$ obtained from the exact expression, Eq. (19), when N increases, for three fixed values of $\Gamma =4, 6, 8$. As before, we have set $\rho _b=1$ and $L=1$. A four parameter fit to an ansatz of the form $U_{exc}=q^2(A N + B + C/N + D/N^2)$ is proposed. As explained in Sect. 4.5, this is the expected finite-size expansion for $U_{exc}$. The fit is done with four consecutive values of N, and the convergence of the parameters A, B, C and D is observed as N increases. This allows us to obtain the bulk value of the excess internal energy and the finite size corrections (Table 3).

Table 3 Excess energy of the 2dOCP on a sphere and its fit to $U_{exc}=q^2(A N + B + C/N + D/N^2)$