A Generalised Airy Distribution Function for the Accumulated Area Swept by N Vicious Brownian Paths

Abstract

In this work exact expressions for the distribution function of the accumulated area swept by excursions and meanders of N vicious Brownian particles up to time T are derived. The results are expressed in terms of a generalised Airy distribution function, containing the Vandermonde determinant of the Airy roots. By mapping the problem to an Random Matrix Theory ensemble we are able to perform Monte Carlo simulations finding perfect agreement with the theoretical results.

Appendices

Appendix 1: Taylor Expansion of Slater’s Determinant

In this secion we discuss the Taylor expansion of a Slater determinant. Starting form the definition of the determinant and doing a Taylor expansion we write:

$$\begin{aligned} \det \limits _{1\le i,j\le N}\varphi _{n_i}(z_j)= & {} \sum _{P\in S_{N}}\text {sign}(P)\prod \limits _{i=1}^N\varphi _{n_P(i)}(z_i)\nonumber \\= & {} \sum _{P\in S_{N}}\text {sign}(P)\sum _{\ell _1,\ldots ,\ell _N\ge 0}\frac{z_1^{\ell _1}\ldots z_N^{\ell _N}}{\ell _1!\ldots \ell _N!}\prod \limits _{i=1}^N\varphi ^{(\ell _i)}_{n_{P(i)}}(0)\nonumber \\= & {} \sum _{\ell _1,\ldots ,\ell _N\ge 0}\frac{z_1^{\ell _1}\ldots z_N^{\ell _N}}{\ell _1!\ldots \ell _N!}\det \limits _{1\le i,j\le N}\varphi ^{(\ell _i)}_{n_{j}}(0)\,. \end{aligned}$$

(25)

Due to the antisymmetric properties of the determinant, only those terms in the multiple sum with different values of $\ell$’s are different from zero. With this in mind we rewrite the sum over the $\varvec{\ell }=(\ell _1,\ldots ,\ell _N)$ as a sum over over $U_{N}$ of ordered indices (let’s say $\ell _1<\ell _2<\cdots <\ell _N$) and then we sum over its permutation, viz.

$$\begin{aligned} \det \limits _{1\le i,j\le N}\varphi _{n_i}(z_j)= & {} \sum _{\ell _1,\ldots ,\ell _N\ge 0}\frac{z_1^{\ell _1}\ldots z_N^{\ell _N}}{\ell _1!\ldots \ell _N!}\det \limits _{1\le i,j\le N}\varphi ^{(\ell _i)}_{n_{j}}(0)\nonumber \\= & {} \sum _{\varvec{\ell }\in U_N}\frac{1}{\ell _1!\ldots \ell _N!}\sum _{P\in S_{N}} z_1^{\ell _{P(1)}}\ldots z_N^{\ell _{P(N)}}\det \limits _{1\le i,j\le N}\varphi ^{(\ell _{P(i)})}_{n_{j}}(0)\nonumber \\= & {} \sum _{\varvec{\ell }\in U_N}\frac{1}{\ell _1!\ldots \ell _N!}\sum _{P\in S_N}\text {sign}(P) z_1^{\ell _{P(1)}}\ldots z_N^{\ell _{P(N)}}\det \limits _{1\le i,j\le N}\varphi ^{(\ell _{i})}_{n_{j}}(0)\nonumber \\= & {} \sum _{\varvec{\ell }\in U_N}\frac{1}{\ell _1!\ldots \ell _N!}\det \limits _{1\le i,j\le N}[z_i^{\ell _j}]\det \limits _{1\le i,j\le N}[\varphi ^{(\ell _{i})}_{n_{j}}(0)]\,, \end{aligned}$$

(26)

as we wanted to show.

Appendix 2: Excursion

For the normalisation factor we need to derive the propagator of N vicious walkers in the semi-infinite line. This propagator is this case can be written as follows

$$\begin{aligned} G_{0}(\varvec{y},T|\varvec{x},0)=\frac{1}{T^{N/2}}\left( \frac{2}{\pi }\right) ^N\int _0^\infty \ldots \int _0^\infty dq_1\ldots dq_N\Phi ^{(0)}_{\varvec{q}}\left( \frac{\varvec{y}}{\sqrt{T}}\right) \Phi ^{(0)}_{\varvec{q}}\left( \frac{\varvec{x}}{\sqrt{T}}\right) e^{-\frac{1}{2}\varvec{q}^2}\,,\nonumber \\ \end{aligned}$$

(27)

where $\Phi ^{(0)}_{\varvec{q}}(\varvec{x})=(1/\sqrt{N!})\det \limits _{1\le i,j\le N}[\sin (q_ix_j)]$. Using the Taylor expansion formula for the Slater determinant we arrive at

$$\begin{aligned} \Phi ^{(0)}_{\varvec{q}}(\varvec{x})=\frac{1}{\sqrt{N!}}\sum _{\varvec{s}\in U_{N}}\frac{1}{(2s_1+1)!\ldots (2s_N+1)!}\det \limits _{1\le i,j\le N}[x_i^{2s_j+1}]\det \limits _{1\le i,j\le N}[q_{i}^{2s_j+1}(-1)^{s_j}]\,,\nonumber \\ \end{aligned}$$

(28)

with $\varvec{s}=(s_1,\ldots ,s_N)$. Noticing that the lowest contribution is obtained by of $s_i=i-1$, we obtain

$$\begin{aligned} \Phi ^{(0)}_{\varvec{q}}\left( \frac{\varvec{x}}{\sqrt{T}}\right)&\simeq \beta _N T^{-\frac{N^2}{2}}\left[ \prod \limits _{i=1}^N x_i\right] \Delta _{N}\left( x_1^2,\ldots ,x_N^2\right) \left[ \prod \limits _{i=1}^N q_i\right] \Delta _{N}(q_1^2,\ldots q_N^2)\,, \end{aligned}$$

(29)

with $\beta _N=(-1)^{\frac{N(N-1)}{2}}/[ 1!\cdots (2N-1)!\sqrt{N!}]$. At this stage we notice that the propagator goes like

$$\begin{aligned} G_{0}(\varvec{\epsilon },T|\varvec{x},\epsilon )\simeq & {} \beta ^2_N T^{-\frac{1}{2}N(2N+1)}\left[ \prod \limits _{i=1}^N \epsilon ^2_i\right] \Delta ^2_{N}\left( \epsilon _1^2,\ldots ,\epsilon _N^2\right) B_N\,,\nonumber \\ B_N= & {} \left( \frac{2}{\pi }\right) ^{N}\int _0^{\infty }\ldots \int _0^\infty dq_1\ldots dq_Ne^{-\frac{1}{2}\varvec{q}^2}\left[ \prod \limits _{i=1}^N q_i^2\right] \Delta _N^2(q_1^2,\ldots ,q^2_N) \nonumber \\ \end{aligned}$$

(30)

where the constant $B_N$ is derived in Appendix 4.

For reflecting boundary conditions the expression for the propagator is the same but with the Slater determinant given by $\Phi ^{(0)}_{\varvec{q}}(\varvec{x})=(1/\sqrt{N!})\det \limits _{1\le i,j\le N}[\cos (q_ix_j)]$. Using the expansion formula for the Slater determinant we arrive at

$$\begin{aligned} \Phi ^{(0)}_{\varvec{q}}(\varvec{x}/\sqrt{T})=\frac{1}{\sqrt{N!}}\sum _{\varvec{s}\in U_{N}}\frac{1}{(2s_1)!\cdots (2s_N)!}\det \limits _{1\le i,j\le N}[T^{-s_{j}}x_i^{2s_j}]\det \limits _{1\le i,j\le N}[q_{i}^{2s_j}(-1)^{s_j}]\,,\nonumber \\ \end{aligned}$$

(31)

with $\varvec{s}=(s_1,\ldots ,s_N)$. Noticing that the lowest contribution is obtained by of $s_i=i-1$, we obtain

$$\begin{aligned} \Phi ^{(0)}_{\varvec{q}}\left( \frac{\varvec{x}}{\sqrt{T}}\right)&\simeq \beta _N T^{-\frac{N(N-1)}{2}}\Delta _{N}\left( x_1^2,\ldots ,x_N^2\right) \left[ \prod \limits _{i=1}^N q_i\right] \Delta _{N}(q_1^2,\ldots q_N^2)\,, \end{aligned}$$

(32)

with $\beta _N=(-1)^{\frac{N(N-1)}{2}}/[ 0!\cdots (2(N-1))!\sqrt{N!}]$. At this stage we notice that the propagator goes like

$$\begin{aligned} G_{0}(\varvec{\epsilon },T|\varvec{x},\epsilon )&\simeq \beta ^2_N T^{-\frac{N(2N-1)}{2}}\Delta ^2_{N}\left( \epsilon _1^2,\ldots ,\epsilon _N^2\right) F_N\,,\nonumber \\ F_N&=\left( \frac{2}{\pi }\right) ^{N}\int _0^{\infty }\cdots \int _0^\infty dq_1\cdots dq_Ne^{-\frac{1}{2}\varvec{q}^2}\Delta _N^2(q_1^2,\cdots ,q^2_N) \end{aligned}$$

(33)

where the constant $F_N$ is derived in Appendix 1.

Appendix 3: Meander

For the case of a meander of N vicious walkers (a star with a wall) we need to integrate over the final position $\varvec{x}$. Using the Taylor expansion of the Slater determinant as explained in the main text we can write

$$\begin{aligned}&\int _{\varvec{W}_N} d^N\varvec{x} G_N^{(1)}(\varvec{x},T|\varvec{\epsilon },0)=\sum _{\varvec{n}}\int _{\varvec{W}_N} d^N\varvec{x}\,\Phi ^{(1)}_{\varvec{n}}(\varvec{x})\overline{\Phi }^{(1)}_{\varvec{n}}(\varvec{\epsilon })e^{-E^{(1)}_{\varvec{n}} T}\nonumber \\&\simeq (2\lambda )^{\frac{N^2}{3}}\gamma _N\left[ \prod \limits _{i=1}^N \epsilon _i\right] \Delta _N(\epsilon _1^2,\ldots \epsilon _N^2)\,\sum _{\varvec{n}}B_N(\alpha _{n_1},\ldots ,\alpha _{n_N})\Delta _N(\alpha _{n_1},\dots \alpha _{n_N})e^{-E^{(1)}_{\varvec{n}} T}\,,\nonumber \\ \end{aligned}$$

(34)

where we have defined

$$\begin{aligned} B_N(\alpha _{n_1},\ldots ,\alpha _{n_N})&=\frac{1}{\prod \limits _{i=1}^N\text {Ai}'(-\alpha _{n_i})}\int _{\varvec{W}_N} d^N\varvec{x}\det \limits _{1\le i,j\le N}\left[ \text {Ai}(x_j-\alpha _{n_i})\right] \,. \end{aligned}$$

(35)

Similarly, the propagator in the denominator goes like

$$\begin{aligned} \int _{\varvec{W}_N} d^N\varvec{x} G_N^{(0)}(\varvec{x},T|\varvec{\epsilon },0)&\simeq T^{-\frac{N^2}{2}}\gamma _N F_{N}\left[ \prod \limits _{i=1}^N \epsilon _i\right] \Delta _N(\epsilon _1^2,\ldots \epsilon _N^2) \end{aligned}$$

(36)

where $F_N$ is a constant with no simple expression.

For reflecting boundary conditions we have instead the following expression for the numerator

$$\begin{aligned}&\int _{\varvec{W}_N} d^N\varvec{x}G_{N}^{(1)}(\varvec{x},T|\varvec{\epsilon },0)=\sum _{\varvec{n}}\int _{\varvec{W}_N} d^N\varvec{x}\Psi _{\varvec{n}}^{(1)}(\varvec{x})\overline{\Psi }_{\varvec{n}}^{(1)}(\varvec{\epsilon })e^{-E_{\varvec{n}}^{(1)}T}\nonumber \\&\quad =\frac{\delta _N}{N!}\Delta _{N}(x_1^2,\ldots ,x_N^2)\sum _{\varvec{n}}\frac{\Delta _N(\beta _{n_1},\ldots ,\beta _{n_N})}{\beta _{n_1}\cdots \beta _{n_N}}e^{-E_{\varvec{n}}^{(1)}T}(2\lambda )^{\frac{N(2N-1)}{6}}(2\lambda )^{N/6}\nonumber \\&\quad \quad \int _{\varvec{W}_N} d^N\varvec{x}\frac{1}{\prod _{i=1}^N\text {Ai}(-\beta _{n_i})}\det \limits _{1\le i,j\le N}[\text {Ai}((2\lambda )^{1/3}x_j-\beta _{n_i})]\nonumber \\&\quad =\frac{\delta _N}{N!}\Delta _{N}(x_1^2,\ldots ,x_N^2)(2\lambda )^{\frac{N(2N-1)}{6}-N/6}\sum _{\varvec{n}}\frac{\Delta _N(\beta _{n_1},\ldots ,\beta _{n_N})}{\beta _{n_1}\cdots \beta _{n_N}}e^{-E_{\varvec{n}}^{(1)}T}\nonumber \\&\quad \quad \int _{\varvec{W}_N} d^N\varvec{z}\frac{1}{\prod _{i=1}^N\text {Ai}(-\beta _{n_i})}\det \limits _{1\le i,j\le N}[\text {Ai}(z_j-\beta _{n_i})]\nonumber \\&\quad =\frac{\delta _N}{N!}\Delta _{N}(x_1^2,\ldots ,x_N^2)(2\lambda )^{\frac{N(N-1)}{3}}\sum _{\varvec{n}}\frac{\Delta _N(\beta _{n_1},\ldots ,\beta _{n_N})}{\beta _{n_1}\cdots \beta _{n_N}}e^{-E_{\varvec{n}}^{(1)}T}\nonumber \\&\quad \quad \frac{1}{\prod _{i=1}^N\text {Ai}(-\beta _{n_i})}\int _{\varvec{W}_N} d^N\varvec{z}\det \limits _{1\le i,j\le N}[\text {Ai}(z_j-\beta _{n_i})] \end{aligned}$$

(37)

Appendix 4: On the Normalisation Constants

Here we derive the expressions for the two normalisation constants for the case of excursions for absorbing and reflecting boundary conditions. For both cases we recall the well-known result of Selberg’s integral [13]:

$$\begin{aligned} \frac{1}{N!}\int _0^\infty \cdots \int _0^{\infty }|\Delta (x_1,\ldots ,x_N)|^{2c}\prod \limits _{i=1}^Nx_i^{a-1}e^{-x_i} dx_i=\prod \limits _{j=0}^{N-1}\frac{\Gamma (a+jc)\Gamma ((j+1)c)}{\Gamma (c)}\nonumber \\ \end{aligned}$$

(38)

For the constant $B_N$ we do the change of variables $q_i^2/2=y_i$ so that $q_i dq_i=dy_i$ or $dq_i=\frac{dy_i}{\sqrt{2y_i}}$, so we can write

$$\begin{aligned} B_N= & {} \left( \frac{2}{\pi }\right) ^{N}\int _0^{\infty }\cdots \int _0^\infty dq_1\cdots dq_Ne^{-\frac{1}{2}\varvec{q}^2}\left[ \prod \limits _{i=1}^N q_i^2\right] \Delta _N^2(q_1^2,\cdots ,q^2_N)\nonumber \\= & {} \left( \frac{2}{\pi }\right) ^{N}2^{-N/2} 2^{N} 2^{N(N-1)}\int _0^{\infty }\cdots \int _0^\infty dy_1\cdots dy_N\prod \limits _{i=1}^Ne^{-y_i} y_i^{-1/2}\left[ \prod \limits _{i=1}^N y_i\right] \Delta _N^2(y_1,\cdots ,y_N)\nonumber \\= & {} \frac{2^{\frac{1}{2}N(2N+1)}}{\pi ^N}N!\prod \limits _{j=0}^{N-1}\Gamma \left( \frac{3}{2}+j\right) \Gamma (j+1)=\frac{2^{\frac{1}{2}N(2N+1)}}{\pi ^N}\prod \limits _{j=0}^{N-1}\Gamma (2+j)\Gamma \left( \frac{3}{2}+j\right) \end{aligned}$$

(39)

Similarly for the case of reflecting boundary conditions we have

$$\begin{aligned} F_N= & {} \left( \frac{2}{\pi }\right) ^{N}\frac{1}{2^{N/2}}2^{N(N-1)}\int _0^{\infty }\cdots \int _0^\infty dy_1\cdots dy_N\Delta _N^2(y_1,\cdots ,y_N)\prod \limits _{i=1}^Ny_i^{-1/2} e^{-y_i}\nonumber \\= & {} N!\frac{2^{\frac{N(2N-1)}{2}}}{\pi ^N}\prod \limits _{j=0}^{N-1}\Gamma \left( \frac{1}{2}+j\right) \Gamma \left( j+1\right) =\frac{2^{\frac{N(2N-1)}{2}}}{\pi ^N}\prod \limits _{j=0}^{N-1}\Gamma \left( \frac{1}{2}+j\right) \Gamma \left( j+2\right) \qquad \end{aligned}$$

(40)

Appendix 5: Derivatives of $F(x,\gamma )$

Starting from the following expression for $F(x,\gamma )$:

$$\begin{aligned} F(x,\gamma )=\frac{\sqrt{3}}{x\sqrt{\pi }}u^{2/3}(x,\gamma )e^{-u(x,\gamma )}U\left( \frac{1}{6},\frac{4}{3},u(x,\gamma )\right) \,,\quad u(x,\gamma )=\frac{2\gamma ^3}{27x^2}\,, \end{aligned}$$

(41)

and using the following property

$$\begin{aligned} x\frac{\partial U(a,b,x)}{\partial x}=(a-b+x)U(a,b,x)-U(a-1,b,x)\,, \end{aligned}$$

(42)

we notice that we can write the expression

$$\begin{aligned} \frac{\partial ^n F(x,\gamma )}{\partial x^n}=\frac{\sqrt{3}}{x^{n+1}\sqrt{\pi }}u^{2/3}(x,\gamma )e^{-u(x,\gamma )}\sum _{\ell =0}^n C^{(n)}_{\ell }U\left( \frac{1}{6}-\ell ,\frac{4}{3},u(x,\gamma )\right) \,. \end{aligned}$$

(43)

for some set of coefficients $\{C^{(n)}_\ell \}$ still to be determined. The expression (43) is certainly correct for $n=1$ and $n=2$. Let us them assume is holds for any n and performe one more derivative with respect to n:

$$\begin{aligned}&\frac{\partial ^{n+1} F(x,\gamma )}{\partial x^{n+1}} =\frac{1}{x^{n+2}\sqrt{3\pi }}u^{2/3}(x,\gamma )e^{-u(x,\gamma )}\nonumber \\&\sum \limits _{\ell =0}^n C^{(n)}_{\ell }\left[ (-7-3n+6u(x,\gamma ))U\left( \frac{1}{6}-\ell ,\frac{4}{3},u(x,\gamma )\right) \right. \nonumber \\&\left. -6u(x,\gamma )U'\left( \frac{1}{6}-\ell ,\frac{4}{3},u(x,\gamma )\right) \right] \,. \end{aligned}$$

(44)

But using the property (42), we can write

$$\begin{aligned}&u(x,\gamma )U'\left( \frac{1}{6}-\ell ,\frac{4}{3},u(x,\gamma )\right) =\left( \frac{1}{6}-\ell -\frac{4}{3}+u\right) \nonumber \\&U\left( \frac{1}{6}-\ell ,\frac{4}{3},u(x,\gamma )\right) -U\left( \frac{1}{6}-\ell -1,\frac{4}{3},u(x,\gamma )\right) \,. \end{aligned}$$

(45)

Gathering results we find

$$\begin{aligned}&\frac{\partial ^{n+1}F(x,\gamma )}{\partial x^{n+1}}=\frac{\sqrt{3}}{x^{n+2}\sqrt{\pi }}u^{2/3}(x,\gamma )e^{-u(x,\gamma )}\nonumber \\&\sum \limits _{\ell =0}^n C^{(n)}_{\ell }\left[ (-n +2\ell )U\left( \frac{1}{6}-\ell ,\frac{4}{3},u(x,\gamma )\right) +2U\left( \frac{1}{6}-\ell -1,\frac{4}{3},u(x,\gamma )\right) \right] \,.\qquad \end{aligned}$$

(46)

On the other hand, we want to write this result as (43) for $n\rightarrow n+1$. This implies to rewrite the sum as:

$$\begin{aligned}&\sum \limits _{\ell =0}^n C^{(n)}_{\ell }\left[ (-n +2\ell )U\left( \frac{1}{6}-\ell ,\frac{4}{3},u(x,\gamma )\right) +2U\left( \frac{1}{6}-\ell -1,\frac{4}{3},u(x,\gamma )\right) \right] \nonumber \\&=\sum \limits _{\ell =0}^n C^{(n)}_{\ell }(-n +2\ell )U\left( \frac{1}{6}-\ell ,\frac{4}{3},u(x,\gamma )\right) +2\sum \limits _{\ell =0}^k C^{(n)}_{\ell }U\left( \frac{1}{6}-\ell -1,\frac{4}{3},u(x,\gamma )\right) \nonumber \\&=\sum \limits _{\ell =0}^n C^{(n)}_{\ell }(-n +2\ell )U\left( \frac{1}{6}-\ell ,\frac{4}{3},u(x,\gamma )\right) +2\sum \limits _{\ell =1}^{n+1} C^{(n)}_{\ell -1}U\left( \frac{1}{6}-\ell ,\frac{4}{3},u(x,\gamma )\right) \nonumber \\&=\sum \limits _{\ell =0}^{n+1} C^{(n+1)}_{\ell }U\left( \frac{1}{6}-\ell ,\frac{4}{3},u(x,\gamma )\right) \,. \end{aligned}$$

(47)

This results in the following set of recurrence relations for the set of coefficents $\{C_{\ell }^{(n)}\}$:

$$\begin{aligned} C^{(n+1)}_0= & {} -nC_{0}^{(n)}\,,\nonumber \\ C_{\ell }^{(n+1)}= & {} C_{\ell }^{(n)}(-n+2\ell )+2C_{\ell -1}^{(n)}\,,\quad \ell =1,\ldots ,n\,,\nonumber \\ C_{n+1}^{(n+1)}= & {} 2C_{n}^{(n)}\,, \end{aligned}$$

(48)

with the initial condition $C_0^{(0)}=1$. By checking explicitly the value of some of these coeffcients, and with the help of the Sloane database², we arrive at the solution:

$$\begin{aligned} C^{(n)}_\ell =\frac{n!}{2^{n-2\ell } (n-\ell )!(2\ell -n)!}\,,\quad \ell =0,\ldots ,n\,. \end{aligned}$$

(49)

A Similar analysis can be perform ny doing derivatives with respect to $\gamma$. Starting from (43) one notices that

$$\begin{aligned}&\frac{\partial ^{k+n} F(x,\gamma )}{\partial \gamma ^k\partial x^n}=\frac{\sqrt{3}}{x^{n+1}\sqrt{\pi }\gamma ^{k}}u^{2/3}(x,\gamma )e^{-u(x,\gamma )}\nonumber \\&\sum _{\ell =0}^{n}\sum _{s=0}^{k}C^{(n)}_{\ell } D^{(k)}_{s}(\ell )U\left( \frac{1}{6}-\ell -s,\frac{4}{3},u(x,\gamma )\right) \,, \end{aligned}$$

(50)

for some set of coefficients $\{D_s^{(k)}(\ell )\}$. Performing one more derivative with respect to $\gamma$ allows us to arrive at the following set of recurrence relations for those, viz.

$$\begin{aligned} D_0^{(k+1)}(\ell )= & {} -\left( \frac{3}{2}+k+3\ell \right) D_{0}^{(k)}(\ell )\,,\nonumber \\ D_{s}^{(k+1)}(\ell )= & {} -\left( \frac{3}{2}+k+3\ell +3s\right) D_{s}^{(k)}(\ell )-3D_{s-1}^{(k)}(\ell )\,,\quad \quad s=1,\ldots ,k\,,\nonumber \\ D_{k+1}^{(k+1)}(\ell )= & {} -3D_{k}^{(k)}(\ell )\,, \end{aligned}$$

(51)

with the initial condition $D_0^{(0)}(\ell )=1$. As we are only interested in the orders $k=0$ and $k=2$, we do not need a general solution. This yields

$$\begin{aligned} D_0^{(0)}(\ell )= & {} 1\,,\nonumber \\ D_{0}^{(1)}(\ell )= & {} -\frac{3}{2}\left( 1+2\ell \right) \,,\quad \quad D_{1}^{(1)}(\ell )=-3\,\nonumber \\ D_0^{(2)}(\ell )= & {} \frac{3}{4} (1 + 2\ell ) (5 + 6\ell )\,,\quad \quad D_1^{(2)}(\ell )=3(7+6\ell )\,,\quad \quad D_2^{(2)}(\ell )=9\,. \end{aligned}$$

(52)

Appendix 6: Moments

We first recall the following identity

$$\begin{aligned} \int _0^{\infty } ds s^{\mu } e^{-xs}=\frac{\Gamma (\mu +1)}{x^{1+\mu }}\,. \end{aligned}$$

(53)

From here we write

$$\begin{aligned} M_{-(1+\mu )}= & {} \int _0^\infty dx x^{-(1+\mu )} F(x)\nonumber \\= & {} \frac{1}{\Gamma (\mu +1)}\int _0^\infty dx F(x)\int _0^{\infty } ds s^{\mu } e^{-xs}\nonumber \\= & {} \frac{1}{\Gamma (\mu +1)}\int _0^{\infty } ds s^{\mu }\int _0^\infty dx F(x) e^{-xs}\nonumber \\= & {} \frac{1}{\Gamma (\mu +1)}\int _0^{\infty } ds s^{\mu }\widehat{F}(s)\nonumber \\= & {} \frac{1}{\Gamma (\mu +1)}\int _0^{\infty } ds s^{\mu +E}e^{-as^{2/3}} \end{aligned}$$

(54)

But

$$\begin{aligned} \int _0^{\infty } ds s^{\mu +E}e^{-as^{2/3}}= & {} \int _0^{\infty } ds (s^{2/3})^{\frac{3}{2}(\mu +E)}e^{-as^{2/3}}\nonumber \\= & {} \frac{3}{2}\int _0^{\infty } dy y^{\frac{3}{2}(\mu +E)+1/2}e^{-a y}\nonumber \\= & {} \frac{3}{2}a^{-\frac{3}{2}(\mu +E+1)}\int _0^{\infty } dy y^{\frac{3}{2}(\mu +E)+1/2}e^{- y}\nonumber \\= & {} \frac{3}{2}a^{-\frac{3}{2}(\mu +E+1)}\Gamma \left( \frac{3}{2}(\mu +E+1)\right) \end{aligned}$$

(55)

Thus denoting $\nu =1+\mu$ we finally have

$$\begin{aligned} M_{-\nu }=\frac{3}{2}a^{-\frac{3}{2}(\nu +E)}\frac{\Gamma \left( \frac{3}{2}(\nu +E)\right) }{\Gamma (\nu )} \end{aligned}$$

(56)