Fractional Boson Gas and Fractional Poisson Measure in Infinite Dimensions

Maria João Oliveira; Rui Vilela Mendes

doi:10.1007/978-3-319-16637-7_11

From Particle Systems to Partial Differential Equations II pp 293-312 | Cite as

Fractional Boson Gas and Fractional Poisson Measure in Infinite Dimensions

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Maria João Oliveira
Rui Vilela Mendes

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First Online: 05 April 2015

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 129)

Abstract

As a consequence of Haag’s theorem, to obtain a non-trivial theory, one either works with a non-Fock representation or with a Fock representation in a finite volume. Calculations in the Fock representation taking the N,V $\rightarrow \infty $ limit with the ratio N/V = $\rho $ fixed, show the equivalence of the free Boson gas and the infinite-dimensional Poisson measure. The N/V limit provides a way to deal with non-trivial infinite systems using the Fock representation. However, by the very nature of the fixed $\rho $ density limit, it is unable to deal with systems with density fluctuations, a shortcoming that is solved by the use of reducible functionals. A particularly interesting reducible functional is the one associated to the infinite-dimensional fractional Poisson measure which we recall in this work.

Keywords

Boson gases Fractional Poisson measure

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Appendix 1: The Infinite-Dimensional Poisson Measure and Configuration Spaces

Here a short summary is given of the properties of the infinite-dimensional Poisson measure and its support on configuration spaces [12, 13, 16, 17, 18, 19, 24].

An infinite-dimensional generalization of the characteristic function of the Poisson measure is obtained by generalizing (5) to

$$\begin{aligned} C\left( \varphi \right) =e^{\int \left( e^{i\varphi \left( x\right) }-1\right) \,d\mu \left( x\right) } \end{aligned}$$

(15)

for test functions $\varphi \in \fancyscript{D}\left( M\right) $ in the space of $C^\infty $-functions of compact support in a manifold $M$. It is easy to prove, using the Bochner-Minlos theorem, that $C$ is indeed the Fourier transform of a measure on the distribution space $ \fancyscript{D}^{\prime }\left( M\right) $.

A support for this measure is obtained in the configuration space $\varGamma :=\varGamma _{M}$ over the manifold $M$, defined as the set of all locally finite subsets of $M$ (simple configurations)

$$\begin{aligned} \varGamma :=\{\gamma \subset M\,:\,|\gamma \cap K|<\infty \;\mathrm {\ for\;any\;compact\;}K\subset M\} \end{aligned}$$

(16)

Here $|A|$ denotes the cardinality of the set $A$. As usual one identifies each $\gamma \in \varGamma $ with a non-negative integer-valued Radon measure,

$$\begin{aligned} \varGamma \ni \gamma \mapsto \sum _{x\in \gamma }\delta _{x}\in \fancyscript{M}(M) \end{aligned}$$

where $\delta _{x}$ is the Dirac measure with unit mass at $x$ and $\fancyscript{ M}(M)$ denotes the set of all non-negative Radon measures on $M$. In this way the space $\varGamma $ can be endowed with the relative topology as a subset of the space $\fancyscript{M}(M)$ with the vague topology, i.e., the weakest topology on $\varGamma $ for which the mappings

$$\begin{aligned} \varGamma \ni \gamma \mapsto \left\langle \gamma ,f\right\rangle :=\int _{M}f(x)d\gamma (x)=\sum _{x\in \gamma }f(x) \end{aligned}$$

are continuous for all real-valued continuous functions $f$ on $M$ with compact support. Denote the corresponding Borel $\sigma $-algebra on $\varGamma $ by $\fancyscript{B}\left( \varGamma \right) $.

For each $Y\in \fancyscript{B}(M)$ let us consider the space $\varGamma _{Y}$ of all configurations contained in $Y$, $\varGamma _{Y}:=\left\{ \gamma \in \varGamma :\left| \gamma \cap (X\backslash Y)\right| =0\right\} $, and the space $\varGamma _{Y}^{(n)}$ of $n$-point configurations,

$$\begin{aligned} \varGamma _{Y}^{(n)}:=\left\{ \gamma \in \varGamma _{Y}:\left| \gamma \right| =n\right\} ,n\in \mathbb {N},\quad \varGamma _{Y}^{(0)}:=\left\{ \emptyset \right\} \end{aligned}$$

A topological structure may be introduced on $\varGamma _{Y}^{(n)}$ through the natural surjective mapping of $\widetilde{Y^{n}}:=\left\{ (x_{1},...,x_{n}):x_{i}\in Y,x_{i}\ne x_{j}\text { if }i\ne j\right\} $ onto $\varGamma _{Y}^{(n)}$,

$$\begin{aligned} \begin{array}{ll} \mathrm {sym}_{Y}^{n}:\widetilde{Y^{n}} &{} \longrightarrow \varGamma _{Y}^{(n)} \\ (x_{1},...,x_{n}) &{} \longmapsto \left\{ x_{1},...,x_{n}\right\} \end{array} \end{aligned}$$

which is at the origin of a bijection between $\varGamma _{Y}^{(n)}$ and the symmetrization $\widetilde{Y^{n}}/S_{n}$ of $\widetilde{Y^{n}}$, $S_{n}$ being the permutation group over $\left\{ 1,...,n\right\} $. Thus, $\mathrm { sym}_{Y}^{n}$ induces a metric on $\varGamma _{Y}^{(n)}$ and the corresponding Borel $\sigma $-algebra $\fancyscript{B}\left( \varGamma _{Y}^{(n)}\right) $ on $ \varGamma _{Y}^{(n)}$.

For $\varLambda \in \fancyscript{B}(M)$ with compact closure ($\varLambda \in \fancyscript{ B}_{c}(M)$), it clearly follows from (16) that

$$\begin{aligned} \varGamma _{\varLambda }=\bigsqcup _{n=0}^{\infty }\varGamma _{\varLambda }^{(n)} \end{aligned}$$

the $\sigma $-algebra $\fancyscript{B}(\varGamma _{\varLambda })$ being defined by the disjoint union of the $\sigma $-algebras $\fancyscript{B}\left( \varGamma _{Y}^{(n)}\right) $, $n\in \mathbb {N}_{0}$.

For each $\varLambda \in \fancyscript{B}_{c}(M)$ there is a natural measurable mapping $p_{\varLambda }:\varGamma \rightarrow \varGamma _{\varLambda }$. Similarly, given any pair $\varLambda _{1},\varLambda _{2}\in \fancyscript{B}_{c}(M)$ with $ \varLambda _{1}\subset \varLambda _{2}$ there is a natural mapping $p_{\varLambda _{2},\varLambda _{1}}:\varGamma _{\varLambda _{2}}\rightarrow \varGamma _{\varLambda _{1}}$. They are defined, respectively, by

$$\begin{aligned} \begin{array}{ll} p_{\varLambda }: &{} \varGamma \longrightarrow \varGamma _{\varLambda } \\ &{} \gamma \longmapsto \gamma _{\varLambda }:=\gamma \cap \varLambda \end{array} \quad \begin{array}{ll} p_{\varLambda _{2},\varLambda _{1}}: &{} \varGamma _{\varLambda _{2}}\longrightarrow \varGamma _{\varLambda _{1}} \\ &{} \gamma \longmapsto \gamma _{\varLambda _{1}} \end{array} \end{aligned}$$

It can be shown that $(\varGamma ,\fancyscript{B}(\varGamma ))$ coincides (up to an isomorphism) with the projective limit of the measurable spaces $(\varGamma _{\varLambda },\fancyscript{B}(\varGamma _{\varLambda }))$, $\varLambda \in \fancyscript{B}_{c}$ $ \left( M\right) $, with respect to the projection $p_{\varLambda }$, i.e., $ \fancyscript{B}(\varGamma )$ is the smallest $\sigma $-algebra on $\varGamma $ with respect to which all projections $p_{\varLambda }$, $\varLambda \in \fancyscript{B} _{c} $ $\left( M\right) $, are measurable.

Let now $\mu $ be a measure on the underlying measurable space $(M,\fancyscript{B }(M))$ and consider for each $n\in \mathbb {N}$ the product measure $\mu ^{\otimes n}$ on $(M^{n},\fancyscript{B}(M^{n}))$. Since $\mu ^{\otimes n}(M^{n}\backslash \widetilde{M^{n}})=0$, one may consider for each $\varLambda \in \fancyscript{B}_{c}(M)$ the restriction of $\mu ^{\otimes n}$ to $(\widetilde{ \varLambda ^{n}},\fancyscript{B}(\widetilde{\varLambda ^{n}}))$, which is a finite measure, and then the image measure $\mu _{\varLambda }^{(n)}$ on $(\varGamma _{\varLambda }^{(n)},\fancyscript{B}(\varGamma _{\varLambda }^{(n)}))$ under the mapping $ \mathrm {sym}_{\varLambda }^{n}$,

$$\begin{aligned} \mu _{\varLambda }^{(n)}:=\mu ^{\otimes n}\circ (\mathrm {sym}_{\varLambda }^{n})^{-1} \end{aligned}$$

For $n=0$ we set $\mu _{\varLambda }^{(0)}:=1$. Now, one may define a probability measure $\pi _{\mu ,\varLambda }$ on $(\varGamma _{\varLambda },\fancyscript{B }(\varGamma _{\varLambda }))$ by

$$\begin{aligned} \pi _{\mu ,\varLambda }:=\sum _{n=0}^{\infty }\frac{\exp (-\mu (\varLambda ))}{n!} \mu _{\varLambda }^{(n)} \end{aligned}$$

(17)

The family $\{\pi _{\mu ,\varLambda }:\varLambda \in \fancyscript{B}_{c}(M)\}$ of probability measures yields a probability measure on $(\varGamma ,\fancyscript{B} (\varGamma ))$ with the $\pi _{\mu ,\varLambda }$ as projections. This family is consistent, that is,

$$\begin{aligned} \pi _{\mu ,\varLambda _{1}}=\pi _{\mu ,\varLambda _{2}}\circ p_{\varLambda _{2},\varLambda _{1}}^{-1},\quad \forall \,\varLambda _{1},\varLambda _{2}\in \fancyscript{B}_{c}(M),\varLambda _{1}\subset \varLambda _{2} \end{aligned}$$

and thus, by the version of Kolmogorov’s theorem for the projective limit space $(\varGamma ,\fancyscript{B}(\varGamma ))$, the family $\{\pi _{\mu ,\varLambda }:\varLambda \in \fancyscript{B}_{c}(M)\}$ determines uniquely a measure $\pi _{\mu }$ on $(\varGamma ,\fancyscript{B}(\varGamma ))$ such that

$$\begin{aligned} \pi _{\mu ,\varLambda }=\pi _{\mu }\circ p_{\varLambda }^{-1},\quad \forall \,\varLambda \in \fancyscript{B}_{c}(M) \end{aligned}$$

The next step is to compute the characteristic functional of the measure $ \pi _{\mu }$. Given a $\varphi \in \fancyscript{D}(M)$ we have supp$\,\varphi \subset \varLambda $ for some $\varLambda \in \fancyscript{B}_{c}(M)$, meaning that

$$\begin{aligned} \langle \gamma ,\varphi \rangle =\langle p_{\varLambda }(\gamma ),\varphi \rangle ,\quad \forall \,\gamma \in \varGamma \end{aligned}$$

Thus

$$\begin{aligned} \int _{\varGamma }e^{i\langle \gamma ,\varphi \rangle }d\pi _{\mu }(\gamma )=\int _{\varGamma _{\varLambda }}e^{i\langle \gamma ,\varphi \rangle }d\pi _{\mu ,\varLambda }(\gamma ) \end{aligned}$$

and the definition (17) of the measure $\pi _{\mu ,\varLambda }$ yields for the right-hand side of the equality

$$\begin{aligned} \sum _{n=0}^{\infty }\frac{\exp (-\mu (\varLambda ))}{n!}\int _{\varLambda ^{n}}e^{i(\varphi (x_{1})+\cdots +\varphi (x_{n}))}d\mu ^{\otimes n}(x)=\sum _{n=0}^{\infty }\frac{\exp (-\mu (\varLambda ))}{n!}\left( \int _{\varLambda }e^{i\varphi (x)}d\mu (x)\right) ^{n} \end{aligned}$$

which corresponds to the Taylor expansion of the characteristic function (15) of the infinite-dimensional Poisson measure

$$\begin{aligned} \exp \left( \int _{\varLambda }(e^{i\varphi (x)}-1)\,d\mu (x)\right) \end{aligned}$$

This shows that the probability measure on $(\fancyscript{D}^{^{\prime }}(M), \fancyscript{C}_{\sigma }(\fancyscript{D}^{^{\prime }}(M)))$ given by (15) is actually supported on generalized functions of the form $\sum _{x\in \gamma }\delta _{x}$, $\gamma \in \varGamma $. Thus, the infinite-dimensional Poisson measure $\pi _{\mu }$ can either be considered as a measure on $(\varGamma , \fancyscript{B}(\varGamma ))$ or on $(\fancyscript{D}^{\prime },\fancyscript{C}_{\sigma }( \fancyscript{D}^{\prime }(M)))$. Notice that, in contrast to $\varGamma $, $ \fancyscript{D}^{\prime }(M)\supset \varGamma $ is a linear space. Since $\pi _{\mu }(\varGamma )=1$, the measure space $(\fancyscript{D}^{\prime }(M),\fancyscript{C} _{\sigma }(\fancyscript{D}^{\prime }(M)),\pi _{\mu })$ can, in this way, be regarded as a linear extension of the Poisson space $(\varGamma ,\fancyscript{B} (\varGamma ),\pi _{\mu })$.

Appendix 2. Complete Monotonicity of the Mittag-Leffler Function for Complex Arguments

A positive $C^{\infty }$-function $f$ is said to be completely monotone if for each $k\in \mathbb {N}_{0}$

$$\begin{aligned} (-1)^{k}f^{(k)}(t)\ge 0,\quad \forall t>0 \end{aligned}$$

According to Bernstein’s theorem (see e.g. [25], Chap. XIII.4 Theorem 1]), for functions $f$ such that $f(0^{+})=1$ the complete monotonicity property is equivalent to the existence of a probability measure $\nu $ on $ \mathbb {R}_{0}^{+}$ such that

$$\begin{aligned} f(t)=\int _{0}^{\infty }e^{-t\tau }\,d\nu (\tau )<\infty ,\quad \forall \,t>0 \end{aligned}$$

Pollard in [26] proved the complete monotonicity of $E_{\alpha } $, $0<\alpha <1$, for non-positive real arguments showing that

$$\begin{aligned} E_{\alpha }(-t)=\int _{0}^{\infty }e^{-t\tau }\,d\nu _{\alpha }(\tau ),\quad \forall \,t\ge 0 \end{aligned}$$

(18)

for $\nu _{\alpha }$ being the probability measure on $\mathbb {R}_{0}^{+}$

$$\begin{aligned} d\nu _{\alpha }\left( \tau \right) :=\alpha ^{-1}\tau ^{-1-1/\alpha }f_{\alpha }(\tau ^{-1/\alpha })\,d\tau \end{aligned}$$

(19)

where $f_{\alpha }$ is the $\alpha $-stable probability density given by

$$\begin{aligned} \int _{0}^{\infty }e^{-t\tau }f_{\alpha }\left( \tau \right) \,d\tau =e^{-t^{\alpha }},\quad 0<\alpha <1 \end{aligned}$$

The complete monotonicity property and the integral representation (18) of $E_{\alpha }$ may be extended to complex arguments.

Lemma 1

For any $z\in \mathbb {C}$ such that $\mathrm {Re}(z)\ge 0$, the following representation holds

$$\begin{aligned} E_{\alpha }(-z)=\int _{0}^{\infty }e^{-z\tau }\,d\nu _{\alpha }(\tau ),\quad 0<\alpha \le 1 \end{aligned}$$

Proof

According to [26], for each $0<\alpha <1$ fixed, for all $t\ge 0$ one has

$$\begin{aligned} E_{\alpha }(-t)= & {} \int _{0}^{\infty }e^{-t\tau }\,d\nu _{\alpha }(\tau ), \nonumber \\= & {} \sum \limits _{n=0}^{\infty }\frac{(-t)^{n}}{n!}\int _{0}^{\infty }\tau ^{n}\,d\nu _{\alpha }(\tau ) \end{aligned}$$

(20)

Comparing (20) with the Taylor expansion (7) of $ E_\alpha $, one concludes that the moments of the measure $\nu _\alpha $ are given by

$$\begin{aligned} m_{n}(\nu _{\alpha }):=\int _{0}^{\infty }\tau ^{n}\,d\nu _{\alpha }(\tau )= \frac{n!}{\varGamma (\alpha n+1)},\quad n\in \mathbb {N}_{0} \end{aligned}$$

For complex values $z$ let

$$\begin{aligned} I(-z):=\int _{0}^{\infty }e^{-z\tau }\,d\nu _{\alpha }(\tau ) \end{aligned}$$

which is finite provided $\mathrm {Re}(z)\ge 0$. For each $z\in \mathbb {C}$ such that $\mathrm {Re}(z)\ge 0$ one then obtains

$$\begin{aligned} I(-z)=\sum _{n=0}^{\infty }\frac{\left( -z\right) ^{n}}{n!}\left( \int _{0}^{\infty }\tau ^{n}\,d\nu _{\alpha }(\tau )\right) =\sum _{n=0}^{\infty }\frac{\left( -z\right) ^{n}}{n!}m_{n}(\nu _{\alpha })=\sum _{n=0}^{\infty }\frac{\left( -z\right) ^{n}}{\varGamma (\alpha n+1)} =E_{\alpha }(-z) \end{aligned}$$

leading to the integral representation

$$\begin{aligned} E_{\alpha }(-z)=\int _{0}^{\infty }e^{-z\tau }d\nu _{\alpha }(\tau ) \end{aligned}$$

for all $z\in \mathbb {C}$ such that $\mathrm {Re}(z)\ge 0$.$\blacksquare $

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Maria João Oliveira
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Rui Vilela Mendes
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1.Universidade Aberta and Centro de Matemática e Aplicações FundamentaisLisbonPortugal
2.Centro de Matemática e Aplicações FundamentaisLisbonPortugal

Fractional Boson Gas and Fractional Poisson Measure in Infinite Dimensions

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