On Bose occupancy problem with randomized energy levels

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On Bose occupancy problem with randomized energy levels

Thierry Huillet, Flora Koukiou

To cite this version:

Thierry Huillet, Flora Koukiou. On Bose occupancy problem with randomized energy levels. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2008, Volume 41 (Number 6), pp.065002.

�hal-00199939�

On Bose occupancy problem with randomized energy levels

T. Huillet and F. Koukiou

Laboratoire de Physique Th´ eorique et Mod´ elisation, CNRS-UMR 8089 et Universit´ e de Cergy-Pontoise, 2, Avenue Adolphe Chauvin, 95302, Cergy-Pontoise, France

September 11, 2007

Abstract

Assuming energy states to be an independent and identically dis- tributed positive sequence, a randomized version of the classical Bose occupancy problem is investigated.

Keywords: Disordered systems (Theory), Bose Einstein condensa- tion (Theory), Energy landscapes (Theory), Extreme value problems.

1 Introduction

Classical Bose occupancy problem consists in assigning indistinguishable parti- cles on a prescribed deterministic energy states sequence. The purpose of this work is to initiate the study of the following related problem: Assuming now en- ergy states to constitute an independent and identically distributed sequence of positive random variables, we wish to investigate the corresponding randomized version of classical Bose occupancy problem. This simple model does not seem to have received much attention in the past, to the best of the authors knowl- edge. Under such conditions, using a grand canonical approach, a Bose-Einstein like condensation phenomenon is shown to take place provided the probability mass at zero of energies is small enough (else, probability mass of energies is not

“too strongly concentrated” at zero); we shall describe this phase transition in some details and give some illustrative examples. The origin of this phenomenon seems to result from the conjunction of two main effects: The randomness of the energy levels (the system is disordered) and the indistinguishability of particles to be allocated. When there is condensation, we shall discuss the problems of understanding the ground state occupancy at critical density, together with the number of distinct occupied states. Some limit laws are obtained in some cases.

Also, some aspects of ground state occupancy in the condensed phase will be investigated. Finally, using a description of occupancies of the ranked energy states (the sampling formula), we shall derive the grand-canonical statistics of states with a prescribed amount of particles. This statistical aspect concerns the fluid phase only. We report on a Poisson factorization of this distribution in a specific low temperature, low density limit.

2 Randomized Bose samples

2.1 The model and some of its statistical features

Let En := (E1, .., En) be a system of independent and identically distributed (iid) positive energies whose law will be specified later on (at this step, we shall however further assume thatE₁ has a densityf₁ with support⊆R+ and that the largest value from E_n goes to∞ almost surely). Letβ >0 be the inverse of temperature. We shall presently study the randomized energy version of the classical allocation model of statistical mechanics where energy levels are originally chosen deterministically. For some related aspects of the randomized problem, see Johnson and Kotz [12]. The problem under study can now be formulated as follows.

Let there be k indistinguishable particles to place on the random energy levels (or states)En. Conditionally onEn (for quenched disorder), letKn,k(m) denote the occupancy of energy level m with Bose equilibrium collective law given by

PE_n(Kn,k(m) =km;m= 1, .., n) = 1 Zk,E_n(β)

n

Y

m=1

e^−βEmkm. (2.1)

With z^k

f(z) the coefficient of z^k in the power-series expansion of f(z) the normalizing term reads

Zk,E_n(β) = X

k⁰₁+..+k⁰_n=k

k0 1,..,k0

n≥0

n

Y

m=1

e^−βEmk0m = z^k

n

Y

m=1

1−ze^−βEm−1

. (2.2)

The distribution defined by Eqs. (2.1, 2.2) favors configurations with minimal (interaction free) energy H_n,k(E_n) := Pn

m=1E_mk_m. Given E_n, it is the one maximizing entropy under the assumption that the average energy is fixed and β stands for the conjugate parameter. Here and throughout, the subscriptEn

in PE_n (orEE_n) will denote conditional probability (or expectation) givenEn. Remarks:

(i) In Eq. (2.1),k_m∈N0 :={0,1,2, ..},m= 1, .., nwith no restriction but k1+..+kn=k. Imposing the additional condition thatkm∈ {0,1},m= 1, .., n (the Pauli exclusion principle), would lead to a Fermi-Dirac occupancy problem which we shall not further develop specifically.

(ii) The classical Bose occupancy problem ([3]) can be obtained by con- sidering an ordered deterministic energy sequence _(n) := ₍₁₎, .., _(n)

where ₍₁₎ < .. < _(n), instead of En used in (2.1, 2.2). Typically, _(m) = m^α for some α >1; see [4], for example. We therefore wish to investigate the role of randomization of the energy sequence, requiring an additional averaging over disorder. ♦

Indeed, given there are k particles, averaging Eq. (2.1) over disorder E_n, the searched Bose unconditional canonical occupancy probability clearly is

P(Kn,k(m) =km;m= 1, .., n) =EPE_n(Kn,k(m) =km;m= 1, .., n). (2.3)

As a symmetric function of thekm, this annealed distribution is exchangeable.

In particular, E(Kn,k(m)) = k/n, m = 1, .., n. Further, for instance, with k1∈ {0, .., k}andE_n\1:= (E2, .., En)

PEn(K_n,k(1) =k₁) = e^−βE1k1 Zk,E_n(β)

X

k2+..+kn=k−k1

n

Y

m=2

e^−βEmkm,

= e^−βE1k1Z_k−k1,E_n\1(β) Z_k,En(β) and

P(K_n,k(1) =k₁) =E

"

e^−βE1k1Z_k−k1,En\1(β) Zk,E_n(β)

#

is the one-dimensional marginal ofK_n,k(m) =k_m;m= 1, .., n. Except for very particular cases as in the example which follows, a closed-form expression of Eq.

(2.3) seems out of reach.

Example: As a limiting example of Bose distribution, whenβ ↓0, the joint law of Kn,k(m) ; m = 1, .., n tends to the uniform Bose-Einstein distribution (see Feller [10], and Holst, [11])

P(Kn,k(m) =km;m= 1, .., n) = 1

n+k−1 k

and the 1−dimensional distribution at high temperature reads

P(Kn,k(1) =k1) =

n+k−k1−2 k−k1

n+k−1 k

,k1= 0, .., k.

Before investigating some additional statistical properties of the Bose distri- bution displayed in Eq. (2.3), let us first stress immediate consequences of the model.

• Energy of a single particle: Assume first that k = 1. Then the sam- ple is made of a single particle. From Bose law, this particle will launch on energy level number m with conditional probability e^−βEm/ZE_n(β) where ZE_n(β) :=Z1,E_n(β) =Pn

1e^−βEm. The corresponding energy attached to this single particle, sayEn,has conditional law given by

PEn(E_n =E_m) =e^−βEm/Z_En(β) =:V_m,n(β) , m= 1, .., n.

The random probabilities V_m,n(β) satisfy Pn

1V_m,n(β)^a.s.= 1 and so constitute a random partition of unity. Let nowEEn e^−λEn

be the conditional Laplace- Stieltjes transform (LST) of En. Then EE_n e^−λEn

= ZE_n(λ+β)/ZE_n(β).

Averaging overEn, E e^−λEn

:=EEEn e^−λEn

=E[Z_En(λ+β)/Z_En(β)]

characterizes the distribution of the single particle energy. In particular, E(En) =∂βE[−logZEn(β)].

In this energy-biased picking procedureE_n → E_n, states with low energy levels are clearly favored. One therefore expects E_n to be stochastically smaller than the typical energy levelEn from the iid sequenceEn.

To study the Sherrington-Kirkpatrick model for spin glasses, similar ques- tions were raised for different purposes by Derrida in the early eighties, under the (ruled out here) hypothesis that energies were normally distributed (see [5]- [7]).

•Total energy: Let now Hn,k :=Pn

m=1EmKn,k(m) stand for total energy of the k particles system where occupancies law is as in Eq. (2.1); its LST is clearly given by

E

e^−λHn,k

=E

Z_k,En(λ+β) Zk,E_n(β)

, λ≥0.

Let us now show that, when the number of energy levels is fixed, the pro- portions of particles tend to concentrate on the ground state as the number of particles increases.

Proposition 1 For eachm∈ {1, .., n}, letE_n\m:={E1, ..E_m−1, E_m+1, ..E_n}. Withnfixed, as the number of particles grows, conditionally given E_n, we have

Kn,k(m)

k ;m= 1, .., n

k↑∞→

Vm,n:=I{^Em<E_n\m};m= 1, .., n (2.4)

in distribution.

Proof:Let us first consider the ordered versionE_(n)of the energy sequence En, namely: E(n) := E(1),n, .., E(n),n

with E(1),n < .. < E(n),n. Let also Ee_(m),n :=E_(m),n−E_(1),n >0, m = 1, .., n.Developing the product partition function

n

Y

m=1

1−ze^−βEm−1

=

n

Y

m=1

1−ze^{−βE(m),n−1}

appearing in Eq. (2.2) into a sum of rational fractions, extracting itsz^k−coefficient and isolating the ground-state term, we get

Z_k,En(β) =Z_k,E(n)(β) =

e^−βE(1),nk



 Y

m6=1

1 1−e^−βEe(m),n

+

n

X

m=2

e^{−βEe(m),nk} Y

l6=m

1

1−e^−β(^E(l),n−E_(m),n)



.

Inserting this expression into Eq. (2.1), with K(n),k(m) ;m = 2, .., n the occupancies of energy levels E(2),n, .., E(n),n and km ∈ N0;m = 2, .., n, such that Pn

m=2k_m≤k, we get

PE_(n) K_(n),k(m) =k_m;m= 2, .., n

= (1−ε(k))

n

Y

m=2

n

e^{−βEe(m),nkm}

1−e^−βEe(m),no (2.5)

where ε(k) = e^{−βEe(2),nk}Q

l6=1

1−e^−βEe(l),n /Q

l6=2

1−e^−β(^E(l),n−E(2),n) is a negative corrective term that goes to 0 exponentially fast with k becoming large. Supposekm=bksmcfor some fixedsm∈(0,1] ;m= 2, .., n.In this case, the probability displayed in Eq. (2.5) goes to 0 when k goes to ∞: in other words, the probabilities of K_(n),k(m)/k;m = 2, .., nall concentrate on 0 and therefore all the probability mass goes to ground state (m= 1). This is state- ment in Eq. (2.4) whereI_{A}stands for the set indicator of the eventA. Note that

Vm,n:=I{^Em<En\m};m= 1, .., n

is an exchangeable partition of unity.

In particular, for eachm,E(V_m,n) = 1/n andσ²(V_m,n) = (1−1/n)/n.

• Some remarks on Maxwell-Boltzmann version of the occupancy problem:

This limiting behavior of occupancies is in sharp contrast with the one arising in a Maxwell-Boltzmann version of the randomized occupancy problem. Let us briefly address this point.

WithPn

1Vm,n(β) ^a.s.= 1, the random numbers Vm,n(β) = P^en^−βEm 1e^−βEm have been shown to be the conditional Bose probabilities that a single particle hits statemwith energyE_m. The partitioning systemV_n(β) := (V_1,n(β), .., V_n,n(β))

constitute an exchangeable random partition of unity (in particular, for eachm, E(Vm,n(β)) = 1/n). The random quantitiesVn(β) are essential in the deriva- tion of the Maxwell-Boltzmann sampling distribution version of the random- ized occupancy problem. It proceeds as follows: let (U1, .., Uk) bekiid uniform throws on [0,1] partitioned byVn(β). Let (Bn,k(1), .., Bn,k(n)) be an integral- valued random vector which counts the number of visits to the different energy levels in a k−sample in the following sense: if Ml is the random state label which thel−th trial hits, then Bn,k(m) :=Pk

l=1I_{Ml=m},m= 1, .., n.Clearly PV_n(β)(Ml=m) =Vm,n(β) and, givenVn(β), statemis chosen proportionally to its “size” Vm,n(β).

With (b1, .., bn)∈Nⁿ0, satisfyingPn

m=1bm=k,(Bn,k(m) =bm;m= 1, .., n) now follows the conditional multinomial distribution:

PV_n(β)(B_n,k(m) =b_m;m= 1, .., n) = k!

Qn m=1bm!

n

Y

m=1

V_m,n^bm (β). Averaging overVn(β) gives the Maxwell-Boltzmann distribution

P(B_n,k(m) =b_m;m= 1, .., n) =EPV_n(β)(B_n,k(m) =b_m;m= 1, .., n). Clearly, proceeding in this way to fill up sequentially the energy levels, particles are assumed distinguishable and, by strong law of large numbers, conditionally givenEn

B_n,k(m)

k ;m= 1, .., n

k↑∞→ (V_m,n(β) ;m= 1, .., n) almost surely and in distribution (compare with Eq. (2.4)).

Note that computational problems also arise in the canonical ensemble, since, averaging overEn, one needs to evaluate the awkward quantities

E

" _n Y

m=1

V_m,n^bm (β)

#

=E

"

e^{−βPnm=1Embm} (Pn

m=1e^−βEm)^k

# .

Due to these computational difficulties, canonical randomized Maxwell Boltz- mann statistics turns out to be as intricate as Bose one. We shall not enter into more details on this question and rather focus on the Bose occupancy problem as defined by Eqs. (2.1-2.2).

•Random allocation scheme representation of Bose distribution: From Eqs.

(2.1-2.2), we observe immediately the distributional identity:

(K_n,k(m) =k_m;m= 1, .., n)= (ξ^d ₁, .., ξ_n|ζ_n =k) (2.6)

where (ξ1, .., ξn) is iid onNⁿ0 conditioned by its sumζn:=Pn

1ξm=k and P(ξ₁=k₁) =E[PEn(ξ₁=k₁)] =E

e^−βE1k1 1−e^−βE1

,k₁∈N0, (2.7)

a mixture of geometric distribution with success probabilitye^−βE1. Such a con- ditional representation of the occupancies is called a random allocation scheme property in Kolchin, [13].

Letρ >0 be the density of particles per state and letκn:=bnρc. Proposition 2 Assume E

h

e^α+βE1−1−1i

= ρ admits a solution for α :=

α(β, ρ)>0. Then

K_n,κn(1)→^d_n↑∞K_ρ, (2.8)

where, withα:=α(β, ρ),Kρ is a mean-ρmixture of geometric distribution with probability generating function

E u^Kρ

=E

1−e^−(α+βE1) 1−ue^−(α+βE1)

, u∈[0,1]. (2.9)

Proof: This follows from the above allocation representation property and the Gibbs conditioning principle. The random variable Kρ is the asymptotic typical cell occupation at densityρof particles per state. Its law can be obtained by exponential-tilting the ones ofξm,m= 1, .., nnamely, withPEn(ξm=km) = e^−βEmkm 1−e^−βEm

, by considering :

P

ξe_m,α=k₁ :=E

PEn

ξe_m,α=k_m

=E

"

e^−αkmPEn(ξ_m=k_m) P

k_m≥0e^−αkmPEn(ξm=km)

# .

Given E_n, ξe_m,α, m = 1, .., n are therefore geometrically distributed with iid success random probabilities e^−(α+βEm) and expected values e^α+βEm−1−1

. Applying Gibbs conditioning principle to obtain Eq. (2.8), one needs to adjust αso as to meet the conditioning requirementζen,α:=Pn

m=1ξem,α=bnρc. This will be possible at the only condition that there exists anα >0 such that

E[Kρ] =E h

e^α+βE1−1⁻¹i

=ρ.

(2.10)

This condition will appear more clearly later to separate, in some cases, between a fluid and a condensed phase. In case there is noαfor a givenρ(see below for examples in the condensed phase), there is no typical limiting cell occupation at densityρ.

•Gibbs randomization of sample size (variable particle number): As it was stressed above, the canonical conditional distributions given sample size iskare extremely difficult to evaluate in general (even when k = 1). To circumvent this drawback, we shall now assume the number of particles is variable and so randomize sample size. Letα >0 (α=βµwhereµ >0 is fugacity). Assume the

number of particlesK_n,αis now random with law given by the Gibbs probability measure

PEn(K_n,α=k) =e^−αkZ_k,En(β) ZE_n(α, β) (2.11)

where the grand canonical partition now reads Z_En(α, β) =X

k≥0

e^−αkZ_k,En(β) =

n

Y

m=1

1 1−e^−(α+βEm). (2.12)

Alternatively, withλ≥0, we have EEn e^−λKn,α

= Z_En(α+λ, β) ZE_n(α, β) =

n

Y

m=1

1−e^−(α+βEm) 1−e^{−(λ+α+βEm)} E e^−λKn,α

=

E

1−e^−(α+βE1) 1−e^{−(λ+α+βE1)}

ⁿ .

GivenEn, the expected number of particles can be obtained through:

EEn(Kn,α) =−∂αlogZEn(α, β).

Indexing now cell occupancies byαrather thank, withk_m∈N0;m= 1, .., n, the joint occupancies probability reads

PEn(K_n,α(m) =k_m;m= 1, .., n) = 1 ZE_n(α, β)

n

Y

m=1

e^{−(α+βEm)km},

andKn,α:=Pn

m=1Kn,α(m).

Letφ1(β) :=Ee^−βE1 be the Laplace-Stieltjes transform of E1. Averaging overEn,the unconditional occupancy probability is

P(Kn,α(m) =km;m= 1, .., n) :=EPEn(Kn,α(m) =km;m= 1, .., n) = E

" _n Y

m=1

e^{−(α+βEm)km}

1−e^−(α+βEm)

#

=

n

Y

m=1

E

he^{−(α+βE1)km}

1−e^−(α+βE1)i

=

n

Y

m=1

e^−αkm

φ₁(βk_m)−e^−αφ₁(β(k_m+ 1)) . (2.13)

It is exchangeable and under the form of a mere product measure. In particular, withk₁∈N0

P(Kn,α(1) =k1) =e^−αk1

φ1(βk1)−e^−αφ1(β(k1+ 1)) .

Stated differently, with ξ_1,α the N0−valued random variable with distribution P(ξ1,α≥k1) =e^−αk1φ1(βk1), we have

P(K_n,α(m) =k_m;m= 1, .., n) =

n

Y

m=1

P(ξ_m,α=k_m)

where (ξ1,α, .., ξn,α) is an iid sequence. This factorized representation turns out to be useful.

1/ For example, letK_n,α⁺ be the grand-canonical occupancy of state with the largest amount of particles. Then

P K_n,α⁺ ≤k

= 1−e^−αkφ1(βk)ⁿ

and one expects some Poisson approximation. However, following Embrechts et al [8], Theorem 3.1.3,p. 117, observing that

P(ξ1,α≥k1)

P(ξ_1,α ≥k₁−1) ≤e^−α∈(0,1) and lim

k1↑∞

P(ξ1,α≥k1) P(ξ_1,α≥k₁−1) <1, there is no sequence k_n such that ne^−αknφ₁(βk_n) → γ ∈ (0,∞). Therefore, whatever the sequence k_n is, there is no non degenerate limit of the form P K_n,α⁺ ≤k_n

→xwherex∈[0,1]\ {0,1}; in other words, for every sequence k_n such that P K_n,α⁺ ≤k_n

→ x, either x = 0 or 1 and there is no way to normalize K_n,α⁺ so as to obtain a non-degenerate limit distribution. The dis- creteness of the distributions involved prevents the maximum from converging properly to some extreme value distribution and instead forces this oscillatory behavior.

2/ The grand canonical distribution of total energy: Let us consider the LST of the random variableHn,α. Withλ≥0 andkm∈N0,it is

E e^−λHn,α

=E



 X

k_m;m=1,..,n

e^{−λPnm=1kmEm}PE_n(Kn,α(m) =km;m= 1, .., n)





=E

n

Y

m=1

X

km

e^{−(α+(λ+β)Em)km}

1−e^−(α+βEm)

!

=E

n

Y

m=1

1−e^−(α+βEm) 1−e^{−(α+(λ+β)Em)}

!

=

E

1−e^−(α+βE1) 1−e^{−(α+(λ+β)E1)}

ⁿ .

This shows that Hn,α is the sum of n iid random variables whose increment has LST given by E

1−e^{−(α+βE1 )} 1−e^{−(α+(λ+β)E}1 )

. In particular, by strong law of large

numbers, H_n,α/nconverges a.s. to the mean energy per state, namely Hn,α

n →n↑∞E

E₁ e^α+βE1−1

.

2.2 A condensation phenomenon

The origin of the condensation phenomenon to be described now results from the conjunction of two main effects of the model under study: the randomness of the energy levels (the system is disordered) and the indistinguishability of particles to be allocated. Phase transitions induced by disorder is well-known in related contexts (see [14]).

•A Bose-Einstein like condensation phenomenon: It remains to interpretα in more details. Observing that

EE_n(Kn,α(m)) =−1

β∂E_mlogZE_n(α, β) = 1 e^α+βEm−1 we have

EE_n(Kn,α) =

n

X

m=1

1 e^α+βEm−1 ν : =EEE_n(Kn,α) =nE

e^α+βE1−1⁻¹ .

Assuming the number of levelsnto be the extensive variable of such a mean-field model, we thus get the particle density per state

ρ:= ν n =E

e^α+βE1−1⁻¹ . (2.14)

Usually, it is therefore of interest to try to deduce α from (ρ, β) which are the physical quantities which are known in practice. Letting z :=e^−α denote fugacity, developing the argument of the expectation, this is also the implicit state equation

ρ=zhβ(z) (2.15)

wherehβ(z) has the power series representation hβ(z) =X

i≥0

zⁱhi(β) withhi(β) =φ1(β(i+ 1)). (2.16)

Letzc >0 be the convergence radius of the series Hβ(z) :=zhβ(z) =X

i≥1

zⁱφ1(βi).

Two cases arise: z_c= 1 orz_c>1.Ifz_c>1,h_β(1) exists, is monotone decreasing and mapsβ ∈(0,∞) onto (∞,0) ; let thereforeβc be defined byρ=hβ_c(1). If zc= 1, letβc be defined as before ifHβ(zc)<∞and βc:=∞ifHβ(zc) =∞.

In any case, forβ < βc, by B˝urmann-Lagrange inversion formula, α=−logz wherez∈(0,1∧zc) and

z= 1 +X

l≥1

ρ^l

l hl(β) with hl(β) :=

z^l−1

hβ(z)^−l. (2.17)

A Bose-Einstein condensation-like phenomenon therefore pops in when the prob- ability that energies take values close to 0 are small enough. We shall now supply three examples.

• Examples: In example (i), the probability mass in 0 of energy is chosen log-algebraically large. In the subsequent example (ii), the probability mass of energy in 0 is exponentially small. These two examples show that a con- densation like phenomenon occurs if the probability distribution of energy in a neighborhood of 0 is small enough (with growth at least linear). In a last extreme example (iii), the support of the law ofE1 is (0,∞),0 >0; ground state₀ is away from 0 with probability 1.

(i) Assume the distribution of energyE₁, sayF₁() :=R

f₁(⁰)d⁰, satisfies:

F1()∼_↓0C·^θ/L(1/)

where C >0,θ ≥0 and L(x) a slowly varying at ∞function (such as logx).

The smaller θ is, the more energy is concentrated at 0. Then by Karamata- Tauberian Theorem (see Bingham et al, [2])

φ₁(β)∼β↑∞C·Γ (1 +θ)·β^−θ/L(β)

andφ₁(β) is regularly varying at∞. For allθ≥0,β >0, the convergence radius of H_β(z) is z_c = 1. Ifθ ≤1,H_β(z_c) =∞ whereas forθ > 1, H_β(z_c)<∞. If θ≤1, there is no condensation whereas whenθ >1,a critical valueβc ofβ can be defined. When θ >1, the order of differentiability of z →Hβ(z) atzc = 1 isbθc −1.

In all such cases,φ1(2β)/φ1(β)∼_β↑∞2^−θdoes not tend to 0 whenβ ↑ ∞.

(ii) Assume density ofE1 satisfies: f1()∼_↓0C·^−3/2e^−1/: a small value of E1 is rather unlikely (exponentially small). Then, φ1(β)∼_β↑∞C·e⁻

√β so thatzc= 1 withHβ(zc)<∞.In this case,φ1(2β)/φ1(β) does tend to 0 when β ↑ ∞.

This example is a particular case of the related one for which: f1()∼↓0C· e^−1/cfor somec >0.In this case, a small value ofE1is exp-algebraically small

and, by applying the saddle point method: φ1(β)∼_β↑∞C·β^−2(c+1)c+2 e^{−B·βc/(c+1)} for some constantB >0.Here again, φ1(2β)/φ1(β)→0 when β↑ ∞.

(iii) Let0>0 denote spectral gap anda >0. Assumef1() =ae^−a(−0), > 0(a truncated exponential model).Thenφ1(β) = _1+β/a^e−β0.Herezc=e^β0 >

1 and H_β(z_c) = ∞. Note that H_β(z) is infinitely differentiable (smooth) at z= 1< zc.In this case,βc is uniquely determined byP

i≥1

e^−βc0i 1+(βci)/a =ρ.

Remark: In this approach, the free parameter is densityρand the Lagrange inversion formula only holds for β < β_c (temperature is high enough). If β is the free parameter, Lagrange inversion formula only is valid forρ < ρ_c :=h_β(1) (the free Bose gas has small enough density). In the (β, ρ) plane, the critical lineρc =:ρc(β) separates a diluted fluid phaseρ < ρc from a condensed phase ρ > ρc.Note thatρc(β) =E

e^βE1−1−1 . ♦ One can summarize shortly the results as follows:

Proposition 3 We have:

(i)If E(1/E1) =∞, there is no condensation and the only available phase is fluid.

(ii)If E(1/E₁)<∞, there is condensation: the critical line separating the condensed (ρ > ρ_c) from fluid (ρ < ρ_c) phases has equation ρ = ρ_c(β) = E

e^βE1−1−1

which is also the convergent series

ρ=ρc(β) =X

i≥1

φ1(βi) involving the LST of energies.

•Thermodynamics of the diluted fluid phase: Internal energy readsU_n(α, β) = E(U_En(α, β)) where

U_En(α, β) =−∂βlogZ_En(α, β) =

n

X

m=1

E_m e^α+βEm−1. Observing thatE

E₁ e^α+βE1−1

<∞we get U_En(α, β)

n

a.s.→

n↑∞E

E₁ e^α+βE1−1

=:u(α, β).

Further, U_n(α, β) = n·u(α, β). The quantity u(α, β) is internal energy per state.

Assume Elog 1−e^−(α+βE1)

<∞ for each (α, β) ∈ R²+. With N_n() :=

Pn

m=1I_{Em≤}, let the empirical distribution function (or density of states) of En be Dn() := n⁻¹Nn(). Let FE_n(α, β) := −_β¹logZE_n(α, β) be the conditional free energy. Then, by the strong law of large numbers

−1

nFE_n(α, β) = 1 nβ

n

X

m=1

log

1−e^−(α+βEm)

= 1

β Z

R+

log

1−e^−(α+β)

Dn(d) ^a.s.→

n↑∞

1 βElog

1−e^−(α+βE1)

=:f(α, β) and

F_n(α, β) :=E(F_En(α, β)) = n βElog

1−e^−(α+βE1)

=n·f(α, β). The quantityf(α, β) is free energy per state. We havef(α, β)>_β¹log (1−e^−α).

As usual, we can define entropySn(α, β) as

Sn(α, β) =β(Un(α, β)−Fn(α, β)).

Clearly,S_n(α, β) =n·s(α, β) wheres(α, β) =β(u(α, β)−f(α, β)) is entropy per state.We have

s(α, β) =E

βE₁ e^α+βE1−1

−Elog

1−e^−(α+βE1) .

Remark: Assumen↑ ∞,ν ↑ ∞in such a way that particle densityρ=ν/n exists. Then, using the equation of state as defined by Eqs. (2.15-2.16), there is an expression ofαin terms ofρandβ,provided the system is in the diluted phase. The thermodynamical quantities per stateu(α, β), f(α, β) ands(α, β) can better be expressed in terms ofρandβ. The functionss(α, β) andβf(α, β) areβ−Legendre conjugates.

The existence of a limit for the thermodynamical quantities per state when the system becomes infinite is clear. See Beghian and Sheldon, [1], for similar considerations. ♦

3 Bose randomized occupancy and order statis- tics

In practice, it is convenient to order the energy levels; by doing so, dependence is naturally introduced. Let thenE_(n):= E_(1),n, .., E_(n),n

be the order statistics of En, with E_(1),n < .. < E_(n),n. As is well known that the joint probability density of this vector is

f_{E(1),n,..,E(n),n}(₁, .., _n) =n!

n

Y

m=1

f₁(_m)·I_{1<..<n}. (3.1)

Thermodynamics under ranked energy levels is the same as the one under un- ranked ones. This is basically because for symmetric in the arguments function- alsϕ(En)^a.s.= ϕ E_(n)

; conditionally onEn, thermodynamic functions involved separable and additive functionals of the form: ϕ(En) =Pn

1g(Em) for differ- ent g. However, Pk_mE_m

a.s.

6= Pk_mE_(m),n and this is why sampling underE_n does not reduce to sampling under E_(n).

For instance, givenE_(n), the joint probability of occupanciesK_(n),α(m) of E_(m),n;m= 1, .., nnow reads

PE_(n) K_(n),α(m) =km;m= 1, .., n

= 1

ZE_(n)(α, β)

n

Y

m=1

e⁻(^α+βE(m),n)^km

Z_E(n)(α, β) =

n

Y

m=1

1

1−e⁻(^α+βE(m),n).

Exchangeability is lost; averaging overE_(n),the unconditional occupancy prob- ability is

P K_(n),α(m) =k_m;m= 1, .., n

:=EPE_(n) K_(n),α(m) =k_m;m= 1, .., n

=

E

" _n Y

m=1

e⁻(^α+βE(m),n)^km

1−e⁻(^α+βE(m),n)

#

=

n!

Z

1<..<n

n

Y

m=1

e^{−(α+βm)km}

1−e^−(α+βm)

f1(m)dm.

The occupation numbers of statemwith energyE_(m),n,namelyE K_(n),α(m) , present some interest in practice.Here for instance, we get a randomized version of Planck statistics

E K(n),α(m)

= X

km≥1

P K(n),α(m)≥km

=E

1 e^α+βE(m),n−1

where, withF1 the distribution function ofE1 (F1 := 1−F1), E_(m),n>0 has density

n!

(m−1)! (n−m)!f1()F1()^m−1F1()^n−m.

Let us investigate the full ground-state occupancy distribution in more details.

Lemma 4 The grand canonical distribution of ground state occupancy is

P K_(n),α(1)≥k1

=e^−αk1

1−β Z ∞

0

1−F1

x k₁

ⁿ

e^−βxdx

. (3.2)

Proof: With E_(n)\1 := E_(2),n, .., E_(n),n

, φ₍₁₎(β) := Ee^−βE(1),n and k₁ ∈ N⁰, the ground stateE_(1),n occupancy is

P K_(n),α(1) =k₁

= E

"

e⁻(^α+βE(1),n)^k1Z_E(n)\1(α, β) ZE_(n)(α, β)

#

= E

h

e⁻(^α+βE(1),n)^k1

1−e⁻(^α+βE(1),n)i

= e^−αk1

φ₍₁₎(βk₁)−e^−αφ₍₁₎(β(k₁+ 1)) .

With F₁(₁) := P(E₁> ₁), F₍₁₎(₁) := P E_(1),n> ₁

, recalling F₍₁₎(₁) = F₁(₁)ⁿ, we get

φ(1)(β) :=E

e^−βE(1),n

= 1−β Z ∞

0

F1(1)ⁿe^−β1d1. Therefore Eq. (3.2) holds.

This last formula allows to extract some limiting information on the ground state occupancy at critical point.

•A limit law at criticality from example(i): AssumeF1()∼↓0C·^θ with θ >1, as in Example (i). Approaching critical point,α↓0, ρ↑ρc(β) =Hβ(1) and so, withk_n,c:=nρ_c(β) the critical number of particles in the system

P

K_(n),0(1)≥k^1/θ_n,cz

= 1−β Z ∞

0

1−F1

x k^1/θn,cz

!!n

e^−βxdx

∼_n↑∞1−β Z ∞

0

e

−nF1 x k1/θ

n,c z

!

e^−βxdx∼_n↑∞1−β Z ∞

0

e^−ρcC(x^z)^−θe^−βxdx=:G(z) where it can be checked thatG(z), z >0, as a scale exponential mixture of the Fr´echet distributione^{−ρcCz−θ}, is a complementary distribution function of some random variableZ >0. Thus, we obtain

Corollary 5 Fix β > 0 and assume F1() ∼_↓0 C·^θ. Then, with kn,c :=

nρc(β) the critical number of particles in the system at temperature β⁻¹, at critical point

K_(n),0(1) kn,c^1/θ

→dn↑∞Z.

(3.3)

Ground state occupancy grows like kn,c^1/θ which is close to nρ_c only as θ ap- proaches 1from above.

•Some remarks on the condensed phase:Fixβ >0 and assume condensation takes place. In the super-critical condensed phaseρ > ρc(β), there is noαsuch that Bose condition holds. Therefore, the grand canonical approach is useless to describe the condensed phase and, with κn =bnρc, one needs to use back the canonical ensemble description Kn,κ_n(m), m = 1, .., n of the occupancy problem. We have the following result:

Proposition 6 Assume E 1/E₁²

<∞, so that E(1/E1)<∞ and condensa- tion takes place. Let ρ > ρc be the density of particles in the condensed phase.

Under this assumption, almost surely

1 n

n

X

m=2

e^−βEe(m),n 1−e^−βEe(m),n

n↑∞→ ρc =ρc(β) =E

e^−βE1 1−e^−βE1

<∞

so that a proportionn(ρ−ρc)of particles goes to ground state, while the residual nρc is scattered on all remaining energy levels.

Sketch of proof: First, the minimal valueE(1),nfromEn goes to 0 almost surely. Under fairly general assumptions indeed (see Embrechts et al, [8]), the almost sure behavior of partial minima reads

lim inf

n↑∞

E_(1),n

_n = 1 a.s.

where_n ↓0 is defined byF₁(_n) =n⁻¹.We shall sayE_(1),n∼_n. Let nowg(x) := e^βx−1−1

.If there is condensation (or ifE(1/E1)<∞), then as n↑ ∞

1

ng E_(1),n

→0

because, in the worst case E_(1),n ∼ n^−1/θ, θ > 1 and then ¹_ng E_(1),n

∼ n^1/θ−1→0.

Recall now from Eq. (2.5) that when k = κ_n is large, the joint law of occupancy numbers K_(n),κn(m) ; m = 2, .., nfor all states but ground state is (up to a constant going exponentially fast to 0 with κ_n ↑ ∞) close to the one of a sequence of independent geometrically distributed random variables with success probabilities e^−βEe(m),n, whereEe_(m),n :=E_(m),n−E_(1),n (see the proof of Proposition 1).Let us therefore investigate the largenbehavior of the sample mean number of particles

1 n

n

X

m=2

e^−βEe(m),n 1−e^−βEe(m),n

=: 1 n

n

X

m=2

g

Ee_(m),n

on all states but ground state. RecallingE_(1),ngoes to 0 almost surely, we have g

Ee_(m),n

∼ 1

e^βE(m),n 1−βE_(1),n

−1

∼ 1 e^βE(m),n−1

1 + βE_(1),n 1−e^−βE(m),n

.

Therefore, 1 n

n

X

m=2

g

Ee_(m),n

∼ 1 n

n

X

m=2

g E_(m),n

+βE_(1),n n

n

X

m=2

e^−βE(m),n 1−e^−βE(m),n2. Under the additional assumption E 1/E₁²

<∞,the corrective term goes to 0 whenn↑ ∞(withE_(1),n↓0). Indeed,

βE_(1),n n

n

X

m=2

e^−βE(m),n 1−e^−βE(m),n2 = βE_(1),n 1

n

n

X

m=1

e^−βE(m),n

1−e^−βE(m),n2− 1 n

e^−βE(1),n 1−e^−βE(1),n2

!

and, by strong law of large numbers, 1

n

n

X

m=1

e^−βE(m),n

1−e^−βE(m),n2 →σ²=E

e^−βE1 (1−e^−βE1)²

!

which is a finite variance ifE 1/E₁²

<∞. Next, βE_(1),n

n

e^−βE(1),n

1−e^−βE(1),n2 ∼ 1

nβE_(1),n →0.

Therefore, _n¹Pn m=2g

Ee(m),n

∼_n¹Pn

m=1g E(m),n

= ¹_nPn

m=1g(Em) and by strong law of large numbers

1 n

n

X

m=2

g

Ee_(m),n

n↑∞→ ρ_c =ρ_c(β) =E[g(E₁)].

In a random scheme of allocating particles, it is said that a giant component appears if, ask, n→ ∞while the ratioρ=k/nis fixed toρ > ρ_c, the maximum K_(n),k(1) of the random variablesK_n,k(m),m= 1, .., n, with probability tend- ing to 1 is of ordernand the second order statistic has a limit distribution with normalizing constant of order less than n. In the condensed phase, K_(n),k(1) (the number of particles attached to ground state) is of order of magnitude n(ρ−ρc). Let us investigate K_(n),k(2) (the number of particles on state just above ground state) and check this from an example.

First,P E_(1),n> ₁

=F(₁)ⁿ and, form= 1, .., n−1 we have the Markov property:

P E_(m+1),n> m+1|E_(m),n=m, .., E_(1),n=1

=

P E_(m+1),n> m+1|E_(m),n=m

=

F1(m+1) F₁(_m)

^n−m

; m+1> m

involving the cumulative probability distribution function of the minimum of n−miid observations drawn fromF1truncated at m. Ifm= 1, with2> 1

P E(2),n> 2|E(1),n=1

=

F1(2) F₁(₁)

ⁿ⁻¹

and so with >0 P

Ee_(2),n> |E_(1),n=1

=

F1(1+) F1(1)

ⁿ⁻¹

=

1−F1(1)−F1(1+) F1(1)

ⁿ⁻¹

From this, whennis large and ifF1()∼↓0C·^θ,θ >1, we can indeed estimate E_(1),nandEe_(2),nto be both of order of magnituden^−1/θ.Therefore, the average occupancy of stateE_(2),n isg

Ee_(2),n

∼n^1/θ which is of order less thann.

4 Sampling formulae

• Sampling formula: Let p ∈ {1, .., n} and 1 ≤ m1 < .. < mp ≤ n be an increasing subsequence of {1, .., n}. Let M1, .., Mp be the labels of occupied energy levels and Pn,α denote the number of such occupied states. Withkq ∈ N:={1,2, ..}, a realization of occupanciesK_(n),α(mq),we clearly have Proposition 7 (sampling formula)

PE_(n) Mq =mq, K(n),α(mq) =kq;q= 1, .., p;Pn,α=p

=

p

Y

q=1

e⁻

α+βE(mq),n

kq

1−e⁻

α+βE(mq),n

Y

q6=1,..,p

1−e⁻

α+βE(mq),n

.

Averaging over E_(n) with joint law given by Eq. (3.1), gives the unconditional probability of the event M_q =m_q, K_(n),α(m_q) =k_q;q= 1, .., p;P_n,α=p.

Summing next the above conditional probability overk_q;q= 1, .., p, we get PE_(n)(Mq=mq;q= 1, .., p;Pn,α=p)

(4.1)

= Y

q6={1,..,p}

1−e⁻

α+βE(mq),n

×

X

k₁,..,k_p≥1 p

Y

q=1

e⁻

α+βE(mq),n

kq

1−e⁻

α+βE(mq),n

= Y

q6={1,..,p}

1−e⁻

α+βE(mq),n

^p Y

q=1

e⁻

α+βE(mq),n

.

Averaging overE(n), using the joint law ofE(n),we get the unconditional prob- abilityP(Mq=mq;q= 1, .., p;Pn,α=p) that onlyp≤nof the ordered energy levels with labelsmq are occupied. In particular

E ( _n

Y

q=2

h

1−e⁻(^α+βE(q),n)i

e⁻(^α+βE(1),n) ) (4.2)

is the unconditional probability that only ground state E_(1),n is occupied.

Summing over the ⁿ_p

sequencesm1< .. < mp, withp∈ {0, .., n}, we obtain a binomial distribution for the number of occupied energy states:

P(P_n,α=p) = n

p

e^−αφ₁(β)^p

1−e^−αφ₁(β)n−p

.

Note thatP(Pn,α= 0) = (1−e^−αφ1(β))ⁿ which is the Gibbs probability that there is no particle in the system: Kn,α= 0.

Corollary 8 Fixβ >0and letk_n,c :=nρ_c(β)be the critical number of particles in the system. Then, as n↑ ∞,α↓0, approaching critical point from the fluid phase

Pn,α

nρ ∼ Pn,0

k_n,c

→d φ1(β) ρ_c(β) <1.

(4.3)

Proof: With λ≥0, the LST of ^P_k^n,0

n,c reads E

e^−λ

Pn,0 kn,c

= 1−

1−e^−kn,cλ

φ1(β)ⁿ

∼e⁻

λφ1 (β) ρc(β) .

From the state equation, ρ_c(β) =H_β(1) =P

i≥1φ₁(βi)> φ₁(β). At critical point, the degenerate (non random) fraction ^φ_ρ^1(β)

c(β) of states is occupied.