Diagrammatical methods within the path integral

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Diagrammatical methods within the path integral representation for quantum systems

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17th International Conference on Recent Progress in Many-Body Theories (MBT17) IOP Publishing Journal of Physics: Conference Series 529 (2014) 012003 doi:10.1088/1742-6596/529/1/012003

Diagrammatical methods within the path integral representation for quantum systems A Alastuey1 1

Laboratoire de Physique, ENS Lyon/CNRS, 69364 Lyon Cedex 07, France

E-mail : [email protected] Abstract The path integral representation has been successfully applied to the study of equilibrium properties of quantum systems for a long time. In particular, such a representation allowed Ginibre to prove the convergence of the low-fugacity expansions for systems with short-range interactions. First, I will show that the crucial trick underlying Ginibre’s proof is the introduction of an equivalent classical system made with loops. Within the Feynman-Kac formula for the density matrix, such loops naturally emerge by collecting together the paths followed by particles exchanged in a given cyclic permutation. Two loops interact via an average of twobody genuine interactions between particles belonging to different loops, while the interactions between particles inside a given loop are accounted for in a loop fugacity. It turns out that the grand-partition function of the genuine quantum system exactly reduces to its classical counterpart for the gas of loops. The corresponding so-called magic formula can be combined with standard Mayer diagrammatics for the classical gas of loops. This provides low-density representations for the quantum correlations or thermodynamical functions, which are quite useful when collective effects must be taken into account properly. Indeed, resummations and or reorganizations of Mayer graphs can be performed by exploiting their remarkable topological and combinatorial properties, while statistical weights and bonds are purely c-numbers. The interest of that method will be illustrated through a brief description of its application to two long-standing problems, namely recombination in Coulomb systems and condensation in the interacting Bose gas.

1

Introduction

The standard formalism for studying the equilibrium properties of interacting quantum systems is the many-body perturbation theory [1]. Thermodynamic quantities are systematically expanded in powers of the interactions, and any contribution in the corresponding series can be represented by a Feynman graph built with loops describing free propagation, which are connected at vertices by interaction lines. The structure of Feynman graphs is determined by well-defined topological

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rules, while their contributions follow from conservation laws prescriptions at the vertices. The many-body perturbation theory is well-suited when interactions can be indeed treated as weak perturbations. This occurs for instance at high temperatures, or at high densities for fermions since the corresponding Fermi kinetic energy then becomes quite large. In this paper, we will present another formalism, based on the Feynman path integral representation of quantum mechanics [2]. Within that representation, one can introduce an equivalent classical gas made of loops with interactions generated by imaginary-time averages of the genuine particle interactions along the paths defining the shapes of the loops. Interestingly, standard methods of classical statistical mechanics can be straightforwardly extended to the gas of loops, in particular the familiar Mayer series which are quite useful at low densities. Furthermore, the classical nature of the loops allows one to proceed to systematic reorganizations and resummations of the diagrammatical series, which are crucial for taking into account collective effects. Eventually, we stress that interactions are not treated perturbatively, an essential feature for dealing with phenomena like recombination for instance. The usefulness of the loop formalism will be illustrated here through its applications to two longstanding problems in statistical mechanics of quantum systems. First, we will study recombination into atoms or molecules in a quantum plasma made with nuclei and electrons (see e.g. Ref. [3]). The second problem concerns the existence of Bose-Einstein (BE) condensation in an interacting Bose gas in the absence of external fields (see ch.2 in Ref. [4]). The paper is organized as follows. In Section 2, we define the Hamiltonian of the general non-relativistic models with two-body interactions which can be studied within the path integral formalism. Then, we present the two examples considered further, namely the hydrogen plasma and an interacting Bose gas. In Section 3, after introducing the Feynman-Kac representation in the simple case of a single particle submitted to an external potential, we sketch the main steps of the transformations which lead to the introduction of a gas made with classical loops, which displays a grand-partition function identical to that of the genuine quantum system of interest. According to the corresponding so-called magic formula, we show that the equilibrium quantities of the quantum system can be represented by Mayer-like series for the equivalent gas of loops. The formalism is first applied to the hydrogen plasma in Section 4, where further resummations and reorganizations of Mayer series provide a series representations in terms of graphs made with particle clusters. The corresponding Screened Cluster Representation allows us to define and compute unambiguously the contributions of any recombined chemical species at finite temperatures. We turn to the second application in Section 5, namely the study of condensation in an interacting Bose gas. We show that, within the celebrated Kac scaling, the mean-field results can be readily retrieved from Mayerlike series. Contributions of fluctuations are also determined, but far from the critical region where a condensate might emerge. Eventually, we give some concluding comments in Section 6.

2


2 2.1

Non-relativistic quantum systems with two-body interactions The model

In many situations encountered on Earth or in Astrophysics, a suitable description relies on the introduction of a quantum system made of point particles with Hamiltonian HN = −

N 1 2 Δi + uαi αj (|xi − xj |) . 2mαi 2 i=1

(1)

i=j

Here, αi denotes the species of the ith particle, while Δi is the Laplacian with respect to its position xi . In addition to the non-relativistic kinetic energy, the total potential energy is a sum of pairwise two-body instantaneous interactions uαi αj (|xi − xj |) which only depend on the relative distance |xi − xj | between particles. Notice that the non-relativistic Hamiltonian (1) does not depend on the spins σαi of the particles, which only determine their bosonic or fermionic nature.

2.2

Examples

A first example is the hydrogen plasma, which can be viewed, within the so-called physical picture, as a mixture of quantum point particles which are either protons or electrons, interacting via the instantaneous Coulomb potential. Protons and electrons have respective charges, masses, and spins, ep = e and ee = −e, mp and me , σp = σe = 1/2, while the two-body interaction reduces to uαi αj (|xi − xj |) = eαi eαj v(|xi − xj |)

(2)

with v(r) = 1/r. Here both species are fermions, and the two-body Coulomb interaction is longranged. A second example is an atomic gas described as a system of neutral point particles with mass m and spin σ = 0. The corresponding two-body interaction potential u(r) includes, in general, both a short-range repulsive part and a long-range attractive part, like the familiar Lennard-Jones potential. Now, the particles are bosons. Notice that, at a more fundamental level, the atomic system should be also viewed as a quantum plasma made of point nuclei and point electrons with two-body Coulomb interactions, similarly to the case of the Hydrogen plasma. Nevertheless, the more phenomenological description in terms of an interacting Bose gas, is a priori sufficient at rather low temperatures and low densities. In fact, the difficult questions about recombination and effective interactions would be precisely investigated for the Hydrogen plasma.

3 3.1

Path integral representation and the equivalent gas of loops The Feynman-Kac formula

For the sake of pedagogy, we introduce the path integral representation in the simple case of a single particle with mass m submitted to a potential V (r). Its Hamiltonian reads H=−

2 Δ + V (r) . 2m

3

(3)


The corresponding density matrix at a given temperature T , namely the matrix element of Gibbs operator exp(−βH), is exactly given by Feynman-Kac formula [5, 6, 7, 8] exp[−(rb − ra )2 /(2λ2 )] rb | exp(−βH)|ra = D(ξ) (2πλ2 )3/2 1 ds V ((1 − s) ra + s rb + λ ξ(s))] , (4) × exp[−β 0

with thermal de Broglie wavelength λ = (β2 /m)1/2 . In the r.h.s. of (4), ξ(s) is a dimensionless Brownian bridge which starts from the origin at dimensionless time s = 0 and comes back at the origin at dimensionless time s = 1, i.e. ξ(0) = ξ(1) = 0. Functional measure D(ξ) is the normalized Gaussian Wiener measure which characterizes the Brownian process, and it is entirely defined by its covariance D(ξ) ξμ (s) ξν (t) = δμν inf(s, t) (1 − sup(s, t)) . (5) The corresponding functional integration is performed over all Brownian bridges ξ(s). Representation (4) is the proper mathematical formulation of genuine Feynman’s idea, which amounts to express the density matrix as a sum over all possible paths going from ra to rb in a time β, of weighting factors exp(−S/) where S is the classical action of a given path computed in potential −V . Here, such paths are parametrized according to ωab (sβ) = (1 − s) ra + s rb + λ ξ(s) ,

(6)

where (1 − s) ra + s rb describes the straight uniform path connecting ra to rb . An example of path is shown in Fig. 1. Also, weighting factor exp(−S/) is splitted into the product of three terms. The first term, which arises from the kinetic energy of the straight uniform path, reduces to the Gaussian prefactor in front of the functional integral. The second term, associated with the kinetic contribution of the Brownian part of the path, is a Gaussian functional of ξ(s) embedded in Wiener measure D(ξ). The third and last term is rewritten as the Boltzmann-like factor associated with time average 1

0

ds V (ωab (sβ))

(7)

of potential V along the considered path ωab . We stress that, independently of the rather poetic introduction of path integrals by Feynman [2], representation (4) can be derived in a straightforward way by starting from the obvious identity exp(−βH) = [exp(−βH/N )]N combined with a suitable insertion of (N − 1) closure relations in position-space (see e.g. Ref. [9]). Feynman-Kac formula (4) then follows by taking the limit N → ∞, as it has been proved for a wide class of potentials [5]. Feynman-Kac (FK) representation (4) perfectly illustrates the intrusion of dynamical features in equilibrium static quantities for quantum systems. Let us consider the diagonal density matrix ra | exp(−βH)|ra . Because of the non-commutativity of the kinetic and potential parts of H, that matrix element does not reduce to its classical counterpart exp(−βV (ra )) , (2πλ2 )3/2

4

(8)


ωab (sβ)

rb

ra 0 Figure 1: A Brownian path ωab (sβ). The dashed straight line is the uniform path connecting ra to rb . so it is not entirely determined by V (ra ). In fact, according to formula (4) specified to rb = ra , ra | exp(−βH)|ra now depends on the potential landscape in some neigbourhood of ra with size λ, which is explored in time-average (7) thanks to Brownian motion. Therefore, ra | exp(−βH)|ra appears to be indeed generated by a dynamical process. Not surprisingly, particle mass m controls the importance of the corresponding dynamical effects. In particular, in the limit of an infinitely heavy particle m → ∞, λ = (β2 /m)1/2 vanishes and ra | exp(−βH)|ra obviously tends to its classical counterpart (8) : dynamical effects do not intervene anymore in the potential contribution which takes its purely static form. Quantum corrections to classical formula (8) can be expanded in powers of 2 according to the well-known Wigner-Kirkwood expansion [10, 11, 12]. That expansion can be easily retrieved within FK representation (4), by expanding time-average (7) in power series of λξ and by applying Wick theorem to the calculation of the resulting moments of ξ. Notice that the present dynamical considerations are the manifestation, in the framework of path integrals, of the Heisenberg uncertainty principle which prevents the particle to stay at ra . Hence, Brownian paths can be interpreted as describing intrinsic quantum fluctuations of position. Remarkably, FK representation (4) involves only classical objects and c-numbers, so the operatorial structure of quantum mechanics is, in some sense, erased. That feature turns out to be particularly useful in the framework of the many-body problem, as described further. However, the intrinsic complexity of quantum mechanics is now hidden in the functional integration over all Brownian bridges, which remains a formidable task. In fact, explicit calculations can be performed in a few number of cases, as reviewed in Ref. [8]. Also, aymptotic Wigner-Kirkwood expansions of ra | exp(−βH)|ra around the classical formula (8), can be derived for situations where λ becomes ∇V (ra )|/|V (ra )|]−1 of variation of V (r) in the neighsmall compared to the characteristic length [|∇ bourhood of ra . Such situations occur for very heavy particles or for high temperatures, and also at large distances for potentials which decay as power laws at infinity. In the opposite limit where λ diverges, direct estimations of FK functional integral become rather cumbersome. Notice that, if H has a single isolated boundstate with energy E0 and wavefunction ψ0 , in the zero-temperature limit, the asymptotic behaviour of ra | exp(−βH)|ra is merely extracted from its spectral representation, i.e. ra | exp(−βH)|ra ∼ |ψ0 (ra )|2 exp(−βE0 ) when T → 0 . (9)

5


As argued in Ref. [13], the relevant paths which provide the low-temperature behaviour (9) occupy a small piece of the whole functional phase space, and they are quite different from the typical paths with divergent size λ. Consequently, an exact direct estimation of their contribution remains an open problem in general.

3.2

The density matrix

Let us come back now to the many-body problem in the framework of the grand-canonical ensemble. The system is enclosed in a box with volume Λ, in contact with a thermostat at temperature T and a reservoir of particles that fixes the chemical potentials μα of the various species. The corresponding grand-partition function reads ΞΛ = TrΛ exp −β(HN − μ α Nα ) . (10) α

The trace TrΛ is taken over a complete basis of N-body wavefunctions, which are symmetrized according to the statistics of each species, and satisfy Dirichlet boundary conditions at the surface of the box, while particle numbers Nα vary from 0 to ∞. The grand-partition function (10) can be expressed in the basis of positions and spins, where a given state is the suitably symmetrized Slater sum of products of one-body states |x σαz . This provides a sum of diagonal and off-diagonal matrix elements of exp(−βHN ). In the case of the hydrogen plasma, an example of above density matrix elements involved in the expression of the grand-partition function, is R1 R3 R2 r2 r3 r1 r4 | exp(−βH3,4 )|R1 R2 R3 r1 r2 r3 r4 ,

(11)

for the Hamiltonian H3,4 of 3 protons and 4 electrons. Here, the positions of two protons are exchanged, as well as those of three electrons. Contributions of spins are factored out in simple degeneracy factors because the Coulomb Hamiltonian does not depend on the spins. For matrix element (11), that multipliyng degeneracy factor is 24 , because the spin-states of the exchanged particles are necessarily identical. The FK representation for each of the matrix elements of exp(−βHN ) takes a form similar to formula (4), with Nα paths ω α for each species α, as well as a Boltzmann-like factor associated with the time average of the potential part of HN . The paths associated with matrix element (11) can be collected into loops. In fact, that property holds for any matrix element of exp(−βHN ), because any permutation can always be decomposed as a product of cyclic permutations. A loop L is constructed by collecting q paths associated with q particles exchanged in a cyclic permutation. Accordingly, L is characterized by its position X, which can be arbitrarily chosen among the extremities of paths ω α , and several internal degrees of freedom which are particle species (α), number q of exchanged particles, and shape λα η obtained as the union of the q paths ω α . It turns out that η(s) is itself a Brownian bridge with flight time q, i.e. η(0) = η(q) = 0, distributed with the corresponding Wiener measure D(η). In the FK representation, the time-average of the total interaction potential can be obviously rewritten as a sum of two-body interactions V between loops, plus a sum of loop self-energies U , namely 1 V(Li , Lj ) + U (Li ) , (12) 2 i

i=j

6


L3

r1

L1 R1

r2

r3 L4

L2

r4

R3

R2

Figure 2: The four loops associated with matrix element (11). Loops L1 and L2 contain respectively 1 proton and 2 protons, while loops L3 and L4 contain respectively 3 electrons and 1 electron. The positions of the particles involved in the matrix element (11) are denoted by black disks with larger radius for the protons, and a smaller one for the electrons. The Brownian paths generated in the FK representation of (11), which go from one particle position to another one, are represented by thick and thin lines for the protons and the electrons respectively. where loops associated with the considered matrix element are labelled as Li with index i running from 1 to their total number N . For instance, four loops can be identified in the FK representation of matrix element (11), as shown in Fig. 2. Two-body potential V(Li , Lj ) between loops Li and Lj reduces to a time-average along their respective shapes of the genuine two-body particle interaction v(|Xi + λαi ηi (s) − Xj − λαj ηj (t)|) evaluated at times which differ by an integer value. Self-energy U (Li ) for loop Li is given by a similar average along its own shape of v(|λαi ηi (s) − λαi ηi (t)|) evaluated at times which differ by a non-zero integer value, with a prefactor 1/2 which avoids double counting of genuine interactions between two exchanged particles.

3.3

The magic formula

At this stage, grand-partition function (10) is rewritten as a sum of Boltzmann-like factors associated with energies (12) multiplied by combinatorial factors and particle fugacities zα /(2πλ2α )3/2 with zα = exp(βμα ), which have to be integrated over positions and shapes of the involved loops. It turns out that the whole sum can be rewritten as the grand-partition function of a classical gas of undistinguishable loops with suitable activities z(L), namely (loop)

ΞΛ = Ξ Λ

=

∞ N 1 dLi z(Li ) exp(−βV(Li , Lj )) . N!

N =0

i=1

(13)

i ρ0,c (β), a macroscopic condensation of particles in state k = 0 takes place. It is well-known that the Bose-Einstein condensation is directly related to the appearance of off-diagonal long-range order (ODLRO) in the one-body density matrix, which can be viewed as the relevant order-parameter for that transition in the interacting system (see e.g. [33]). In the ideal case, it has been shown that lim

(1)

|r−r |→∞

and lim

r|D0 |r = 0

(1)

for

r|D0 |r = ρ − ρ0,c (β)

|r−r |→∞

18

ρ < ρ0,c (β)

for ρ > ρ0,c (β) ,

(33)

(34)


(1)

where D0 is the ideal one-body density matrix. For the interacting system, the eigenstates of the Hamiltonian no longer reduce to tensorial products of one-body states, so the possible persistence of the BE condensation has to be related to the existence of ODLRO for the one-body density matrix (1) r|D0 |r above some critical density. (1) Within the equivalent gas of loops, the off-diagonal matrix element r|D0 |r reads [34] ∞ (1) r|D0 |r = D(η) q exp[−(r − r )2 /(2qλ2 )]ρ(Lr,r ) . (35) q=1

Here ρ(Lr,r ) is the density of open loops Lr,r starting from r and going to r along a Brownian path (1 − (s/q)) r + (s/q) r + λ η(s) , (36) (1)

with s varying from 0 to q. Accordingly, r|D0 |r can be represented by Mayer-like graphs analogous to those appearing in series (15) for the particle density where now the root loop L0 is replaced by the open loop Lr,r . The emergence of ODLRO is related to the behaviour of ρ(Lr,r ) for large open loops with large values of q.

5.2

The mean-field Kac limit

Let us introduce a family of two-body potentials uγ (r) defined by the following parametrization uγ (r) = γ 3 v(γr)

(37)

where γ is a real positive parameter and v(r) is a given positive, integrable, and spherically symmetric pair potential. When dimensionless parameter γ approaches zero, the potential v being kept fixed, the potential uγ (r) becomes weak and long range, whereas the value of the integral ∞ dr uγ (r) = 4π dr r2 v(r) = a > 0 (38) 0

remains constant. In the strict limit γ = 0, one can expects that particles feel a mean-field constant potential aρ. In other words, the system should then be driven by the so-called mean-field Hamiltonian [35]. That heuristic prediction has been rigorously proved by Pulé and Zagrebnov [36]. More precisely, ODLRO still persists, while the ideal critical density remains unchanged. Furthermore, below ρ0,c (β), the density is the unique solution of the self-consistent equation 1 1 . (39) dk ρ = ρ0 (β, μ − aρ) = 3 (2π) exp(β( (k) + aρ − μ)) − 1 which defines the mean-field equation of state ρ = ρmf (β, μ). For ρ > ρ0,c (β), that equation reduces to the simple linear law ρ = μ/a. Notice the simple shift aρ in the one-body energy spectrum involved in the right hand side of formula (39), in agreement with the mean-field Hamiltonian picture. Interestingly, the Mayer-like series (15) for ρ provide a self-consistent equation which takes the general form ([37]) ρ = Fγ (β, μ − aρ) (40)

19


where function Fγ (β, ν) is defined by a series of multiconnected Mayer graphs, which do converge for ν sufficiently negative, and uniformly with respect to γ. For γ = 0, F0 reduces to the sum of tree graphs, which is nothing but the ideal density function ρ0 (β, ν), so relation (40) does coincide with equation (39) as it should.

5.3

Contributions of fluctuations beyond the mean-field limit

Now the central question concerns the effects of fluctuations beyond the mean-field limit, and in particular their influence on the possible emergence of ODLRO. The answer requires to study the case of γ small but finite. Such a program has been partially accomplished by using the Mayer-like series for the equivalent gas of loops. For finite negative values of ν, Fγ (β, ν) can be expanded in powers of γ 3 . The first correction to F0 (β, ν) of order γ 3 , is non-trivial, because various resummations of infinite sets of graphs have to be performed in order to properly take into account collective effects arising from the long range nature of the two-body potential uγ (r). That correction reduces to the sum of contributions from tree and ring graphs [34], (1)

(1)

γ 3 [Ftree + Fring ] with

(1)

Fring

βv(0) (1) (1) [f (ν) − f (2) (ν)] , Ftree = 2 [ˆ v (|q|)]2 β2 dq f (1) (ν) f (2) (ν) = 2(2π)3 1 + β f (1) (ν) vˆ(|q|)

(41)

(42) (43)

and

∂ρ0 ∂ 2 ρ0 (β, ν) , f (2) (ν) = β −2 (β, ν) . (44) ∂ν ∂ν 2 Inserting the correction (41) into the self-consistent equation (40), one can compute the γ 3 corrections to the mean-field equation of state for fixed strictly negative values of ν = μ − aρ. Unfortunately, the coefficients of the γ 3 -terms diverges when ν → 0, i.e. when one approaches the critical region where a condensate might emerge. Thus, the above perturbative expansion in powers of γ 3 does not allow one to answer the question of existence of condensation for an arbitrarily small but fixed value of γ [38]. In order to analyze the behaviours of interest in the critical region for γ small but finite, various strategies using Mayer-like series for the equivalent loop gas have been proposed. A first insight should be gained by investigating an approximate version of the self-consistent equation (40), where Fγ (β, ν) is replaced by the contributions of tree graphs only, ∞ exp(βqν) 1 D(η) exp(−βU(L)) . (45) Fγ(tree) (β, ν) = q (2πλ2 )3/2 f (1) (ν) = β −1

q=1

The central problem would then be to determine the behaviour, for simultaneously large values of q and small values of γ, of the average over all loop shapes η of the Bolzmann factor associated with the self-energy U (L) of loop L made with q exchanged particles. Another strategy, more ambitious, would be to analyze exactly the possible emergence of ODLRO within formula (35). In that approach, one would have to determine the behaviour of the density of open loops ρ(Lr,r ), for both large values of |r − r | and q, and small values of γ.

20


6

Concluding remarks and perspectives

The classical nature of the loop gas allows one to perform exact and useful reorganizations of diagrammatical series with : X Non-perturbative treatment of interactions X Incorporation of collective effects X Systematic account of fluctuations beyond mean-field approaches

Various applications are actually in progress. First, the ionic and molecular internal partition functions for hydrogen are determined within Quantum Monte Carlo numerical computations of their path integral reprsentations. Furthermore, the Screened Cluster Representation will be applied to the hydrogen-helium mixture in order to study the corresponding equation of state at low temperatures and low densities. For the interacting Bose gas, the preliminary results described in Section 5 are promising, and there is some reasonable hope to estimate contributions of large loops, by analogy with the exact results derived for self-repelling closed polymers. However, we notice that there exist alternative methods, like that based on the analysis of the hierarchy equations for the imaginary-time Green functions [39] which might be more efficient.

References [1] Fetter A L and Walecka J D 1971 Quantum theory of many particle systems (McGraw-Hill, New York) [2] Feynman R P and Hibbs A R 1965 Quantum mechanics and path integrals (McGraw-Hill, New York) [3] Ebeling W, Kraeft D and Roepke G 2012 Ann. Phys. 524 311 [4] Seiringer R 2010 Hot topics in cold gases XVIth International Congress on Mathematical Physics (Ed. Pavel Exner, World Scientific, Prague) pp 231-245. [5] Simon B 1979 Functional Integration and Quantum Physics (Academic, New York) [6] Schulman L S 1981 Techniques and Applications of Path Integrals (Wiley, New York) [7] Roepstorff G 1994 Path Integral Approach to Quantum Physics (Springer, Berlin) [8] Kleinert H 2004 Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets (World Scientific) [9] Alastuey A, Magro M and Pujol P 2008 Physique et outils mathématiques : méthodes et exemples (EDP Sciences, CNRS Editions) [10] Wigner E P 1932 Phys. Rev. 40 749 [11] Kirkwood J G 1934 Phys. Rev. 45 116

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[12] Landau L D and Lifchitz E M 1958 Quantum mechanics vol.5 of the course of Theoretical Physics (Pergamon Press, Oxford) [13] Paulin S, Alastuey A and Dauxois T 2007 J. Stat. Phys. 128 1391 [14] Ginibre J 1971 Statistical Mechanics and Quantum Field Theory edited by C. DeWitt and R. Stora (les Houches, Gordon and Breach) [15] Cornu F 1996 Phys. Rev. E 53 4562 [16] Martin P A 2003 Acta Physica Polonica B 34 3629 [17] Mayer J E and Montroll E W 1941 J. Chem. Phys. 9 2 [18] Hansen J P and Mc Donald I R 1986 Theory of simple liquids (Academic Press, London) [19] Lieb E and Lebowitz J 1972 Adv. Math. 9 316 [20] Saha M 1920 Philos. Mag. 40 472 [21] Macris N and Martin P A 1990 J. Stat. Phys. 60 619 [22] Fefferman C 1985 Rev. Math. Iberoamericana 1 1 [23] Brillouin L 1930 Les Statistiques Quantiques et leurs Applications (Presse Univ. de France, Paris) [24] Larkin A I 1960 Zh. exp. teor. Fiz. 38 1896 [25] Balleneger V. 2012 Ann. Phys. 524 103 [26] Ballenegger V, Martin P A and Alastuey A 2002 J. Stat. Phys. 108 169 [27] Alastuey A, Ballenegger V, Cornu F and Martin P A 2003 J. Stat. Phys. 113 455 [28] Rogers F J 1974 Phys. Rev. A 10 2441 [29] Alastuey A, Ballenegger V, Cornu F and Martin P A 2008 J. Stat. Phys. 130 1119 [30] Alastuey A and Ballenegger V 2012 Cont. Plasma. Phys. 52 95 [31] Ebeling W 1967 Ann. Phys. Leipzig 19 104 [32] Alastuey A and Ballenegger V 2012 Phys. Rev. E 86 066402 [33] Ziff R M, Uhlenbeck G E and Kac M 1977 Physics Reports 32C 169 [34] Martin P A and Piasecki J 2005 Phys. Rev. E 71 016109 [35] Lewis J T 1986 Lecture Notes on Physics 257 pp 234 (Springer, New York) [36] Pulé J V and Zagrebnov V A 2004 J. Phys. A: Math. Gen. 37 8929

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[37] Martin P A and Piasecki J 2003 Phys. Rev. E 68 016113 [38] Martin P A 2005 International Journal of Theoretical Physics 44 2021 [39] Alastuey A and Piasecki J 2011 Phys. Rev. E 84 041122

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Diagrammatical methods within the path integral

Diagrammatical methods within the path integral

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