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Some applications of the Lambert W function to classical statistical mechanics

We apply the recently defined Lambert W function to some problems of classical statistical mechanics, i.e. the Tonks gas and a fluid of classical particles interacting via repulsive pair potentials. The latter case is considered both from the point of view of the standard theory of liquids and in the framework of a field theoretical description. Some new mathematical properties of the Lambert W function are obtained by passing.


I. INTRODUCTION
The Lambert W function is defined as the multivalued inverse of the function w → we w .
Its mathematical properties have been explored only quite recently after its implementation in the mathematical library of the computer algebra program Maple. The history, applications and properties of Lambert W are reviewed in Corless et al. 1 Mathematical developments and applications to physics (mainly in quantum mechanics and electrostatics) can be found in refs. 1,2,3,4. Since W is a very simple function, all its applications to physics or other fields are not exhausted, and, after a short enquiry, I have discovered that many of my colleagues, aware or unaware of its name, have met the function in their own works. 5 Here we reformulate some old and new problems of the statistical mechanics of classical liquids in terms of Lambert W.
Our paper is organized as follows. In sec. II we commence by refreshing the reader ideas with some known mathematical properties of W (z). We focus on the principal branch W 0 (z) of the Lambert function and the related function U 0 (z) = W 0 (z) + W 2 0 (z)/2, both being of prime importance for their applications to the problems of statistical mechanics considered in this article. Some new mathematical properties of W 0 (z) and U 0 (z) used in the paper are presented in the appendix (the Legendre transforms of the real functions x ∈ R , x → W 0 (exp(x)) and x → U 0 (exp(x)) are computed and the dispersion relations for W 0 (z) and U 0 (z) established).
In sec. III we reformulate the properties of the Tonks gas, 6 i.e. a one-dimensional (1D) classical fluid of hard rods, in terms of the principal branch W 0 (z) of the Lambert function.
Our presentation is compared with the seminal work of Hauge and Hemmer on this system. 7 The introduction of Lambert W 0 (z) allows to recover all the properties of the model in a unified treatment.
In sec. IV we focus on the second system considered By Hauge and Hemmer in ref. 7, i.e. a classical 3D fluid made of particles with intermolecular pair repulsions of the form where φ is a positive integrable function. The extension to arbitrary dimension is trivial. In the limit γ → 0 which will be considered, we deal with infinitely weak and infinitely long range repulsive interactions. Once again, the use of the functions W 0 (z) and U 0 (z) is of a great help for describing in a unified manner all the known properties of this model. A brief survey of the properties of the principal branch W 0 (z) will be sufficient for our purpose. Firstly, W 0 (z) is analytic at z = 0. This follows from the Lagrange inversion theorem 9 . Its power series about the point z = 0 reads as The radius of convergence of the series (2.2) is equal to e −1 as it is easily shown using the Cauchy test and, within the circle of convergence, |W (z)| < 1. As a consequence of the relation (2.1) one has , where the radius of convergence of the series (2.3b) is again equal to e −1 . Lagrange theorem gives more than eq. (2.2); it allows to obtain power series for W α 0 (z), 1/(1 + W 0 (z)), exp(αW 0 (z)), etc for z within the circle of convergence of W 0 (z). 3 In particular, the power series of the function U 0 (z) defined to be W 0 (z) + W 0 (z) 2 /2 is given by Note that if follows from eq. (2.3a) that where p = (2(ez + 1) (the series converges for |p| < √ 2).
In physics, it is quite common (and sometimes enlighting) to write dispersion relations for functions having a branch cut such that W 0 (z) and U 0 (z). 10 We show in the appendix that, in the present case, these relations can be recast under the form where z is an arbitrary complex number which however is not on the branch cut. These equations convey the interpretation of W 0 (z) (resp. U 0 (z)) as the two-dimensional complex electric potential created by a distribution of charges g W (s) (resp. g U (s)) located on the branch cut. Moreover, we show in the appendix that the distribution g W (s) is normalized to unity (cf eq. (A.14)) while the total charge of the charge distribution g U (s) is infinite. Other physical interpretations of the relations (2.7) will be discussed in forthcoming developments.

III. THE TONKS GAS
In this section we consider the Tonks gas, i.e. a 1D fluid of hard rods of length σ. 6 The equation of state (EOS) is known exactly: (P pressure, β = 1/kT , T temperature, ρ number density). Of course 0 < ρσ < 1. For ρσ → 0 one recovers the EOS of the ideal gas, while for ρσ → 1, which corresponds to the close packing of the hard rods, the pressure diverges. Less trivial is the expression of the pressure as a function of the activity z. Recall that z = exp(βµ) where µ is the chemical potential and we have assumed that the deBroglie thermal wavelength Λ = 1. Starting from the thermodynamic relation it is not difficult to obtain the relation 7 σχe σχ = σz . did not know the Lambert function as we do; clearly one has since the other branches W k of Lambert W will not give a real pressure for real chemical potentials µ. The Mayer expansions for the pressure and the density of the Tonks gas are therefore obtained by copying out eqs. (2.2) and (2.3b): The results of sec. II on Lambert W enable us to conclude that 1. The radius of convergence of the Mayer series (3.5a) and (3.5b) is R = 1/eσ. Within the circle of convergence |χ(z)| < σ −1 .
2. The pressure χ(z) is singular at the point z = −R of the circle of convergence, i.e.
the branch point of W 0 (σz), in agreement with one of the conclusions of the second theorem of Groeneveld. 11 3. χ(z) has a branch cut on the negative real axis: −∞ < s ≤ −R. It can be identified with the distribution of zeros of the grand partition function in the thermodynamic limit. 12,13 The distribution of the Yang-Lee zeros is given by g(s) = g W (σs) = −ℑW ′ 0 (σs)/π (cf eqs. (2.7a) and (A.13a)). g(s) is a positive and increasing function over the interval −∞ < s < −R which behaves as g(s) ∼ −1/(σs ln |s|) 2 ) for s → −∞ and as g(s) ∼ e/2/(πx 1/2 ) for x = −R − x, x → 0+; therefore the integral of g(s) over the interval −∞ < s < −R is convergent and equal to σ (cf eq. (A.14)). 4. For an arbitrary z (however not on the branch cut) the pressure χ(z) may be seen as the 2D complex electrostatic potential created by the charge distribution g(s). It follows from eqs. (2.7a) and (A.13a) that χ(z) takes the form proposed by Yang and Lee 12,13 A close examination of their paper reveals that the derivation of eq. (3.6) given by Hauge and Hemmer is restricted to a point z inside the circle of convergence.
It is amusing to check the various conclusions of second theorem of Groeneveld. 11 Recall that for a d-dimensional classical fluid of particles interacting via positive pair potentials ϕ(r ij ) ≥ 0 such that (twice) the second virial coefficient converges, the radius of convergence R of the virial series (3.5a) and the Mayer coefficients b n satisfy to the following inequalities In the present case f = 2b 2 = −σ, and one easily checks that R = 1/eσ and b n as given by eq. (3.5c), do satisfy to the Groeneveld inequalities (3.8).
We end this section by determining the Helmoltz free energy per unit volume βf (ρ) as the Legendre transform of βP viewed as a function of the reduced chemical potential ν ≡ βµ = ln z (see eg ref. 14). It follows from the eq. (A.5) of the appendix that Note that we have restored Λ to make the argument of the ln dimensionless. βf (ρ) is a strictly convex function of the density defined for 0 < ρσ < 1. The first term in the RHS of the equation is the Helmoltz free energy per unit volume of the ideal gas while the second one is the excess free energy. The analytic continuation of βf (ρ) to complex densities is obtained by defining βf (ρ) according to eq. (3.9) where ln is the principal branch of the natural logarithm for βf (ρ) must take on real values for ρ ∈ R, 0 < ρσ < 1. βf (ρ) is then a multivalued function with two branch cuts −∞ < ρ ≤ 0 and σ −1 ≤ ρ < +∞.

IV. WEAK AND LONG RANGE REPULSION
The EOS of a gas of particles interacting via the pair potential (1.1) is known exactly and given by 7 is a positive, γ-independent constant. Here ρ can take on all non negative real values. It is not difficult to obtain the relation between the density and the activity which reads as 7 z = ρe aρ . Therefore, since for a real z the density must be real and, by integration of the thermodynamic relation (3.2) where g(s) = g U (as) = − 1 πas ℑW 0 (as) . βf (ρ) is a strictly convex function of ρ on the interval 0 < ρ < +∞.

V. REPULSIVE INTERACTIONS: A FIELD THEORETICAL APPROACH
We consider the statistical mechanics of a system made of N classical point particles interacting via pair potentials of the form v(r) = ϕ 0 (r) + ϕ(r) , where ϕ 0 (r) is some reference potential (eg a hard sphere repulsion for instance) and ϕ(r) admits a positive, well defined Fourier transform ϕ(k) ≥ 0. ϕ 0 (r) and ϕ(r) are supposed to meet all the requirements which are necessary for the existence of a thermodynamic limit (TL). 15 We denote by Ω the domain occupied by the molecules of the fluid. It will be convenient to assume that Ω is a cube of side L with periodic boundary conditions (PBC). In a given configuration ω = (N; r 1 . . . r N ) the microscopic density of particles reads as and its Fourier transform The configurational potential energy of the system can be decomposed as where V 0 (ω) is the configurational energy of the reference system, ν S ≡ βϕ(0) a self-energy contribution and We shall work in the grand canonical (GC) ensemble. We denote by µ the chemical potential and by ψ( r) an external potential. The local chemical potential will be noted ν( r) = β(µ − ψ( r)). Performing a Kac-Siegert-Stratonovich-Hubbard-Edwards (KSSHE) transform 16,17,18,19,20 one can show that the GC partition function can be recast under the form of a functional integral 8 where Dξ is the functional measure and the action H[ξ] of the KSSHE field theory reads as where ν = ν + ν S and Ξ 0 is the GC partition function of the reference system. The field ξ which enter eqs. (5.6) and (5.7) is a real scalar field. Note that, in the general case where the sign of ϕ(k) is abitrary, the action H involves two real scalar fields ξ + and ξ − (or a complex field ξ). 8 Finally, the normalization constant N is given by In ref. 8 the potential ϕ 0 (r) of the reference system was chosen to be a hard core potential of diameter σ. Here we specialize to the case σ → 0 (or equivalently ϕ 0 ≡ 0), i.e. we take for the reference system the ideal gas. All the conclusions of ref. 8 remain valid provided that the TL of our system is well behaved. Note that the configuration energy (5.4) can be rewritten as Therefore if ϕ(k) ≥ 0 for all k and ϕ(0) > 0 (which will be assumed henceforth) then can be recast under the form of the implicit integral equation Moreover the MF density of the fluid is given by the density of the reference fluid -here the ideal gas-at the local chemical potential ν( r) + iξ( r) 8 , i.e.
ρ M F ( r) = e ν( r)+iξ( r) . (5.14) In the case of a homogeneous system to which we stick from now, (i.e. ψ( r) ≡ 0) the solution of eq. (5.13) is clearly where W k is some branch of Lambert W, z the activity and 2. We have shown elsewhere 8 that the one-loop order approximation of the KSSHE field theory coincides with the random phase approximation (RPA) of the theory of liquids. 21 As well known, for long range potentials of the form (1.1) the RPA corrections to the pressure vanishes as γ → 0. 21 Note that, in this limit, ν S → 0 and ϕ(0) = φ(0); thence, in the limit γ → 0, one recovers for the pressure, the free energy etc the expressions derived in sec. IV. Stated otherwise, the MF-KSSHE theory is exact for infinitely weak and long range repulsive potentials.
3. We have shown in ref. 8 that the MF pressure and free energy constitute exact bounds.
More precisely we have In this paper we have discussed some applications of the Lambert W function to the theory of liquids. In the case of 1D hard rods or infinitely weak long range repulsive potentials, one obtains a close expression for the complex pressure χ(z) as a function of complex activities in terms of either W 0 (z) or the related function U 0 (z). The dispersion relations derived for W 0 (z) and U 0 (z) in the appendix give a rigorous justification to the heuristic formula proposed by Lee and Yang for χ(z). 12,13 We have also shown that, in the framework of the KSSHE field theory of liquids, the MF pressure χ(z) of a gas of particles interacting via soft repulsive potentials can be expressed in terms of the function U 0 (z).
APPENDIX: NEW MATHEMATICAL PROPERTIES OF LAMBERT W

Legendre transforms
Let us first consider the real function x → f (x) = W 0 (e x ). By applying twice the formula (2.3a) one finds Note that L h (ρ) is defined on 0 < ρ < ∞). Over this interval, L h (ρ) is strictly convex.

Dispersion relations
We establish here the dispersion relations for the functions W 0 (z) and U 0 (z). Let us first consider W 0 (z). It follows from the Cauchy theorem that where C is the contour shown in fig. (1) and z is not on the cut (−∞, −e −1 ). Indeed W 0 (0) = 0 as follows from eq. (2.2). We consider now eq. (A.7) in the limit ǫ → 0 and R → ∞. The asymptotic formula for large (complex) z, i.e. 1,4 W 0 (z) ∼ ln(z) − ln(ln(z)) , (A.8)