Density Renormalization Group for Classical Liquids

We study response of liquid to a scale transformation, which generates a change of the liquid density, and obtain a set of differential equations for correlation functions. The set of equations, which we call density renormalization group equations (DRGEs), is similar to the BBGKY hierarchy as it relates different multiple-point correlation functions. In particular, we derive DRGEs for one-particle irreducible vertex functions of liquid by performing Legendre transformations, which enables us to calculate properties of liquid at higher density in terms of correlation functions at lower density.


Introduction
Thermodynamical properties of gas and liquid are described by the equation of state, and the microscopic derivation is one of the most important issues in the liquid theory. Liquid is microscopically described by a collection of interacting particles and the thermodynamical quantities such as pressure, internal energy, and isothermal compressibility can be calculated from density correlation functions [1,2].
Statistical mechanics of liquid has a long history. The simplest microscopic study of liquid is the virial expansion method, which gives a systematic expansion around the ideal gas. But its convergence is very slow 1 and the method is limited to low density region. In order to describe high density region of liquid, we need to go beyond the virial expansion and take into account effects of strong multi-point density correlations. For this purpose, various integral equations and their approximations have been studied. The Orstein-Zernike equation, which is an integral equation for two-point correlation functions (see Eq.(81)), is often used and various approximations such as the Percus-Yevick approximation [4,5,6,7,8] or the Hyper-Netted chain approximation [9,10,11,12,13] are proposed. Another type of integral equations is the BBGKY hierarchy [14,15,16,17,18], which is a set of integral equations relating different multi-point correlation functions. Thus, in order to solve them, we need to cut the chain of equations at some orders. In the Kirkwood's superposition approximation [16,17,19,20], 3-point correlation function is assumed to be expressed in terms of a product of 2-point correlation functions so that the hierarchical equations are closed. As explained above, there are various different equations and approximations, but it is not clear when they are justified. No systematic methods to improve these approximations are known. (See [1,2] and references therein for more details.) The situation reminds us of the renormalization group (RG) in quantum field theories (QFT): understanding the energy dependence of physical quantities in QFT. Just as an effective coupling varies as a function of the energy scale (i.e. renormalization scale) in QFT, pressure changes as a function of liquid density in liquid theory. In both theories, physical quantities are calculated from correlation functions. Thus we can infer the following analogies between classical liquid theories and quantum field theories: • Helmholtz free energy ←→ Effective potential Γ
• Density ←→ Renormalization scale µ Such analogies between statistical mechanics and quantum field theories have been occasionally pointed out, but our present study is strongly stimulated by Nambu's seminar paper [21] in which an analogy between the renormalization group (RG) equation in gauge theories and thermodynamic equation of state was discussed. (See also [22,23,24] and the references therein where the ordinary concept of the RG is applied in the classical liquid/vapor system.) The purpose of the present paper is to make the analogy more concrete and to propose a set of (density) differential equations for correlation functions by studying response of the liquid to a scale transformation. Since a scale transformation generates a change of liquid density, the resultant equations describe response of correlation functions to a small change of density; thus we call them density renormalizaton group equations (DRGEs). The physical meaning of DRGEs are the following. In (classical) gas and liquid, when its density is low, the system is well described by the ideal gas, and the density correlation functions are exactly calculated (see Appendix A). As the density increases, correlation functions start to behave nontrivially due to two different reasons. One is, of course, a direct consequence of intermolecular (2-body) potential between particles. This causes a nontrivial behavior for the 2-point density correlation function. But there is another effect. Nontrivial (multi-) correlation (more than 2-point functions) will appear due to finite density effects. Namely, e.g. density fluctuations at 3 different points get correlated mediated by finite density effect at a single point in the middle. This effect becomes stronger in higher density liquid and also when 2-point correlation becomes stronger. Therefore, if we can resum (or accumulate) these effects from low to high density, we will be able to describe the dynamics of high density liquid by solving DRGEs.
The paper is organized as follows. In Section 2, we first briefly review the statistical mechanics of classically interacting particles and then study how a partition function of such a system responds to a scale transformation. In this way, we derive a set of partial differential equations for correlation functions. In Section 3, by using a field theoretical method by Hubbard and Schofield, we calculate perturbative corrections to correlation functions by a small change of density and obtain explicit forms of the density renormalization group equations (DRGEs). We also briefly mention how we can solve the DRGEs to obtain thermodynamical behaviors of liquid. Details are left for future investigations. In Section 4, we perform Legendre transformations to obtain the Helmholtz free energy and derive another type of DRGEs. The Helmholtz free energy generate 1PI diagrams; thus the correlation functions generated by the Helmholtz free energy correspond to the 1PI vertices in QFT. Section 5 is devoted to summary and discussion.

Response to Scale Transformations
In this section, we investigate how classical liquid (or gas), i.e. a set of classically interacting particles, responds to scale transformations. A scale transformation generates a change of the liquid density and accordingly we can obtain partial differential equations describing how the system changes according to the change of the density. Thus we call them density renormalization group equations (DRGE). 2 In section 2.1, we briefly review various properties of classically interacting particles, and then in section 2.2 obtain the DRGEs for density correlation functions.

Brief review of classically interacting particles
We consider statistical mechanics of d-dimensional classically interacting particles whose Hamiltonian is given by where v(x, y) represents two-body interactions. In this paper we assume that v(x, y) is a function of the relative distance v(x, y) = v(|x−y|), which reflects the translational symmetry of the system and the absence of polarizations (i.e. simple liquids). Furthermore we neglect many-body interactions for simplicity. The grand-canonical partition function is given by where β = 1/T is the inverse temperature and z(µ) is defined by We have introduced the external source U (x) for later convenience. By using the density operator the grand-canonical partition function can be rewritten as where The functional derivatives of the grand potential, with respect to βU (x) produce correlation functions of the density fluctuation δρ(x) = ρ(x) − ρ(x) : Here we assumed that the translational symmetry is not spontaneously broken at U (x) = 0 and the particle density ρ(x) does not depend on its position x, which is written as n.
Note that these correlation functions F l correspond to the connected Green functions in the Hubbard's field theoretic formulation of classical liquids [25]. The correlation functions are related to the thermodynamical quantities such as the isothermal compressibility. In order to see this, we write the density in presence of the external source U = ∆U (x) as n(x) and expand ∆n(x) = n(x) − n with respect ∆U (x); Setting ∆U (x) = ∆µ = const, we have from which we obtain the following relations between the thermodynamical quantities and integrals of the correlation functions; where κ T is called the "isothermal compressibility" because it can be rewritten as by using the thermodynamical relations. 3 It is an indicator of the response of the fluid density against a small change of the external pressure. The first equation (13) is well known as the "isothermal compressibility equation". It is usually written as the following form, where the "two-point distribution function" n (2) (x, y) and the "total correlation function" h 2 (x, y) are defined by Note that the total correlation function h 2 (x, y) vanishes for the ideal gas (see Appendix A), and therefore κ T = 1. We can similarly define the "l-point distribution function" n (l) , and the "l-point total correlation function" h l for higher l by n (l) (x 1 , · · · , x l ) ≡ n l (h l (x 1 , · · · , x l ) + 1) 3 When T and V are fixed, the differential form of the grand potential becomes Because of W = −pV , the relation V dp = N dµ follows for fixed T and V . Thus we have which relates (13) and (15).
These approaches based on integral equations can be applicable to high density regions beyond the ordinary virial expansion method. It is, however, difficult to treat them analytically and most studies have relied on numerical computations with various approximations whose validity are not well understood. In the studies of classical liquids, we often ask the following questions: How does the pressure or the isothermal compressibility vary as a function of the liquid density? How does the phase transition or the critical phenomena occur as we change the density? In order to answer these questions, it will be desirable if we can investigate the system based on a systematic formulation and approximations, and understand the evolution of the thermodynamical quantities as we gradually increase the density. The underlying idea is similar to the renormalization group (RG) in quantum field theories (QFTs) since, in both cases, we are interested in the response of a system against scale transformations. But there is a big difference. In quantum field theory at zero temperature and zero (e.g. baryon) density, scale transformations induce a change of the energy scales; the renormalization scale µ is changed. On the other hand, in classical liquids at nonzero temperature and nonzero density, the transformations induce a change of the magnitude of the liquid density. Therefore scale transformations in liquid theory lead to differential equations to describe response of thermodynamical quantities against a change of the density n (or the chemical potential µ) instead of the renormalization scale. 4

Scale transformations
In order to investigate response of the system against a scale transformation, we consider a transformation of the two-body potential: v(x) → v(ax). Then the partition function changes as Therefore we see that the change of the potential under the scale transformation, v(x) → v(ax), is equivalent to the changes of the chemical potential µ, the volume of the system V , and the external source U (x). For an infinitesimal scale transformation a = 1 + , we have or equivalently where δv(x) = x µ ∂ µ v(x). Thus, by differentiating it with respect to βU (x), we obtain the following relation of the correlation functions, 5 where ∆F l (x 1 , · · · , x l ) represents perturbative corrections due to δv(x), which are explicitly evaluated in the next section. Finally, by taking the → 0 limit, we obtain the following set of partial differential equations for the correlation functions: The lhs contains a derivative with respect to the chemical potential µ. Thus they describe how the classical liquids respond to a change of the chemical potential. By performing the Fourier transform 5 The derivation is the following; By taking functional derivatives l times, we obtain we have Furthermore, because the chemical potential and the density are related each other through Eq.(13), Eq.(29) can be also written by which describes how the system (in particular its correlation functions) changes as we change the density. In the next section, we explicitly evaluate the corrections ∆F l to the correlation functions due to change of the potential δv(x) = x µ ∂ µ v(x). We adopt the field theoretical approach to the classical interacting particles proposed by J. Hubbard and P. Schofield [25], and we will see that the corrections ∆F l are written in terms of higher-body correlation functions such as F l+1 . Thus the above equations generate a hierarchical structure similar to the BBGKY hierarchy. Note that the volume derivative at fixed (µ, T ) vanishes in the large V limit.

Density Renormalization Group Equations
In the previous section, we have derived the density evolution equations (27)(29)(30) from the scale transformation. In this section, we explicitly calculate the perturbative corrections ∆F l .
In particular, we adopt the field theoretical approach proposed by J. Hubbard and P. Schofield [25]. One of the benefits of the approach is that we can use the ordinary field theoretical techniques for calculating the correlation functions such as the Feynman diagram method. The resultant corrections contain contributions from higher-body correlation functions, so the equations generate a hierarchical structure like the ordinary BBGKY equations. In section 3.1, we briefly review [25] and calculate the perturbative corrections of the correlation functions. In section 3.2, we derive new hierarchical equations by combing all the results of this section and the previous one.

Field theory of classically interacting particles
One of the successful perturbative approaches in the classical liquid/vapor theory is the high temperature expansion from some reference system whose properties are supposed to be already known or exactly solved [26,27]. In particular, it was shown that this approach can be even applicable to the lower temperature and high density regions. However, this method is usually used to derive global thermodynamical quantities and we need a new framework which enables us to calculate various local quantities such as the correlation functions. One of useful approaches toward understanding such local quantities is the field theoretical method proposed by J. Hubbard and P. Schofield [25]. In this method, the grand canonical partition function is cleverly transformed into a path integral formulation of a scalar field theory. In the following, we first review the method [25], and then calculate the corrections to the correlation functions.
Suppose that a reference system is described by a two-body potential v R (x) and then perturbed as where is a small parameter and v 1 (x) is an arbitrary potential. Note that, in the case of the scale-transformation discussed in the previous section, Under the shift (31), the partition function Eq. (6) becomes where · · · R represents the thermodynamical expectation value in the reference system: In order to rewrite the partition function Eq.(32) in a path integral form, we use the following mathematical identity: where a > 0 (< 0) corresponds to an attractive (repulsive) potential respectively, i.e., v 1 (x) < 0 (> 0). In the following, we consider an attractive case for simplicity. 6 By completing the square in the exponent in Eq.(32), we obtain is an inverse operator acting on functions. Eq.(32) is rewritten in terms of the scalar field path integral: where N is a normalization factor. Then, the cumulant expansion of e ρ|φ R leads to the following result: whereF (R) l (k 1 , · · · , k l ) is the Fourier transform of the correlation function of the reference system. This result shows that a classical theory of liquid is equivalent to a quantum field theory with an infinite number of the multiple point operators, each of which corresponds to the correlation function of the reference system.
In this expression, we see that the external source U (x) originally introduced for the density ρ(x) now plays a role of the source term for the quantum field φ(x). Therefore φ(x) can be essentially identified with ρ(x). 7 In particular, it is apparent that the correlation function F l (x 1 , · · · , x l ) of the density fluctuations corresponds to the connected part of φ(x 1 ) · · · φ(x l ) because it is generated by the generating functional −βW [U ] = log Ξ[U ]. Therefore, we can calculate the perturbative corrections of these correlation functions in the same manner as the ordinary quantum field theories. Note that this procedure is widely applicable to any perturbed (or transformed) system as long as the resultant potential is given by Eq.(31).
Let us now apply this method to our scale-transformed system, i.e. v R (x) = v(x), v 1 (x) = δv(x). In this case, the correlation functions of the reference systemF (R) l (k 1 , · · · , k l ) is given 7 There is an extra factor v −1 1 / in the quadratic and a linear term of φ in the Lagrangian (37) but these factors are cancelled in = 0 limit. Thus U derivatives give the correlation functions of φ(x). by the exact correlation functionF l (k 1 , · · · , k l ) at some fixed density. They are, of course, not yet known. Instead, we investigate how they change under a scale transformation, namely under a change of the density.
In calculating corrections to the correlators ∆F l , it is convenient to use Feynman diagrammatic representations. Associated with the propagator and l-point vertices, we introduce the following graphical representations: • For each internal propagator for φ(x), • For each l-th vertex, whereF l (k 1 , · · · , k l ) is symmetric under permutations of momenta and vanishes unless momentum conservation l i=1 k i is satisfied. The Fourier transform of δv(x) is given by Thus we have δṽ(0) = −dṽ(0). In the graphical representation, there is an important property derived from Eq.(37). When we take the functional derivatives of Eq.(37) with respect to βŨ (p), an additional factor −( δṽ(p)) −1 is added to each of the external legs. However, this additional factor is completely canceled by the propagator − δṽ(p). This property is depicted as where the blob is any diagram connected with this external line. As a result, the tree-level correlation functionsF l (p 1 , · · · , p l ), namely correlation functions without perturbation δv, can be correctly reproduced. By using them, we can diagrammatically evaluate the perturbative corrections of the correlation functions, i.e. ∆F l (x 1 , · · · , x l ) in Eqs. (27)(29)(30). Because we are interested in the → 0 limit, it is sufficient to consider the leading order contributions with respect to . Such contributions are represented by the diagrams which contain only one internal propagator because it is the only place where an additional factor appears. In other words, if there are more than one internal propagators, the diagram vanishes in the → 0 limit. Therefore, we obtain the following results: (0) Zero point function (=Grand potential): (1) One-point function: (2) Two-point correlation function: (3) Three-point correlation function: (4) Four-point correlation function: (5) l-point correlation functions (l ≥ 5): In general, ∆F l+1 (k 1 , · · · , k l+1 ) can be automatically obtained by taking the functional derivative of the l-th diagrams and using the following vertex relation: which is diagrammatically represented by δ δ(βŨ (k l+1 )) .
In fact, we can straightforwardly check that all the results (l ≤ 4) can be rederived by taking the functional derivatives of ∆F 0 = −β∆W .

DRGEs as hierarchical equations
By substituting the results of the previous section into Eqs. (27)(29)(30), we obtain a sequence of the differential equations that govern the changes of the correlation functions against a variation of the density (or chemical potential). In order to separate the momentum conservation from the correlation functions, we introduce the following notations: Especially, the 2-point function at zero momentum κ(k = 0) is identified with κ T . For higher l ≥ 3, we definẽ k i nλ l (k 1 , · · · , k l ). (52) Note that λ, κ and λ l (l ≥ 3) are dimensionless. Especially, their initial values at n = 0 (µ = −∞), namely the values for the ideal gas, are given by as can be easily checked from the correlation functions for the ideal gas (see Appendix A).
By substituting Eqs.(49)-(52) into Eq.(29) or (30), we obtain the following set of hierar-chical equations. 8 For l = 0, we get an equation of state of the classical liquid: which relates the pressure with an integral of two-point correlation function. Here we have used δṽ(0) = −dṽ(0). For l ≥ 1 we have the following set of partial differential equations. First we define the differential operator D as Here we note that the volume derivative in the differential operator D can be neglected in the large V limit since local quantities such as the density n or the correlation functions do not depend on the total volume when (µ, T ) are fixed. For l = 1, we get Because of the assumption of the translational invariance, it vanishes unless k = 0. Since µ derivative in the left hand side for k = 0 is written in terms of κ T = κ(0), the equation relates κ T with an integral of the 3-point function λ 3 . For l = 2, we have 8 The momentum derivative in the LHS of Eq.(29) or (30) is given by which relates a density response of κ(k) with an integral of 3-and 4-point functions. For l = 3 and l = 4, we have and where is the real part and (k a ↔ k b ) denotes interchanging the momenta, k a and k b . These hierarchical equations describe the response of the system to a small change of the density. Though we have used perturbative technique, they are the exact (non-perturbative) equations and offer us an alternative formulation of the classical liquid/vapor system. Our next step is to solve them by using physically reasonable approximations or assumptions.

Towards solving DRGEs
In this section, we briefly discuss how we can attack to solve the set of differential equations (DRGE) derived in the previous section; more details are left for future investigations.
There are two difficulties in solving the DRGEs. The first one is a mixture with higherpoint correlation functions, and some approximations are necessary to close the hierarchical equations. Another is the momentum integration, which originates in the loop diagrams. The second difficulty can be avoided by noticing that various integrals have similar forms; integrants always contain δṽ(p). Thus, we can regard a special set of integrals as couplings that govern the system. One of the most important examples is given by the following integral It is interpreted as a "coupling" of the liquid system as well as κ(0) = κ T . Other quantities appearing in the integrals of Eqs.(58)-(61) at zero external momenta k i = 0 are related to κ I as follows; 1 2d 1 2d 1 2d In deriving these equations, we used the relation which can be proved by using the fact that the chemical potential µ and the zero mode of U (x) can be identified. In other words, the relation is satisfied, and then Eq.(66) is derived. By using these relations in Eq.(55) and Eq.(58), we get the following equations: Setting the external momenta k i = 0 and using these relations for l ≥ 2, e.g., in Eq.(59), we get a similar equation But it is not independent from Eq.(69) since Eq.(70) can be derived by taking a (βµ) derivative of Eq.(69). Eq.(68) and Eq. (69) are not independent either, and there is a single independent equation for vanishing external momenta. Thus another relation between κ T and κ I is necessary to solve the equation. An independent equation can be obtained, e.g., by multiplying βδṽ(k) on Eq. (59) with k 1 = −k 2 = k and integrating over k. Then, defining two new couplings by the following integrals and we get the following equation In this way we can generate independent differential equations, but at the same time more new couplings are introduced and we need some approximations to close the equations; it is the destiny of the hierarchical equations and further investigations are left for future publications.
In the next section we will make Legendre transformations to one-particle irreducible (1PI) diagrams in which multipoint 1PI vertices are expected to become local and closure of the hierarchical equations becomes more tractable and reliable.

Legendre Transformation and 1PI Potential
In quantum field theory, it is usually much more convenient to discuss the dynamics of a system based on the effective action Γ[φ] which is obtained by the Legendre transformation of iW [J] = log Z[J]; Γ[φ] represents the generating functional of the 1PI diagrams. The 1PI effective action Γ[φ] is especially useful and inevitable when a spontaneous symmetry breaking occurs. Similarly 2PI effective action is important when we discuss a nontrivial behavior of the propagator [28]. In this paper, we concentrate on Γ[φ] and leave analysis of 2PI actions for future. In the case of the classically interacting particles, such a transformation corresponds to the thermodynamical Legendre transformation of the grand potential −βW [U ] to the Helmholtz free energy −βΓ[ρ] where the thermodynamical parameters are transformed from (T, V, µ) to (T, V, N ). Either thermodynamical potential has its advantage and we can use them as the situation demands. Here we generalize the thermodynamical Legendre transformation including the local external source term U (x). Thus, −βW v [U ] is transformed to a generating functional of 1PI correlations functions of density, i.e. −βΓ [ρ]. Up to trivial contributions from the ideal gas, such 1PI correlation functions are called the direct correlation functions in the liquid theory.
In the following, in order to make the discussions simpler, we absorb the chemical potential µ into the zero mode of the external source U (x) and denote −βW v [T, µ, V ; U ] as −βW v [T, V ; U ]. Also we introduce the correlation functions of density fluctuations in the presence of external source terms and denote them as Setting U (x) = µ, they coincide with the previous ones F l (x 1 , · · · , x l ).

Direct correlation functions
We define the (generalized) Helmholtz free energy −βΓ v [T, V ; ρ] by the Legendre transfor- where ρ(x) represents a density field, and Then, we can define new correlation functions by taking the functional derivatives of −βΓ v [T, V ; ρ] with respect to ρ(x): In the following, we call them the l−point 1PI vertices for l ≥ 3. In particular, c 2 (x, y) and c 3 (x, y, z) satisfy the following relations where F 2 and F 3 are the (connected) correlation functions defined in the previous section. These relations are the direct consequences of the Legendre transformation. The first relation is called the "Ornstein-Zernike equation" in the liquid/vapor theory. In order to rewrite it in a standard form, we note that the two-point "direct correlation function" c D 2 (x, y) is defined by The direct correlation function c D 2 vanishes for the ideal gas (see Appendix A). Then using the definition of the total correlation function h 2 in Eq. (19) which also vanishes for the ideal gas, Eq.(78) becomes which is the standard form of the Orstein-Zernike equation.
Similarly we define l-point direct 1PI vertices (for l ≥ 3), c D l , by The direct 1PI vertices are defined so as to vanish for the ideal gas (see Appendix A).

Scale transformations of 1PI potential
where we have regardedŨ (x) = U (x(1 − )) − dT as a new external source and performed the Legendre transformation with respect to it. Thus, we obtain and, by differentiating it with respect to ρ(x), we obtain the following differential equation of the 1PI vertices: where ∆c l (x 1 , · · · , x l ) represents the perturbative corrections caused by δv(x). They are explicitly calculated in the next section. Note that the V derivative in the LHS is equivalent to the n(= N /V ) derivative because N is fixed here. Thus the equation describes a response of various 1PI quantities against a small change of density. For the ideal gas, the LHS is shown to vanish (see Appendix A).

Derivations of ∆c
In this section, we will explicitly calculate the corrections ∆c l (x 1 , · · · , x l ). In order for this, we use the previous result of the correction to the grand potential; where −β∆W v [T, V ; U ] is given by Eq. (42). Then, from the definition of the Legendre transformation, we have Thus, by denoting the difference between U v+ δv where we used Eq.(76). From Eq.(42), the second term −β∆W v [T, V ; U v ρ ] is given by 9 Therefore, by functionally differentiating Eq.(91) with respect to ρ(x) and putting ρ(x) = n, we can obtain ∆c l (x 1 , · · · , x l ). In order for systematic calculations, we introduce the following graphical representations: 9 Here, F • For the perturbative potential, we use the wavy line: βδv(x − y) = (94) • For the two-point (exact) correlation function, a straight line with a blob is used: • For the 1PI vertices, shaded polygons are used: These graphical representations are used to express ∆c l in the following.
Here we note that the functional derivative of c l (x 1 , · · · , x l ) with respect to the density field ρ is given by which is graphically represented by Finally, because the functional derivative with respect to ρ(x) can be also written as where we have used Eqs.(78)(79). It is graphically represented by δ δρ(z) .

DRGEs for 1PI correlation functions
We now summarize the differential equations that the Legendre transformed correlation functions satisfy. By substituting the corrections ∆c l into Eq.(86) and performing the Fourier transform, we obtain the hierarchical equations for the 1PI correlation functions. In the following, we use g l (k 1 , · · · , k l ) defined bỹ in which the momentum conservation is factorized. For the ideal gas (see Appendix A), g l 's are constants Eq.(120) and do not have dependence on the external momenta.
For l = 0, we have where we have used (∂(−βΓ v [T, V, N ])/∂ ln V ) T,N = pV /T . Noticing that g 2 (p, −p) = −1/κ(p, −p), it is equivalent to Eq.(55). For l = 1, we have which is equivalent to Eq.(58) since and we have from which we can easily check that Eq.(108) coincides with Eq.(58). For l = 2, namely the response of the 1PI 2-point correlation to a small change of the liquid density is given by This can be rewritten as (δṽ(k 1 ) + δṽ(k 2 )) + 1 2d d d p (2π) d g 4 (k 1 , k 2 , p, −p)κ(p, −p) 2 βδṽ(p) The DRGEs for l = 3 and l = 4 point 1PI vertices are given in Appendix C. In Eq.(112), we can see an advantage of using 1PI quantities to Eq.(59). Even when higher-point vertex functions such, g 3 and g 4 , are replaced by the ideal gas vertices, longrange correlations in the multi-point correlation functions F 3 or F 4 can be at least partially taken into account through a product of two-point function F 2 (x, y). Indeed, if we replace g 3 and g 4 by the ideal gas vertices and set k 2 = −k 1 , Eq.(112) becomes (we write κ(k) = κ(k, −k) and g 2 (k) = g 2 (k, −k)) ∂ ∂ ln n T,N It is a closed equation for the two-point correlation function. Validity of the approximation to replace multi-point vertices by the ideal gas needs to be checked by studying the DRGEs for these vertices. Further details are studied in a separate paper.

Summary and Discussion
In this paper, we proposed a new formulation of statistical mechanic of classical liquid based on a scale transformation method. Scale transformations generate an analogue equation to the Ward-Takahashi identity of scale transformations in QFTs; and consequently we have obtained the density renormalization group equations (DRGEs). The set of equations describes response of various physical quantities and correlation functions to a change of the liquid density. The response itself depends on the density. Thus if we can integrate the equations from low to high density, we can accumulate the effects of finite density, which corresponds to a resummation of quantum effects in the renormalization group method in QFT. The DRGEs are a set of differential equations which contain multi-point correlation functions. It is similar to the BBGKY hierarchy. Namely the equations must be appropriately closed at some orders. Hence, our next necessary step is to introduce reasonable approximations for higher order correlation functions. A simple but physically reasonable approximation is to replace higher order (more than 2-point functions) 1PI vertices by those of the ideal gas (times a density-dependent function). It will be reasonable because in this approximation multiple-correlation effects of the liquid can be partially taken into account. This approximation will be systematically improved by slightly taking nonlocal effects of multiple-point 1PI vertices. Also it is interesting to see how we can perform resummation of the virial expansion by solving DRGEs. We will investigate these issues in separate papers. from which we obtain the following direct correlation functions: n , · · · , c l (x 1 , · · · , x l ) = (−1) l+1 (l − 2)! n l−1 Thus, the log V derivative in Eq.(86) gives d(l − 1)c l (x 1 , · · · , x l ) which is canceled by the third term in the LHS. For the ideal gas, g l (k 1 , · · · , k l )'s defined by Eq. (106) are given by constants g l (k 1 , · · · , k 2 ) = (−1) l+1 (l − 2)! n l−1 (120) and do not have dependence on the external momenta.

Appendix B Corrections to l = 3, 4 1PI vertices
The correction to the l = 3 1PI vertex ∆c 3 (x, y, z) is graphically given by The correction to the l = 4 1PI vertex ∆c 4 (x, y, z, w) is graphically given by