On the Stationary BBGKY Hierarchy for Equilibrium States

Abstract

We consider infinite classical systems of particles interacting via a smooth, stable and regular two-body potential. We establish a new direct integration method to construct the solutions of the stationary BBGKY hierarchy, assuming the usual Gaussian distribution of momenta. We prove equivalence between the corresponding infinite hierarchy and the Kirkwood–Salsburg equations. A problem of existence and uniqueness of the solutions of the hierarchy with appropriate boundary conditions is thus solved for low densities. The result is extended in a milder sense to systems with a hard core interaction.

Appendices

Appendix A: Positivity of the Activity

In this appendix we check that the constant introduced by (31) in the proof of Theorem 1 is well defined and positive. We put x _j =(y _j ,p _j ). By assumption (11) the denominator in (31) can be written as

(63)

where B _R is the ball centered in 0 and with radius R>0. Using the definition of correlation functions, Eq. (6), we can easily rewrite the first line as

(64)

Expanding the product, the integral in this expression is

(65)

where the equality holds by symmetry of $\mu_{B_{R}}^{(k)}$, with

$$ C^{(k,n)}_{B_R} := \int_{(B_R\times\mathbb{R}^\nu)^k} dx_1\cdots dx_k \Biggl(\prod _{i=1}^n e^{-\beta\varphi(y_{i})} \Biggr) \mu_{B_R}^{(k)}(x_1,\ldots,x_k) . $$

(66)

Putting (65) into (64) and interchanging the sums, we have

(67)

having used the normalization condition, Eq. (9), in the last step. Condition (9) implies also that this quantity is bounded away from zero uniformly in R for R larger than some R ₀>0. Since the term in the second line of (63) is made arbitrarily small by taking R large enough, the proof is complete.

Appendix B: Integration of the Hard Rod Hierarchy

In this appendix we shall find the unique and explicit solution to the one-dimensional hard core hierarchy (hard rod BBGKY hierarchy)

(68)

with the assumptions of invariance under translation and permutation of particles, sufficiently small ρ≡ρ ₁ (precisely ρ<1/d), cluster property (17), continuity over $\overline{\mathbb{R}}^{n}_{d}$, piecewise C ¹ regularity on $\mathbb{R}^{n}_{d}$ and boundedness of the derivative. The special feature of this case is the existence of an explicit form for the equilibrium correlation functions, e.g. [16]. In what follows, we derive these expressions from the hierarchy by direct integration and without going through the corresponding Kirkwood–Salsburg equations.

To do this, we can follow the procedure of [8] in a rather natural way by ordering the particles from left to right: q _i ≤q _i+1−d; hence we start rewriting

(69)

Now we choose q ₀≪q ₁ and we integrate from q ₀ to q ₁:

(70)

where we used again the symmetry in the particle labels to split the integral in the second equality. Sending q ₀ to −∞ gives

(71)

having used the cluster property and the translation invariance.

Call R:=ρ+dρ ₂(d). Iterating once the above equation we have

(72)

We stress again that the above explained procedure does not lead directly to the Kirkwood–Salsburg equations. The extracted constant R is different from the activity of the hard rod gas (which is known to be given by z=Re ^Rd, see for instance [16]). Nevertheless the set of Eq. (72) can be solved explicitly for every n, starting from n=2 (the equation for n=1 is of course useless in this model), as we show below. Actually, the simple structure of Eq. (72) allows to construct easily ρ _n from ρ _n−1: this structure is due to the strong symmetry used to split the integral in the second equality of (70), and it seems to have no analogue in higher dimensions.

We start with the n=2 case. Call x=|q ₂−q ₁|, x≥d. Formula (72) implies $\rho_{2}(d) = \frac{\rho^{2}}{1-\rho d}$, $R=\frac{\rho}{1-\rho d}$ and

(73)

Solving these set of equations iteratively in the intervals (kd,(k+1)d), k=1,2,… , using the continuity assumption, leads to

(74)

In a similar way, using (72) and (74) and proceeding by induction on n, one finds that the solution of (68) for n≥2 is

(75)

Appendix C: The Method of [8]

In this appendix we discuss the method established in [8] for the integration of the hierarchy (15), pointing out an error in the formula for the activity and sketching how to correct it (we refer to [23] for details). At the end of the section we make comparisons between this method and the one established in the present paper.

We will need somewhat stronger assumptions than those of Theorem 1, namely the potential is a function φ∈C ¹(ℝ^ν) which is radial, stable and with compact support, while the smooth Maxwellian state has positional correlation functions ρ _n ∈C ¹(ℝ^νn)³ with the following properties:

(76)

where the truncated correlation functions $\rho_{n}^{T}$ can be defined by

$$\everymath{\displaystyle} \left \{ \begin{array}{@{}l} \rho_2^{T}(q_1,q_2) = \rho_2(q_1,q_2) - \rho(q_1)\rho(q_2) , \\[5pt] \rho_3^{T}(q_1,q_2,q_3) = \rho_3(q_1,q_2,q_3) - \rho_2(q_1,q_2)\rho (q_3)-\rho (q_1)\rho_2(q_2,q_3)+\rho(q_1)\rho(q_2)\rho(q_3) , \end{array} \right . $$

etc. (see [5] and [8, page 279]).

Let us start by integrating Eq. (15) along a straight line connecting q ₀ (arbitrary) to q ₁: using the same notations introduced for (22), we have

(77)

which is nothing but a rewriting of Eq. (22). In the assumption (b), the case n=1 is a trivial identity, hence we shall assume n≥2 in the following.

The strategy consists in taking the limit as |q ₀|→+∞ right away, before iteration of formulas. This is also the essential difference with respect to the method discussed in Sect. 3, where such a limit is taken at the very end of the proof, after infinitely many iterations. Using (c), from (77) we get

(78)

where the double integral in the second term on the right hand side is well defined (by the exponential clustering (76) and the assumed rotation symmetry of the potential and of ρ ₂), though not absolutely convergent. Interchanging the integrations we find

(79)

where we put

(80)

Here B(q) is the ball centered in q and with radius equal to the range of φ, and the second equality is again true by the rotation symmetry.

The authors in [8] proceed by iteration of formula (79). Call for simplicity

$$ \zeta= \rho-\gamma . $$

(81)

The first iteration gives

(82)

If we iterate once again (79) in (82), the last term of (82) becomes

(83)

which is not equal to the formula in step (8) of [8] with N=2, since the first term of (83) is not equal to $\zeta e^{-\beta W_{q_{1}}(q_{2},\dots,q_{n})}$ times

(84)

In fact, integrating in $\bar{q}_{2}$ the first term of (83), we have $\zeta e^{-\beta W_{q_{1}}(q_{2},\dots,q_{n})}$ times

$$ \int_{\mathbb{R}^\nu} dy_1 \int_{\mathbb{R}^\nu} dy_2 \int_{-\infty}^{q_1} d \bar{q}_1 \bigl(1-e^{-\beta\varphi(\bar{q}_1-y_2)}\bigr) \frac{\partial(1-e^{-\beta\varphi(\bar{q}_1 -y_1)})}{\partial\bar{q}_1} \rho_{n+1}(q_2,\dots,q_n,y_1,y_2) $$

(85)

and, since the integrals in y ₁ and y ₂ are not interchangeable, integration by parts of this formula does not lead to the single desired term (84), in spite of the symmetry of ρ _n+1. We refer to [23] for a proof of the last assertion.

The additional terms missing in the formula in step (8) of [8], obtained by repeated iteration of (79), give higher order corrections to the constant ζ, and all these (infinitely many) corrections lead to a definition of the activity. A rather convenient way to modify the proof (leaving essentially unchanged the procedure) in order to obtain the correct expression for the activity, is to keep the inverse order of integration in formula (79): that is to iterate Eq. (78). This computation is reported in detail in [23]. The convergence of the iteration procedure is handled in assumptions (a), (b) and (c): after N iterations one gets a remainder R _n,N that can be bounded as

$$ \bigl|R_{n,N}(q_1,\dots,q_n)\bigr| \leq(A \xi)^{n+N+1} , $$

(86)

where A is a suitable constant depending on β,C,κ,φ and the configuration q ₁,…,q _n (but not on ξ), so that it goes to zero when N→∞ if ξ is small enough. For these values of ξ, the method provides convergence to the Kirkwood–Salsburg equations with exponential rate.

Comparisons

We have the following differences with respect to the method presented in Sect. 3:

• 1. The rate of convergence of the iteration is exponential, Eq. (86) (instead of factorial, see Eq. (27)): this implies convergence for sufficiently small values of ξ.

• 2. The radius of convergence of the procedure is at least 1/A, where A is not uniformly bounded in the maximum of the potential; a bound which is uniform in the hard core limit could be obtained by assuming estimates for the smooth state as those in (12)–(13), instead of (a) of (76).

• 3. The exponential strong cluster property (76) (instead of the weak cluster (17)), as well as the short range assumption on the potential (instead of the weak decrease (3)), are needed to control the convergence of the integrals over the unbounded domains of integration;

• 4. the same can be said for the assumption (b) in (76), that is for the rotation invariance of the state, which is not needed in Theorem 1. Furthermore we shall notice that, in the proof of Theorem 1, translation invariance is only used in the very last step, i.e. to perform the limit |q ₀|→+∞ of expression (28), obtained after infinitely many iterations. This makes the method suitable to extend the result to situations in which there is no symmetry and different kinds of boundary condition are considered; an example of such a situation has been given in Theorem 2.