Critical correlations and hierarchy equations Jarosaw Piasecki - - PowerPoint PPT Presentation

Critical correlations and hierarchy equations

Jarosław Piasecki Institute of Theoretical Physics Faculty of Physics University of Warsaw

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 1/19

Correlation functions

Correlation functions hk are defined by cluster decompositions of the reduced number densities nk(r1, r2, ..., rk) = nkgk(r1, r2, ..., rk), k = 1, 2, 3, ... The decompositions of dimensionless densities gk read g2(r1, r2) = 1 + h2(r1, r2) g3(r1, r2, r3) = 1 + h2(r1, r2) + h2(r1, r3) + h2(r2, r3) + h3(r1, r2, r3) ... ...

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 2/19

Equilibrium: Yvon-Born-Green hierarchy

The equation relating h2(r1, r2) and h3(r1, r2, r3) reads kBT ∂ ∂r12 h2(r1, r2) + ∂V (r12) ∂r12 [1 + h2(r1, r2)] +

• dr3

∂V (r13) ∂r13 h2(r2, r3) = −

• dr3

∂V (r13) ∂r13 h3(r1, r2, r3)

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 3/19

Evoking memories: year 1986

Proposed closure of the YGB hierarchy for the two-dimensional one-component plasma n

• dr3

ˆ r13 r13 h3(r12, r13|Γ) = 1 2 ∂ ∂r12 h2(r12|Γ) The resulting two-particle distribution: provides exact results at Γ = 2, and for Γ → 0 satisfies three sum rules: perfect screening, Stillinger-Lovett rule, compressibility sum rule predicts the transition from monotonic exponential (Γ < 2) to oscillatory algebraic (Γ > 2) decay of correlations

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 4/19

Multi-particle critical correlations (with Andres Santos)

The infinite hierarchy derived by R.J. Baxter in 1964

• nχT

∂ ∂n − k

• nkhk(r1, . . . , rk)

= nk+1

• drk+1 hk+1(r1, . . . , rk, rk+1),

k = 1, 2, ... reflects the structure of equilibrium correlation functions hk(r1, . . . , rk) in any dimension. Putting k = 1 we find the compressibility equation χT = kBT ∂n ∂p

• T

= 1 + n

• dr2 h2(r1, r2)

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 5/19

Generalized compressibility equations

According to the the Baxter hierarchy the correlation integrals Ik(n, T) ≡ nk

• dr2 · · ·
• drk hk(r1, r2, . . . , rk)

satisfy the relations Ik+1 = LkIk, where Lk ≡ nχT ∂ ∂n − k implying the generalized compressibility equations Ik+1 = LkLk−1 · · · L2L1n (k = 1, 2, ...)

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 6/19

Critical exponent δ: correlation integral I2

If a liquid–vapour critical point exists at T = Tc and n = nc, then along the critical isotherm (when n → nc) p − pc nckBTc ≈ (±1) Aδ

• n

nc − 1

• δ

, δ > 2 implying the divergence of the compressibility integral I2 ≈ ncχT ≈ ncA

• n

nc − 1

• −(δ−1)

, T = Tc with amplitude A. The divergence is linked to the development of long-range pair correlations.

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 7/19

Critical exponent δk : correlation integral Ik

In general we define the critical exponents δk by Ik ≈ (±1)k Aknc

• n

nc − 1

• −(δk−1)

, T = Tc From the recursion relation Ik+1 = LkIk we find δk+1 = δk + δ, Ak+1 = −AAk(δk − 1) with the solution δk = (k − 1)δ Ak = (−1)kAk−1 (δ − 1) (2δ − 1) · · · [(k − 2)δ − 1] k = 3, 4, ...

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 8/19

Divergence of correlation integrals Ik near the critical point

critical exponent corresponding to Ik increases linearly with k critical amplitude alternates in sign and its absolute value grows almost exponentially with k Ik with k odd diverges to either −∞ or +∞ depending

• n whether the critical point is reached along the

critical isotherm with n > nc or n < nc, respectively.

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 9/19

Critical point: fundamental role of multi-particle correlations

If hk0+1 ≡ 0 then the Baxter hierarchy yields the equation

• nχT

∂ ∂n − k0

• Ik0 = Ik0+1 ≡ 0

But if χT ≈ |n/nc − 1|−(δ−1) then the left-hand side the equation diverges as |n/nc − 1|−(k0δ−1). The assumption hk0+1 ≡ 0 is thus inconsistent with the existence of a critical point. Conclusion: correlations of all orders are crucial for the appearance of a critical point

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 10/19

Kirkwood’s superposition approximation

g3(r12, r13, r23) = g2(r12)g2(r13)g2(r23) The Kirkwood superposition approximation expresses three-particle correlations in terms of two-particle ones h3(r12, r13, r23) = h2(r12)h2(r13)h2(r23) +h2(r12)h2(r13) + h2(r13)h2(r23) + h2(r12)h2(r23)

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 11/19

Kirkwood’s approximation and Baxter’s hierarchy: no critical point

The Kirkwood approximation applied within the Baxter hierarchy yields the relation

• dr

1 1 + h2(r) ∂h2(r) ∂n =

• dr h2(r)

2 1 + n

• dr h2(r)

inconsistent with the existence of a critical point.

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 12/19

Generalization of Kirkwood’s approximation

Generalized superposition approximation at the level of the four-particle number density g4(1, 2, 3, 4) = g3(1, 2, 3)g3(1, 2, 4)g3(1, 3, 4)g3(2, 3, 4) g2(1, 2)g2(1, 3)g2(1, 4)g2(2, 3)g2(2, 4)g2(3, 4) permits to express h4 in terms of lower order correlations h3 and h2. It can be shown that the resulting closure of the hierarchy is incompatible with the existence of a critical point.

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 13/19

Conjectures concerning closures of Baxter’s hierarchy

any theory which expresses three-particle correlations in terms of two-particle ones loses the possibility of describing a critical point strong version: once the higher order correlations are assumed to be functionals of the lower order ones the resulting theory becomes inconsistent with critical behaviour

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 14/19

Square-well fluid

It is convenient to define the distribution y2 by g2(r) = χB(r)y2(r) where χB(r) is the Boltzmann factor corresponding to particles with a hard core of diameter σ interacting via an attractive well of depth E and range λσ, λ > 1. χB(r) = exp(E/kBT) if σ < r < λσ χB(r) = 1 if r > λσ The function y2 is supposed to be continuous and differentiable.

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 15/19

Kirkwood’s approximation applied to Yvon-Born-Green hierarchy

(with Piotr Szymczak and John J. Kozak) Asymptotic decay of correlations via exponential modes h2(r) ≈ exp(κr) r The constant κ vanishes if and only if Γ ≡ 1 + 8φ [ y2(σ)B − λ3y2(λσ)(B − 1) ] = 0 where B = exp(E/kBT), and φ = πnσ3/6 is the volume fraction. No obvious contradiction with the existence of a critical isotherm.

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 16/19

Appearance of a near-critical region: 0 < Γ ≪ 1

• 0.1

0.2 0.3 0.4 0.5

• 0.5

1 1.5

• Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP

, November 2015 – p. 17/19

Kirkwood’s closure: summary

Baxter’s hierarchy: no critical point in any dimension YBG hierarchy d=3: no true critical point but existence

• f a near-critical region (numerical results,

δ = 4.65 ± 0.2) YBG hierarchy d>4: prediction of critical behavior of classical Ornstein-Zernike scaling form (M. E. Fisher and S. Fishman) Challenging problem: provide a complete analytic proof of the existence or non-existence of a critical point within a theory based on YBG under Kirkwood’s closure

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 18/19

Bibliography

• J. Piasecki, D. Levesque J. Stat. Phys. 47, 489 (1987).
• R. J. Baxter J.Chem. Phys. 41, 553 (1964).
• A. Santos, J. Piasecki Mol. Phys. 113, 2855 (2015).

DOI: 10.1080/00268976.2015.1021397

• J. Piasecki, P

. Szymczak, and J.J. Kozak J.Chem.

• Phys. 139, 141101 (2013).
• M. E. Fisher, S. Fishman J. Chem. Phys. 78, 4227

(1983).

Conference in memory of Bernard Jancovici, Analytical Results in Statistical Physics, IHP , November 2015 – p. 19/19