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 h k are defined by cluster decompositions of the reduced number densities n k ( r 1 , r 2 , ..., r k ) = n k g k ( r 1 , r 2 , ..., r k ) , k = 1 , 2 , 3 , ... The decompositions of dimensionless densities g k read g 2 ( r 1 , r 2 ) = 1 + h 2 ( r 1 , r 2 ) g 3 ( r 1 , r 2 , r 3 ) = 1 + h 2 ( r 1 , r 2 ) + h 2 ( r 1 , r 3 ) + h 2 ( r 2 , r 3 ) + h 3 ( r 1 , r 2 , r 3 ) ... ... 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 h 2 ( r 1 , r 2 ) and h 3 ( r 1 , r 2 , r 3 ) reads ∂ h 2 ( r 1 , r 2 ) + ∂V ( r 12 ) k B T [1 + h 2 ( r 1 , r 2 )] ∂ r 12 ∂ r 12 ∂V ( r 13 ) � + d r 3 h 2 ( r 2 , r 3 ) ∂ r 13 ∂V ( r 13 ) � = − d r 3 h 3 ( r 1 , r 2 , r 3 ) ∂ r 13 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 ˆ h 3 ( r 12 , r 13 | Γ) = 1 ∂ � r 13 n d r 3 h 2 ( r 12 | Γ) r 13 2 ∂ r 12 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 � � ∂ n k h k ( r 1 , . . . , r k ) nχ T ∂n − k � = n k +1 d r k +1 h k +1 ( r 1 , . . . , r k , r k +1 ) , k = 1 , 2 , ... reflects the structure of equilibrium correlation functions h k ( r 1 , . . . , r k ) in any dimension. Putting k = 1 we find the compressibility equation � ∂n � � χ T = k B T = 1 + n d r 2 h 2 ( r 1 , r 2 ) ∂p T 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 � � I k ( n, T ) ≡ n k d r 2 · · · d r k h k ( r 1 , r 2 , . . . , r k ) satisfy the relations ∂ I k +1 = L k I k , where L k ≡ nχ T ∂n − k implying the generalized compressibility equations I k +1 = L k L k − 1 · · · L 2 L 1 n ( 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 I 2 If a liquid–vapour critical point exists at T = T c and n = n c , then along the critical isotherm (when n → n c ) δ � � p − p c ≈ ( ± 1) n � � − 1 , δ > 2 � � n c k B T c Aδ n c � � implying the divergence of the compressibility integral − ( δ − 1) � � n � � I 2 ≈ n c χ T ≈ n c A − 1 , T = T c � � n c � � 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 δ k : correlation integral I k In general we define the critical exponents δ k by − ( δ k − 1) � � n I k ≈ ( ± 1) k A k n c � � − 1 , T = T c � � n c � � From the recursion relation I k +1 = L k I k we find δ k +1 = δ k + δ, A k +1 = − AA k ( δ k − 1) with the solution δ k = ( k − 1) δ A k = ( − 1) k A k − 1 ( δ − 1) (2 δ − 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 I k near the critical point critical exponent corresponding to I k increases linearly with k critical amplitude alternates in sign and its absolute value grows almost exponentially with k I k with k odd diverges to either −∞ or + ∞ depending on whether the critical point is reached along the critical isotherm with n > n c or n < n c , 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 h k 0 +1 ≡ 0 then the Baxter hierarchy yields the equation � � ∂ nχ T ∂n − k 0 I k 0 = I k 0 +1 ≡ 0 But if χ T ≈ | n/n c − 1 | − ( δ − 1) then the left-hand side the equation diverges as | n/n c − 1 | − ( k 0 δ − 1) . The assumption h k 0 +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 g 3 ( r 12 , r 13 , r 23 ) = g 2 ( r 12 ) g 2 ( r 13 ) g 2 ( r 23 ) The Kirkwood superposition approximation expresses three-particle correlations in terms of two-particle ones h 3 ( r 12 , r 13 , r 23 ) = h 2 ( r 12 ) h 2 ( r 13 ) h 2 ( r 23 ) + h 2 ( r 12 ) h 2 ( r 13 ) + h 2 ( r 13 ) h 2 ( r 23 ) + h 2 ( r 12 ) h 2 ( r 23 ) 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 � 2 �� d r h 2 ( r ) 1 ∂h 2 ( r ) � d r = � 1 + h 2 ( r ) ∂n 1 + n d r h 2 ( 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 g 3 (1 , 2 , 3) g 3 (1 , 2 , 4) g 3 (1 , 3 , 4) g 3 (2 , 3 , 4) g 4 (1 , 2 , 3 , 4) = g 2 (1 , 2) g 2 (1 , 3) g 2 (1 , 4) g 2 (2 , 3) g 2 (2 , 4) g 2 (3 , 4) permits to express h 4 in terms of lower order correlations h 3 and h 2 . 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 y 2 by g 2 ( r ) = χ B ( r ) y 2 ( r ) where χ B ( 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 . χ B ( r ) = exp( E/k B T ) if σ < r < λσ χ B ( r ) = 1 if r > λσ The function y 2 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 h 2 ( r ) ≈ exp( κr ) r The constant κ vanishes if and only if Γ ≡ 1 + 8 φ [ y 2 ( σ ) B − λ 3 y 2 ( λσ )( B − 1) ] = 0 where B = exp( E/k B T ) , and φ = πnσ 3 / 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 � 1.5 � � � � 1 � � � � � 0.5 � � � � � � � � � 0 � � � � � � � � � � � � � � � 0 0.1 0.2 0.3 0.4 0.5 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 of 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
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