Fractional Liouville and BBGKI Equations

Vasily E. Tarasov

REVIEW

Fractional Liouville and BBGKI Equations

Vasily E. Tarasov

Chaotic Dynamics · Submitted 28 Feb 2006 · 25 references

Pith's verdict: The fractional BBGKI derivation is formally smooth, but the normalization condition it rests on is origin-dependent and cannot hold for all y; that flaw undermines the paper's central physical claim.

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arXiv IDnlin/0602062 (version 1)

Pith IDBB4BG32Y, Pith's permanent record of this paper and its review

CitationVasily E. Tarasov. "Fractional Liouville and BBGKI Equations." arXiv:nlin/0602062 (2006). https://pith.science/paper/BB4BG32Y

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Claim paper

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Editor's note

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Bottom line: the fractional BBGKI hierarchy is a clean formal derivation, but it rests on a fractional normalization condition that is origin-dependent and, as stated, impossible. Fixing that flaw may be possible, but as written the paper's central physical claim doesn't hold up.

What's new: the fractional BBGKI equations (78) and (84) and the fractional Vlasov equation (83). The author repeats the classical derivation with the fractional measure dμ_α, and that part is internally consistent. The step from the fractional Liouville equation to the hierarchy is standard calculus; if you grant the measure, the hierarchy follows. I also appreciate that the paper is explicit about interpreting fractional phase space as fractal, and it connects to Zaslavsky's fractional kinetic program.

Soft spots: the main one is Eq (4). For α≠1, (I^α_+ + I^α_-)ρ(y) is 1/Γ(α)∫ |y-x|^{α-1} ρ(x) dx. For a normalizable ρ, this tends to 0 at |y|→∞ when 0<α<1 and diverges when α>1. It cannot be 1 for all y. The paper introduces y as an arbitrary split point but never restricts it. If you fix y=0, as the paper later does, then the fractional measure is origin-dependent and all averages become functions of that origin. That's a real physical claim—a preferred point in phase space—that the paper doesn't justify. This isn't a minor technicality; it determines whether the derived equations are transport equations or integral-transform identities for a chosen origin.

Second, there's a notational conflation: Eqs (21)-(22) use K = dq/dt, while Eq (24) redefines the role with dq^α/dt = K. The Liouville operator in Eq (31) mixes the two, and the N-particle equations need this disambiguated. Third, the paper offers no application or falsifiable prediction; it's purely formal.

Who's it for: people already working in fractional kinetic theory or non-Hamiltonian statistical mechanics. It's a useful derivation if you're already committed to the fractional measure program. As a standalone claim about fractal systems, it overreaches.

Recommendation: I'd send it to peer review. The derivations are careful and the approach is active in the literature, and the normalization issue, while serious, is conceivably fixable by fixing y and translating it into a modeling assumption. A referee can decide if that rescue works. But if the author can't justify the y-dependence, the paper remains a formal artifact rather than physics.

Referee report

Significance

The algebraic chain from the fractional normalization to the BBGKI hierarchy is internally consistent, and the paper is commendably explicit: the definitions of fractional reduced distributions, the structure of the collision term I(ρ~2), and the Vlasov limit are clearly written. If the fractional measure and the symmetrized distributions were physically justified, the resulting hierarchy would provide a useful formal framework for deriving fractional kinetic equations from an N-particle Liouville-type equation. As it stands, the main results are conditional on a postulate whose normalization properties are problematic and whose physical interpretation is not established; the paper therefore offers a formal calculus rather than a validated statistical-mechanical derivation.

Major objections (4)

[Sec. 2, Eqs. (4)-(6)] The fractional normalization condition is not a normalization for arbitrary y. For 0<α<1, for any rapidly decaying ρ∈L1, dominated convergence gives (I^α_+ρ)(y)+(I^α_-ρ)(y)→0 as |y|→∞; for α>1 the first term typically grows like y^{α-1}∫ρ/Γ(α). Hence Eq. (4) cannot hold for all y, although y is introduced in Eq. (1) as an arbitrary split point. The fractional normalization therefore defines a different condition for each y, and the entire subsequent derivation—fractional Liouville equation (18), fractional averages (64), reduced distributions (67), and the BBGKI hierarchy (78)-(84)—inherits this arbitrary origin. The paper must either fix y and justify the resulting breaking of translation invariance, or show that the results are y-independent; neither is done.
[Sec. 7, Eq. (71)] In the reduction to the first BBGKI equation, the boundary term is set to zero because ρ~N vanishes at infinity. The fractional integral, however, requires the product |q_k|^{α-1}K_kρ~N (or the analogous product appearing after the change of variables to q^α_k) to vanish at both infinities. For α>1 the weight |q_k|^{α-1} grows, so a stronger decay condition is needed; for α<1 the integrability at infinity must be checked separately. These conditions are not stated or verified, and they are needed for the derivation of Eqs. (78) and (84).
[Sec. 5.4, Eq. (54)] Equation (54) is not well defined on the whole phase space: the factor (q_k p_k)^{1-α} is not real for negative q_k or p_k when α is not an integer, and it diverges on the axes. The correct transformation from Eq. (52) to usual phase-space variables involves |q_k p_k|^{1-α} together with sign factors. The paper neither states these restrictions nor discusses them, which affects the claimed interpretation of fractional Hamiltonian systems as non-Hamiltonian systems in the usual phase space.
[Secs. 4, 5, 8] The paper interprets α as a fractal mass dimension and asserts that fractional kinetic equations are better approximations for systems on fractals, but it does not connect the derived hierarchy to any concrete model, such as Zaslavsky's fractional Fokker-Planck equation. In particular, there is no argument that the T-symmetrized distribution ρ~=Tρ is the physical phase-space density, nor a prescription for determining α and y from a physical system. A concrete consistency test would be to show that, in a suitable scaling limit, Eq. (78) reduces to a known fractional kinetic equation; absent such a test, the results remain formal.

Minor objections (4)

[Sec. 2, Eq. (7)] The phase-space measure should use the absolute value, dµ_α(q,p)=|qp|^{α-1}/Γ^2(α)dq∧dp, and the wedge product should be dµ_α(q)∧dµ_α(p); as written, the expression (qp)^{α-1} is not real for negative q or p when α is non-integer, and the repeated dµ_α(q) in the text is a misprint.
[Secs. 3 and 7, Eqs. (21)-(22), (24), (27)-(28)] The symbol K denotes dq/dt in Eqs. (21)-(22) but denotes the right-hand side of dq^α/dt in Eqs. (24) and (27)-(28). Since these are different velocity fields, the same letter should not be used for both; this conflation obscures the relation between Eq. (22) and Eq. (28).
[Title and throughout] The acronym should be BBGKY (Bogoliubov-Born-Green-Kirkwood-Yvon), not BBGKI; the reference list also omits Yvon's contribution to the hierarchy.
[Sec. 2 and Sec. 4] The parameter range is left unclear: Sec. 2 assumes 1<p<1/α and hence α<1, while Sec. 4 uses α values up to 3 for the fractal dimension. The domain of α for which the fractional Liouville and BBGKI equations are claimed to hold should be stated explicitly and consistently.

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The paper

The paper claims

This paper derives fractional Liouville and BBGKI equations from a fractional normalization condition, giving fractional kinetic theory a classical statistical-mechanics foundation.

The reading

This paper sets out to show that the two foundational equations of classical kinetic theory—Liouville's equation and the BBGKI hierarchy—survive unchanged in form when the ordinary phase-space measure is replaced by a fractional one. The starting point is a fractional normalization condition, in which the usual integral over phase space is replaced by fractional integrals with measure $d\mu_\alpha = |qp|^{\alpha-1}/\Gamma^2(\alpha)\,dq\,dp$. From this condition and fractional analogs of Hamilton's equations, the paper derives a fractional Liouville equation $d\tilde\rho/dt + \Omega_\alpha \tilde\rho = 0$, and then, using fractional reduced distribution functions, a fractional BBGKI hierarchy whose first member is Eq. (78) and whose independent-particle closure is a fractional Vlasov equation, Eq. (83). If the derivation is sound, fractional kinetic equations such as Zaslavsky's fractional Fokker-Planck equation can be obtained from first principles rather than postulated, and systems living on fractals acquire a statistical-mechanical foundation. The paper also interprets the fractional phase space as a space with fractal dimension or fractional measure, and shows that ordinary Hamiltonian systems are dissipative in this space.

What carries the argument

The machinery has three parts. First, the fractional measure $d\mu_\alpha(x) = |x|^{\alpha-1}/\Gamma(\alpha)\,dx$ and the symmetrization operator $T_x f(x) = \frac12(f(y-x)+f(y+x))$ convert the fractional normalization condition into an ordinary-looking integral over a new measure, so the classical normalization argument can be rerun. Second, fractional powers $x^\alpha = \operatorname{sgn}(x)|x|^\alpha$ turn fractional integrals into derivatives with respect to $q^\alpha$ and $p^\alpha$, so the classical derivation's integration-by-parts steps go through in the fractional setting. Third, the omega function $\Omega_\alpha = d/dt\ln\{q^\alpha_t, p^\alpha_t\}_0$ encodes the rate of fractional phase-volume change and plays the role of the divergence of the velocity field in the ordinary continuity equation. Together these allow each step of the classical Liouville and BBGKI derivation—normalization, continuity, integration by parts, and symmetric reduction—to be rerun with only the measure changed.

Extended reading notes

Core claim

The paper's central claim is that the fractional normalization condition, written as $(I^\alpha_+ + I^\alpha_-)\rho = 1$ with the fractional measure $d\mu_\alpha = |qp|^{\alpha-1}/\Gamma^2(\alpha)\,dq\,dp$ and the symmetrization operator $T_qT_p$, is enough to reproduce the entire structure of classical statistical mechanics. Concretely, the paper derives the fractional Liouville equation $d\tilde\rho/dt + \Omega_\alpha \tilde\rho = 0$, where $\Omega_\alpha = d/dt\ln\{q^\alpha_t, p^\alpha_t\}_0$ measures the rate of fractional phase-volume change; for ordinary nondissipative systems like the harmonic oscillator, $\Omega_\alpha \neq 0$ when $\alpha \neq 1$, so such systems become dissipative in the fractional phase space. The paper then defines fractional average values and fractional reduced distribution functions, and from the fractional Liouville equation obtains the first BBGKI equation (78), the second BBGKI equation (84), and, under the independent-particle product ansatz, the fractional Vlasov equation (83). The conclusion is that fractional kinetic equations inherit the deductive structure of classical statistical mechanics rather than being ad hoc generalizations.

Load-bearing premise

The whole derivation stands or falls on one premise: that a fractal system's phase-space distribution should be normalized with the fractional measure $|qp|^{\alpha-1}/\Gamma^2(\alpha)\,dq\,dp$ rather than with the ordinary measure $dq\,dp$; the paper asserts this premise in Section 2 without testing it on any physical example.

Editorial extensions

If this is right

Fractional kinetic equations, including Zaslavsky's fractional Fokker-Planck equation, can be derived from the fractional Liouville and BBGKI equations instead of being postulated by analogy.
Ordinary, nondissipative Hamiltonian systems are dissipative in the fractional phase space, so their fractional Liouville equation describes phase-volume change; this extends the formalism of statistical mechanics to a broad class of non-Hamiltonian systems.
The first fractional BBGKI equation with the independent-particle product ansatz closes into the fractional Vlasov equation, Eq. (83), giving a mean-field kinetic equation for systems with fractional phase space.
The fractional BBGKI hierarchy can serve as a starting point for fractional hydrodynamic equations and Enskog-type transport equations, and for stationary fractional Gibbs distributions with non-Gaussian statistics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable consequence the paper leaves implicit: for a real chaotic system confined to a fractal attractor, one should observe that the fractional normalization integral is constant in time for exactly $\alpha$ equal to the attractor's mass or correlation dimension, and drifts for other $\alpha$; this could be checked by direct time-series analysis.
Because the derivation's only structural input is the measure, the entire standard toolbox—molecular chaos closure, Boltzmann-Enskog approximations, Grad moment hierarchies—could be transplanted to the fractional setting; the paper only develops the Vlasov closure.
If the fractional Liouville operator is non-Hermitian in the fractional inner product, the hierarchy may naturally produce irreversibility without added collisions; the paper notes dissipation but does not discuss entropy production or H-theorems in the fractional setting.

Original abstract

We consider the fractional generalizations of Liouville equation. The normalization condition, phase volume, and average values are generalized for fractional case.The interpretation of fractional analog of phase space as a space with fractal dimension and as a space with fractional measure are discussed. The fractional analogs of the Hamiltonian systems are considered as a special class of non-Hamiltonian systems. The fractional generalization of the reduced distribution functions are suggested. The fractional analogs of the BBGKI equations are derived from the fractional Liouville equation.

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We consider the fractional generalizations of Liouville equation. The normalization condition, phase volume, and average values are generalized for fractional case.The interpretation of fractional analog of phase space as a space with fractal dimension and as a space with fractional measure are discussed. The fractional analogs of the Hamiltonian systems are considered as a special class of non-Hamiltonian systems. The fractional generalization of the reduced distribution functions are suggested. The fractional analogs of the BBGKI equations are derived from the fractional Liouville equation.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider the fractional generalizations of Liouville equation.	contribution 0.88	0.77	0.06	asserted 0.86
s1	The normalization condition, phase volume, and average values are generalized for fractional case.The interpretation of fractional analog of phase space as a space with fractal dimension and as a space with fractional measure are discussed.	contribution 0.50	0.75	0.12	asserted 1.00
s2	The fractional analogs of the Hamiltonian systems are considered as a special class of non-Hamiltonian systems.	contribution 0.74	0.65	0.08	asserted 1.00
s3	The fractional generalization of the reduced distribution functions are suggested.	contribution 0.98	0.85	0.13	hedged possibility 0.75
s4	The fractional analogs of the BBGKI equations are derived from the fractional Liouville equation.	result 0.61	0.92	0.47	asserted 1.00

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References

Works cited (25)

[1]

Zaslavsky G M 2002 Phys. Rep. 371, 461

View on Pith 2002
[2]

Zaslavsky G M 1994 Physica D 76, 110

View on Pith 1994
[3]

Zaslavsky G M and Weitzner H 2002 Some Applications of Fractional Equations Preprint nlin.CD/0212024

View on Pith arXiv 2002
[4]

Mandelbrot B 1983 The Fractal Geometry of Nature (New York: W. H. Freeman and Co.)

View on Pith 1983
[5]

Schroeder M 1990 Fractals, Chaos, Power Laws (New York: W.H. Freeman Co.) sec. 10

View on Pith 1990
[6]

Isihara A 1971 Statistical Physics (New York: Academic Press) Appendix IV, and sec. 7.5

View on Pith 1971
[7]

Resibois P and De Leener M 1977 Classical Kinetic Theory of Fluids (New York: John Wyley and Sons) sec. IX.4

View on Pith 1977
[8]

Forster D 1975 Hydrodynamics Fluctuations, Broken Symmetry, and Correla tion Functions (London: W.E. Benjamin Inc.) sec. 6.4

View on Pith 1975
[9]

Kornfeld I P, Sinai Ja G and Fomin S V 1980 Ergodic Theory (Moscow: Nauka) sec. 2.2

View on Pith 1980
[10]

Tarasov V E 2004 Chaos 14, 123 ( Preprint nlin.CD/0312044)

View on Pith arXiv 2004
[11]

Christensen R M 1979 Mechanics of Composite Materials (New York: Wiley)

View on Pith 1979
[12]

Fu-Yao Ren, Jin-Rong Liang, Xiao-Tian Wang, and Wei-Yuan Qiu 20 03 Chaos, Solitons and Fractals 16, 107

View on Pith
[13]

Cottrill-Shepherd K and Naber M 2001 J. Math. Phys. 42, 2203 ( Preprint math- ph/0301013)

View on Pith arXiv 2001
[14]

Samko S G, Kilbas A A and Marichev O I 1987 Integrals and Derivatives of Fractional Order and Applications (Minsk: Nauka i Tehnika) or 1993 Fractional Integrals and Deriva- tives Theory and Applications (New York: Gordon and Breach)

View on Pith 1987
[15]

Tarasov V E 1997 Theor. Math. Phys. 110, 57

View on Pith 1997
[16]

Collins J C 1984 Renormalization (Cambridge University Press) sec 4.1

View on Pith 1984
[17]

Bogolybow N 1946 J. Phys. USSR 10, 265 19

View on Pith 1946
[18]

Gurov K P 1966 Foundation of Kinetic Theory. Method of N.N. Bogoliubov. (Moscow: Nauka)

View on Pith 1966
[19]

Petrina D Ya, Gerasimenko V I and Malishev P V 1985 Mathematical Basis of Classical Statistical Mechanics (Kiev: Naukova dumka)

View on Pith 1985
[20]

Born M and Green H S 1947 Proc. Roy. Soc. London A 188, 168

View on Pith 1947
[21]

Bogoliubov N N and Bogoliubov N N (Jr.) 1984 Introduction to Quantum Statistic Me- chanics (Moscow: Nauka, Moscow) sec. 1.4

View on Pith 1984
[22]

Tarasov V E 2002 Phys. Rev. E 66, 056116 ( Preprint quant-ph/0311177)

View on Pith Pith review arXiv 2002
[23]

Tarasov V E 2003 Mod. Phys. Lett. B 17, 1219 ( Preprint cond-mat/0311536)

View on Pith Pith review arXiv 2003
[24]

Tarasov V E 2001 Phys. Lett. A 288, 173 ( Preprint quant-ph/0311159)

View on Pith Pith review arXiv 2001
[25]

Tarasov V E 2001 Moscow Univ. Phys. Bull. 56/6, 5 20

View on Pith 2001

How the literature uses it

Papers whose main result relates to this one's

How each label was checked · JSON

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

The fractional analogs of the Bogoliubov hierarchy equations are derived from the fractional Liouville equation.

Fractional Systems and Fractional Bogoliubov Hierarchy Equations · 0.92

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Zaslavsky G M 2002 Phys. Rep. 371, 461

View on Pith 2002

[2] [2]

Zaslavsky G M 1994 Physica D 76, 110

View on Pith 1994

[3] [3]

Zaslavsky G M and Weitzner H 2002 Some Applications of Fractional Equations Preprint nlin.CD/0212024

View on Pith arXiv 2002

[4] [4]

Mandelbrot B 1983 The Fractal Geometry of Nature (New York: W. H. Freeman and Co.)

View on Pith 1983

[5] [5]

Schroeder M 1990 Fractals, Chaos, Power Laws (New York: W.H. Freeman Co.) sec. 10

View on Pith 1990

[6] [6]

Isihara A 1971 Statistical Physics (New York: Academic Press) Appendix IV, and sec. 7.5

View on Pith 1971

[7] [7]

Resibois P and De Leener M 1977 Classical Kinetic Theory of Fluids (New York: John Wyley and Sons) sec. IX.4

View on Pith 1977

[8] [8]

Forster D 1975 Hydrodynamics Fluctuations, Broken Symmetry, and Correla tion Functions (London: W.E. Benjamin Inc.) sec. 6.4

View on Pith 1975

[9] [9]

Kornfeld I P, Sinai Ja G and Fomin S V 1980 Ergodic Theory (Moscow: Nauka) sec. 2.2

View on Pith 1980

[10] [10]

Tarasov V E 2004 Chaos 14, 123 ( Preprint nlin.CD/0312044)

View on Pith arXiv 2004

[11] [11]

Christensen R M 1979 Mechanics of Composite Materials (New York: Wiley)

View on Pith 1979

[12] [12]

Fu-Yao Ren, Jin-Rong Liang, Xiao-Tian Wang, and Wei-Yuan Qiu 20 03 Chaos, Solitons and Fractals 16, 107

View on Pith

[13] [13]

Cottrill-Shepherd K and Naber M 2001 J. Math. Phys. 42, 2203 ( Preprint math- ph/0301013)

View on Pith arXiv 2001

[14] [14]

Samko S G, Kilbas A A and Marichev O I 1987 Integrals and Derivatives of Fractional Order and Applications (Minsk: Nauka i Tehnika) or 1993 Fractional Integrals and Deriva- tives Theory and Applications (New York: Gordon and Breach)

View on Pith 1987

[15] [15]

Tarasov V E 1997 Theor. Math. Phys. 110, 57

View on Pith 1997

[16] [16]

Collins J C 1984 Renormalization (Cambridge University Press) sec 4.1

View on Pith 1984

[17] [17]

Bogolybow N 1946 J. Phys. USSR 10, 265 19

View on Pith 1946

[18] [18]

Gurov K P 1966 Foundation of Kinetic Theory. Method of N.N. Bogoliubov. (Moscow: Nauka)

View on Pith 1966

[19] [19]

Petrina D Ya, Gerasimenko V I and Malishev P V 1985 Mathematical Basis of Classical Statistical Mechanics (Kiev: Naukova dumka)

View on Pith 1985

[20] [20]

Born M and Green H S 1947 Proc. Roy. Soc. London A 188, 168

View on Pith 1947

[21] [21]

Bogoliubov N N and Bogoliubov N N (Jr.) 1984 Introduction to Quantum Statistic Me- chanics (Moscow: Nauka, Moscow) sec. 1.4

View on Pith 1984

[22] [22]

Tarasov V E 2002 Phys. Rev. E 66, 056116 ( Preprint quant-ph/0311177)

View on Pith Pith review arXiv 2002

[23] [23]

Tarasov V E 2003 Mod. Phys. Lett. B 17, 1219 ( Preprint cond-mat/0311536)

View on Pith Pith review arXiv 2003

[24] [24]

Tarasov V E 2001 Phys. Lett. A 288, 173 ( Preprint quant-ph/0311159)

View on Pith Pith review arXiv 2001

[25] [25]

Tarasov V E 2001 Moscow Univ. Phys. Bull. 56/6, 5 20

View on Pith 2001