Abstract

Dynamic light-scattering techniques provide noninvasive probes of diverse media such as colloidal suspensions, granular materials, and foams. Traditional analysis relies on the Gaussian properties of the scattering process found in most experimental situations and uses second-order intensity-correlation functions. This approach fails in the presence of, among other things, the collective intermittent dynamics found in systems such as granular materials. By extending the existing formalism and introducing higher-order intensity-correlation functions, we show how to detect and quantify the intrinsic dynamics and switching statistics of intermittent processes. We then explore two systems: (1) an auger-driven granular column for which the granular dynamics are controlled and the formalism is tested and (2) a granular heap whose dynamics are a priori unknown but may, now, be characterized.

© 2001 Optical Society of America

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References

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  1. B. Berne, R. Pecora, Dynamic Light Scattering, with Applications to Chemistry, Biology, and Physics (Wiley, New York, 1976).
  2. H. Cummins, E. Pike, eds., Photon Correlation Spectroscopy and Velocimetry, Vol. B23 of the NATO Advanced Study Institutes Series (Plenum, New York, 1977).
  3. B. Chu, Laser Light Scattering: Basic Principles and Practice (Academic, New York, 1991).
  4. W. Brown, ed., Dynamic Light Scattering: The Method and Some Applications (Clarendon, Oxford, U.K., 1993).
  5. P. A. Lemieux, D. J. Durian, “Investigating non-Gaussian scattering processes by using nth-order intensity correlation functions,” J. Opt. Soc. Am. A 16, 1651–1664 (1999).
    [CrossRef]
  6. M. E. Cates, J. P. Wittmer, J. P. Bouchaud, P. Claudin, “Jamming, force chains, and fragile matter,” Phys. Rev. Lett. 81, 1841–1844 (1998).
    [CrossRef]
  7. A. J. Liu, S. R. Nagel, “Nonlinear dynamics—jamming is not just cool any more,” Nature 396, 21–22 (1998).
    [CrossRef]
  8. H. M. Jaeger, S. R. Nagel, R. P. Behringer, “Granular solids, liquids, and gases,” Rev. Mod. Phys. 68, 1259–1273 (1996).
    [CrossRef]
  9. P.-A. Lemieux, D. J. Durian, “From avalanches to fluid flow: a continuous picture of grain dynamics down a heap,” Phys. Rev. Lett. 85, 4273–4276 (2000).
    [CrossRef] [PubMed]
  10. E. Weeks, J. Crocker, A. Levitt, A. Schofield, D. Weitz, “Three-dimensional direct imaging of structural relaxation near the colloidal glass transition,” Science 287, 627–631 (2000).
    [CrossRef] [PubMed]
  11. A. Gopal, D. Durian, “Shear-induced ‘melting’ of an aqueous foam,” J. Colloid Interface Sci. 213, 169–178 (1999).
    [CrossRef] [PubMed]
  12. A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  13. A. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [CrossRef] [PubMed]
  14. S. Chopra, L. Mandel, “Higher-order correlation properties of a laser beam,” Phys. Rev. Lett. 30, 60–63 (1973).
    [CrossRef]
  15. C. D. Cantrel, M. Lax, W. A. Smith, “Third- and higher-order intensity correlation in laser light,” Phys. Rev. A 7, 175–181 (1973).
    [CrossRef]
  16. M. Corti, V. Degiorgio, “Intrinsic third-order correlations in laser light near threshold,” Phys. Rev. A 14, 1475–1478 (1976).
    [CrossRef]
  17. J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures, 2nd ed. (Wiley-Interscience, New York, 1985).
  18. R. F. Voss, J. Clarke, “Flicker (1/f) noise: equilibrium temperature and resistance fluctuations,” Phys. Rev. B 13, 556–573 (1976).
    [CrossRef]
  19. M. Weissman, “Low-frequency noise as a tool to study disordered materials,” Ann. Rev. Mater. Sci. 26, 395–429 (1996).
    [CrossRef]
  20. P. N. Pusey, “Number fluctuations of interacting particles,” J. Phys. A 12, 1805–1818 (1979).
    [CrossRef]
  21. H. M. Jaeger, L. Chu-heng, S. R. Nagel, “Relaxation at the angle of repose,” Phys. Rev. Lett. 62, 40–43 (1989).
    [CrossRef] [PubMed]

2000 (2)

P.-A. Lemieux, D. J. Durian, “From avalanches to fluid flow: a continuous picture of grain dynamics down a heap,” Phys. Rev. Lett. 85, 4273–4276 (2000).
[CrossRef] [PubMed]

E. Weeks, J. Crocker, A. Levitt, A. Schofield, D. Weitz, “Three-dimensional direct imaging of structural relaxation near the colloidal glass transition,” Science 287, 627–631 (2000).
[CrossRef] [PubMed]

1999 (2)

1998 (2)

M. E. Cates, J. P. Wittmer, J. P. Bouchaud, P. Claudin, “Jamming, force chains, and fragile matter,” Phys. Rev. Lett. 81, 1841–1844 (1998).
[CrossRef]

A. J. Liu, S. R. Nagel, “Nonlinear dynamics—jamming is not just cool any more,” Nature 396, 21–22 (1998).
[CrossRef]

1996 (2)

H. M. Jaeger, S. R. Nagel, R. P. Behringer, “Granular solids, liquids, and gases,” Rev. Mod. Phys. 68, 1259–1273 (1996).
[CrossRef]

M. Weissman, “Low-frequency noise as a tool to study disordered materials,” Ann. Rev. Mater. Sci. 26, 395–429 (1996).
[CrossRef]

1989 (1)

H. M. Jaeger, L. Chu-heng, S. R. Nagel, “Relaxation at the angle of repose,” Phys. Rev. Lett. 62, 40–43 (1989).
[CrossRef] [PubMed]

1983 (1)

1979 (1)

P. N. Pusey, “Number fluctuations of interacting particles,” J. Phys. A 12, 1805–1818 (1979).
[CrossRef]

1976 (2)

M. Corti, V. Degiorgio, “Intrinsic third-order correlations in laser light near threshold,” Phys. Rev. A 14, 1475–1478 (1976).
[CrossRef]

R. F. Voss, J. Clarke, “Flicker (1/f) noise: equilibrium temperature and resistance fluctuations,” Phys. Rev. B 13, 556–573 (1976).
[CrossRef]

1973 (2)

S. Chopra, L. Mandel, “Higher-order correlation properties of a laser beam,” Phys. Rev. Lett. 30, 60–63 (1973).
[CrossRef]

C. D. Cantrel, M. Lax, W. A. Smith, “Third- and higher-order intensity correlation in laser light,” Phys. Rev. A 7, 175–181 (1973).
[CrossRef]

1970 (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Behringer, R. P.

H. M. Jaeger, S. R. Nagel, R. P. Behringer, “Granular solids, liquids, and gases,” Rev. Mod. Phys. 68, 1259–1273 (1996).
[CrossRef]

Bendat, J. S.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures, 2nd ed. (Wiley-Interscience, New York, 1985).

Berne, B.

B. Berne, R. Pecora, Dynamic Light Scattering, with Applications to Chemistry, Biology, and Physics (Wiley, New York, 1976).

Bouchaud, J. P.

M. E. Cates, J. P. Wittmer, J. P. Bouchaud, P. Claudin, “Jamming, force chains, and fragile matter,” Phys. Rev. Lett. 81, 1841–1844 (1998).
[CrossRef]

Cantrel, C. D.

C. D. Cantrel, M. Lax, W. A. Smith, “Third- and higher-order intensity correlation in laser light,” Phys. Rev. A 7, 175–181 (1973).
[CrossRef]

Cates, M. E.

M. E. Cates, J. P. Wittmer, J. P. Bouchaud, P. Claudin, “Jamming, force chains, and fragile matter,” Phys. Rev. Lett. 81, 1841–1844 (1998).
[CrossRef]

Chopra, S.

S. Chopra, L. Mandel, “Higher-order correlation properties of a laser beam,” Phys. Rev. Lett. 30, 60–63 (1973).
[CrossRef]

Chu, B.

B. Chu, Laser Light Scattering: Basic Principles and Practice (Academic, New York, 1991).

Chu-heng, L.

H. M. Jaeger, L. Chu-heng, S. R. Nagel, “Relaxation at the angle of repose,” Phys. Rev. Lett. 62, 40–43 (1989).
[CrossRef] [PubMed]

Clarke, J.

R. F. Voss, J. Clarke, “Flicker (1/f) noise: equilibrium temperature and resistance fluctuations,” Phys. Rev. B 13, 556–573 (1976).
[CrossRef]

Claudin, P.

M. E. Cates, J. P. Wittmer, J. P. Bouchaud, P. Claudin, “Jamming, force chains, and fragile matter,” Phys. Rev. Lett. 81, 1841–1844 (1998).
[CrossRef]

Corti, M.

M. Corti, V. Degiorgio, “Intrinsic third-order correlations in laser light near threshold,” Phys. Rev. A 14, 1475–1478 (1976).
[CrossRef]

Crocker, J.

E. Weeks, J. Crocker, A. Levitt, A. Schofield, D. Weitz, “Three-dimensional direct imaging of structural relaxation near the colloidal glass transition,” Science 287, 627–631 (2000).
[CrossRef] [PubMed]

Degiorgio, V.

M. Corti, V. Degiorgio, “Intrinsic third-order correlations in laser light near threshold,” Phys. Rev. A 14, 1475–1478 (1976).
[CrossRef]

Durian, D.

A. Gopal, D. Durian, “Shear-induced ‘melting’ of an aqueous foam,” J. Colloid Interface Sci. 213, 169–178 (1999).
[CrossRef] [PubMed]

Durian, D. J.

P.-A. Lemieux, D. J. Durian, “From avalanches to fluid flow: a continuous picture of grain dynamics down a heap,” Phys. Rev. Lett. 85, 4273–4276 (2000).
[CrossRef] [PubMed]

P. A. Lemieux, D. J. Durian, “Investigating non-Gaussian scattering processes by using nth-order intensity correlation functions,” J. Opt. Soc. Am. A 16, 1651–1664 (1999).
[CrossRef]

Gopal, A.

A. Gopal, D. Durian, “Shear-induced ‘melting’ of an aqueous foam,” J. Colloid Interface Sci. 213, 169–178 (1999).
[CrossRef] [PubMed]

Jaeger, H. M.

H. M. Jaeger, S. R. Nagel, R. P. Behringer, “Granular solids, liquids, and gases,” Rev. Mod. Phys. 68, 1259–1273 (1996).
[CrossRef]

H. M. Jaeger, L. Chu-heng, S. R. Nagel, “Relaxation at the angle of repose,” Phys. Rev. Lett. 62, 40–43 (1989).
[CrossRef] [PubMed]

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Lax, M.

C. D. Cantrel, M. Lax, W. A. Smith, “Third- and higher-order intensity correlation in laser light,” Phys. Rev. A 7, 175–181 (1973).
[CrossRef]

Lemieux, P. A.

Lemieux, P.-A.

P.-A. Lemieux, D. J. Durian, “From avalanches to fluid flow: a continuous picture of grain dynamics down a heap,” Phys. Rev. Lett. 85, 4273–4276 (2000).
[CrossRef] [PubMed]

Levitt, A.

E. Weeks, J. Crocker, A. Levitt, A. Schofield, D. Weitz, “Three-dimensional direct imaging of structural relaxation near the colloidal glass transition,” Science 287, 627–631 (2000).
[CrossRef] [PubMed]

Liu, A. J.

A. J. Liu, S. R. Nagel, “Nonlinear dynamics—jamming is not just cool any more,” Nature 396, 21–22 (1998).
[CrossRef]

Lohmann, A.

Mandel, L.

S. Chopra, L. Mandel, “Higher-order correlation properties of a laser beam,” Phys. Rev. Lett. 30, 60–63 (1973).
[CrossRef]

Nagel, S. R.

A. J. Liu, S. R. Nagel, “Nonlinear dynamics—jamming is not just cool any more,” Nature 396, 21–22 (1998).
[CrossRef]

H. M. Jaeger, S. R. Nagel, R. P. Behringer, “Granular solids, liquids, and gases,” Rev. Mod. Phys. 68, 1259–1273 (1996).
[CrossRef]

H. M. Jaeger, L. Chu-heng, S. R. Nagel, “Relaxation at the angle of repose,” Phys. Rev. Lett. 62, 40–43 (1989).
[CrossRef] [PubMed]

Pecora, R.

B. Berne, R. Pecora, Dynamic Light Scattering, with Applications to Chemistry, Biology, and Physics (Wiley, New York, 1976).

Piersol, A. G.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures, 2nd ed. (Wiley-Interscience, New York, 1985).

Pusey, P. N.

P. N. Pusey, “Number fluctuations of interacting particles,” J. Phys. A 12, 1805–1818 (1979).
[CrossRef]

Schofield, A.

E. Weeks, J. Crocker, A. Levitt, A. Schofield, D. Weitz, “Three-dimensional direct imaging of structural relaxation near the colloidal glass transition,” Science 287, 627–631 (2000).
[CrossRef] [PubMed]

Smith, W. A.

C. D. Cantrel, M. Lax, W. A. Smith, “Third- and higher-order intensity correlation in laser light,” Phys. Rev. A 7, 175–181 (1973).
[CrossRef]

Voss, R. F.

R. F. Voss, J. Clarke, “Flicker (1/f) noise: equilibrium temperature and resistance fluctuations,” Phys. Rev. B 13, 556–573 (1976).
[CrossRef]

Weeks, E.

E. Weeks, J. Crocker, A. Levitt, A. Schofield, D. Weitz, “Three-dimensional direct imaging of structural relaxation near the colloidal glass transition,” Science 287, 627–631 (2000).
[CrossRef] [PubMed]

Weigelt, G.

Weissman, M.

M. Weissman, “Low-frequency noise as a tool to study disordered materials,” Ann. Rev. Mater. Sci. 26, 395–429 (1996).
[CrossRef]

Weitz, D.

E. Weeks, J. Crocker, A. Levitt, A. Schofield, D. Weitz, “Three-dimensional direct imaging of structural relaxation near the colloidal glass transition,” Science 287, 627–631 (2000).
[CrossRef] [PubMed]

Wirnitzer, B.

Wittmer, J. P.

M. E. Cates, J. P. Wittmer, J. P. Bouchaud, P. Claudin, “Jamming, force chains, and fragile matter,” Phys. Rev. Lett. 81, 1841–1844 (1998).
[CrossRef]

Ann. Rev. Mater. Sci. (1)

M. Weissman, “Low-frequency noise as a tool to study disordered materials,” Ann. Rev. Mater. Sci. 26, 395–429 (1996).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

J. Colloid Interface Sci. (1)

A. Gopal, D. Durian, “Shear-induced ‘melting’ of an aqueous foam,” J. Colloid Interface Sci. 213, 169–178 (1999).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

J. Phys. A (1)

P. N. Pusey, “Number fluctuations of interacting particles,” J. Phys. A 12, 1805–1818 (1979).
[CrossRef]

Nature (1)

A. J. Liu, S. R. Nagel, “Nonlinear dynamics—jamming is not just cool any more,” Nature 396, 21–22 (1998).
[CrossRef]

Phys. Rev. A (2)

C. D. Cantrel, M. Lax, W. A. Smith, “Third- and higher-order intensity correlation in laser light,” Phys. Rev. A 7, 175–181 (1973).
[CrossRef]

M. Corti, V. Degiorgio, “Intrinsic third-order correlations in laser light near threshold,” Phys. Rev. A 14, 1475–1478 (1976).
[CrossRef]

Phys. Rev. B (1)

R. F. Voss, J. Clarke, “Flicker (1/f) noise: equilibrium temperature and resistance fluctuations,” Phys. Rev. B 13, 556–573 (1976).
[CrossRef]

Phys. Rev. Lett. (4)

H. M. Jaeger, L. Chu-heng, S. R. Nagel, “Relaxation at the angle of repose,” Phys. Rev. Lett. 62, 40–43 (1989).
[CrossRef] [PubMed]

P.-A. Lemieux, D. J. Durian, “From avalanches to fluid flow: a continuous picture of grain dynamics down a heap,” Phys. Rev. Lett. 85, 4273–4276 (2000).
[CrossRef] [PubMed]

M. E. Cates, J. P. Wittmer, J. P. Bouchaud, P. Claudin, “Jamming, force chains, and fragile matter,” Phys. Rev. Lett. 81, 1841–1844 (1998).
[CrossRef]

S. Chopra, L. Mandel, “Higher-order correlation properties of a laser beam,” Phys. Rev. Lett. 30, 60–63 (1973).
[CrossRef]

Rev. Mod. Phys. (1)

H. M. Jaeger, S. R. Nagel, R. P. Behringer, “Granular solids, liquids, and gases,” Rev. Mod. Phys. 68, 1259–1273 (1996).
[CrossRef]

Science (1)

E. Weeks, J. Crocker, A. Levitt, A. Schofield, D. Weitz, “Three-dimensional direct imaging of structural relaxation near the colloidal glass transition,” Science 287, 627–631 (2000).
[CrossRef] [PubMed]

Other (5)

B. Berne, R. Pecora, Dynamic Light Scattering, with Applications to Chemistry, Biology, and Physics (Wiley, New York, 1976).

H. Cummins, E. Pike, eds., Photon Correlation Spectroscopy and Velocimetry, Vol. B23 of the NATO Advanced Study Institutes Series (Plenum, New York, 1977).

B. Chu, Laser Light Scattering: Basic Principles and Practice (Academic, New York, 1991).

W. Brown, ed., Dynamic Light Scattering: The Method and Some Applications (Clarendon, Oxford, U.K., 1993).

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures, 2nd ed. (Wiley-Interscience, New York, 1985).

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Figures (4)

Fig. 1
Fig. 1

Experimental apparatus for the two sample systems considered: (a) A toy system composed of a granular column driven in a predictable fashion by an auger. It exhibits, by design, ideal on–off intermittent dynamics. (b) A granular-heap system driven by the addition of grains at its apex at a controlled flow rate Q. It exhibits intrinsically intermittent dynamics for Q smaller than a certain threshold.

Fig. 2
Fig. 2

Intermittency statistics: The switching-probability functions P 00 and P 0 are shown along with the underlying state-duration probability function p 0 for a variety of switching processes [Eqs. (32)–(34)]. For the sake of simplicity, we chose identical functional forms for both p 1 and p 0, all with t 0 = 2 s and t 1 = 3 s.

Fig. 3
Fig. 3

Second- and fourth-order intensity-correlation functions for the auger system shown in Fig. 1(a). The first decay of g (2) is due to the rapid motion of the grains of sand while the auger is turning, whereas the second decay of g (2) is a sign of the intermittent nature of the dynamics. The periodic structure present in the baseline of g (4) and absent from g (2) is a telltale sign of periodic intermittency. To verify the combined equations (18), (20), and (33), we generate predictions (shown as the solid curves) for the remaining correlation functions by using solely g (2) from the continuous case and the known values of t 0 and t 1.

Fig. 4
Fig. 4

Intermittency statistics for the granular-heap flow shown in Fig. 1(b). P 0(τ) is the probability of being in the same static state after a time interval τ, and t 0 is the average time between successive avalanches. Data are shown for flow rates ranging from 0.03 (lighter-shade symbols) to 0.3 (darker-shade symbols) g/s. Over this range t 0 varies from 6 to 96 s. Note that there is no systematic dependence between the shape of P 0(τ) and the flow rate. The scatter in the data results from the relatively small number of statistical samples at large τ and, accordingly, vanishes as τ → 0. For comparison, we plot P 0(τ) for the quasi-periodic [Eq. (34)] and the random-telegraph switching processes [Eqs. (32)] as solid and dashed curves, respectively.

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

g(2)(τ)=1+|E(0)E*(τ|2|EE*|21+|γ(τ)|2,
g2τ=1+1-f0|γτ|2+f0P0τ.
gT4τ=I0ITIτIτ+T/I4
gnτ1, τ2,, τn-1=I0Iτ1Iτ2  Iτn-1/In
g2τ1=1+β|γ01|2,
g3τ1, τ2=1+β|γ01|2+|γ12|2+|γ20|2+2β2 Reγ01γ12γ20,
g4τ1, τ2, τ3=1+β|γ01|2+|γ02|2+|γ03|2+|γ12|2+|γ13|2+|γ23|2+β2|γ01|2|γ23|2+|γ02|2|γ13|2+|γ03|2|γ12|2+2β2 Reγ01γ12γ20+γ01γ13γ30+γ02γ23γ30+γ12γ23γ31+2β3 Reγ01γ12γ23γ30+γ02γ23γ31γ10+γ02γ21γ13γ30.
In/In=11+β1+2β1+n-1β
It=I1txt+Ix¯t.
g2τ1=y0y1+β|γ01|2x0x1,
g3τ1, τ2=y0y1y2+β|γ01|2x0x1y2+|γ12|2y0x1x2+|γ20|2x0y1x2+2β2 Reγ01γ12γ20x0x1x2,
g4τ1, τ2, τ3=y0y1y2y3+β|γ01|2x0x1y2y3+|γ02|2x0y1x2y3+|γ03|2x0y1y2x3+β|γ12|2y0x1x2y3+|γ13|2y0x1y2x3+|γ23|2y0y1x2x3+β2|γ01|2|γ23|2+|γ02|2|γ13|2+|γ03|2|γ12|2x0x1x2x3+2β2 Reγ01γ12γ20x0x1x2y3+γ01γ13γ30x0x1y2x3+γ02γ23γ30x0y1x2x3+γ12γ23γ31y0x1x2x3+2β3 Reγ01γ12γ23γ30+γ02γ23γ31γ10+γ02γ21γ13γ30x0x1x2x3.
x0x1x2=f1P11τ01P11τ12.
ImInf1P11τ+ImInf1P10τ+ImInf0P01τ+ImInf0P00Στ+Im+nf0P0τ.
I0IτI2y0yτ=f1P11τ+f1P10τ+f0P01τ+f0P00Στ+1+βf0P0τ=1+βf0P0τ.
I0ITIτIτ+TI4y0yTyτyτ+T=1+βf02+βP00τ+4+10β+6β2P0τ.
g2τ=1+βf0P0τ+βf1P11τ|γτ|2.
g2τ1=1+βf0P0τ01+βf1|γ01|2,
g3τ1, τ2=1+βf0P0τ01+P0τ12+1+2βP0τ02+βf1|γ01|2+|γ12|2+|γ20|2+2β2f1 Reγ01γ12γ20,
g4τ1, τ2, τ3=1+βf0P0τ01+P0τ12+P0τ23+1+2βP0τ02+1+2βP0τ13+βf0βP0τ01P00τ12P0τ23+1+6β+6β2P0τ03+βf1|γ01|21+βP10τ12P0τ23+|γ02|2+|γ03|2+|γ12|2+|γ13|2+|γ23|21+βP0τ01P10τ12+β2f1|γ01|2|γ23|2+|γ02|2|γ13|2+|γ03|2|γ12|2+2β2f1 Reγ01γ12γ20+γ01γ13γ30+γ02γ23γ30+γ12γ23γ31+2β3f1 Reγ01γ12γ23γ30+γ02γ23γ31γ10+γ02γ21γ13γ30.
gT4τ=1+βf02+βP00τ+4+10β+6β2P0τ+βf1|γτ|22+β|γτ|2+|γτ-T|21+β|γτ-T|2.
I0Iτ/I2=I10I1τx0xτ/I2=1+β|γτ|2P11τ/f1.
g2τ=1+β|γτ|2f1P11τ+β|γ¯τ|2f0P00τ,
g3τ1, τ2=1+β|γ01|2f1P11τ01+|γ12|2f1P11τ12+|γ20|2f1P11τ20+|γ¯01|2f0P00τ01+|γ¯12|2f0P00τ12+|γ¯20|2f0P00τ20+2β2Reγ01γ12γ20f1P11τ01P11τ12+2β2Reγ¯01γ¯12γ¯20f0P00τ01P00τ12.
P0τ=0dut0u+τ p0τ0dτ0
=τ p0τ0τ0-τdτ0.
P00τ=P0τ+n=1 P002nτ.
P002τ=0dut0uu+τ p0τ0dτ00u+τ-τ0 p1τ1dτ1×u+τ-τ0-τ1-τ2 p0τ2dτ2.
P˜002s=0 P002τexp-sτdτ=p˜1s1-p˜0s2/t0s2.
P˜002n>0s=p˜1snp˜0sn-11-p˜0s2/t0s2.
P˜00s=P˜0s+n=1 P˜002ns=1s2t0st0-1+p˜0s+p˜1s1-p˜0s21-p˜0sp˜1s,
P0τ=exp-τ/t0, P00τ=f0+f1 exp-τ/tA, p0τ=exp-τ/t0/t0, p1τ=exp-τ/t1/t1,
P0τ=1-τ/t0if τt00if τt0,P00τ=1-τ/t0if τ<mint0, t10if t0<τ<t11-t1/t0if t1<τ<t0,p0τ=δτ-t0,p1τ=δτ-t1,
P0τ=8τ/t03+12τ/t02+9τ/t02+3τ/t0exp-τ/t0/43, p0τ=τ3/t0/44exp-τ/t0/46,  p1τ=τ3/t1/44exp-τ/t1/46,

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