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A dynamic theory for contact angle hysteresis on chemically rough boundary
Computer-assisted equilibrium validation for the diblock copolymer model
Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA |
The diblock copolymer model is a fourth-order parabolic partial differential equation which models phase separation with fine structure. The equation is a gradient flow with respect to an extension of the standard van der Waals free energy functional which involves nonlocal interactions. Thus, the long-term dynamical behavior of the diblock copolymer model is described by its finite-dimensional attractor. However, even on one-dimensional domains, the full structure of the underlying equilibrium set is not fully understood. In this paper, we develop a rigorous computational approach for establishing the existence of equilibrium solutions of the diblock copolymer model. We consider both individual solutions, as well as pieces of solution branches in a parameter-dependent situation. The results are presented for the case of one-dimensional domains, and can easily be implemented using standard interval arithmetic packages.
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, San Diego -London, 1975.
![]() ![]() |
[2] |
M. Bahiana and Y. Oono,
Cell dynamical system approach to block copolymers, Physical Review A, 41 (1990), 6763-6771.
doi: 10.1103/PhysRevA.41.6763. |
[3] |
P. W. Bates and P. C. Fife,
Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening, Physica D, 43 (1990), 335-348.
doi: 10.1016/0167-2789(90)90141-B. |
[4] |
P. W. Bates and P. C. Fife,
The dynamics of nucleation for the Cahn-Hilliard equation, SIAM Journal on Applied Mathematics, 53 (1993), 990-1008.
doi: 10.1137/0153049. |
[5] |
D. Blömker, B. Gawron and T. Wanner,
Nucleation in the one-dimensional stochastic Cahn-Hilliard model, Discrete and Continuous Dynamical Systems, Series A, 27 (2010), 25-52.
doi: 10.3934/dcds.2010.27.25. |
[6] |
C.-K. Chen and P. C. Fife,
Nonlocal models of phase transitions in solids, Advances in Mathematical Sciences and Applications, 10 (2000), 821-849.
|
[7] |
R. Choksi, M. A. Peletier and J. F. Williams,
On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM Journal on Applied Mathematics, 69 (2009), 1712-1738.
doi: 10.1137/080728809. |
[8] |
R. Choksi and X. Ren,
On the derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113 (2003), 151-176.
doi: 10.1023/A:1025722804873. |
[9] |
E. B. Davies, Linear Operators and Their Spectra, vol. 106 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2007. |
[10] |
S. Day, J.-P. Lessard and K. Mischaikow,
Validated continuation for equilibria of PDEs, SIAM Journal on Numerical Analysis, 45 (2007), 1398-1424.
doi: 10.1137/050645968. |
[11] |
J. P. Desi, H. Edrees, J. Price, E. Sander and T. Wanner,
The dynamics of nucleation in stochastic Cahn-Morral systems, SIAM Journal on Applied Dynamical Systems, 10 (2011), 707-743.
doi: 10.1137/100801378. |
[12] |
E. Doedel,
AUTO: A program for the automatic bifurcation analysis of autonomous systems, Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, Vol. Ⅰ (Winnipeg, Man., 1980), 30 (1981), 265-284.
|
[13] |
P. C. Fife and M. Kowalczyk,
A class of pattern-forming models, Journal of Nonlinear Science, 9 (1999), 641-669.
doi: 10.1007/s003329900081. |
[14] |
P. C. Fife,
Models for phase separation and their mathematics, Electronic Journal of Differential Equations, 2000 (2000), 1-26.
|
[15] |
P. C. Fife, Pattern formation in gradient systems, in Handbook of dynamical systems, North-Holland, Amsterdam, 2 (2002), 677-722. |
[16] |
P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, 2003,153-191. |
[17] |
P. C. Fife and D. Hilhorst,
The Nishiura-Ohnishi free boundary problem in the 1D case, SIAM Journal on Mathematical Analysis, 33 (2001), 589-606.
doi: 10.1137/S0036141000372507. |
[18] |
P. C. Fife, H. Kielhöfer, S. Maier-Paape and T. Wanner,
Perturbation of doubly periodic solution branches with applications to the Cahn-Hilliard equation, Physica D, 100 (1997), 257-278.
doi: 10.1016/S0167-2789(96)00190-X. |
[19] |
M. Gameiro and J.-P. Lessard,
Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs, Journal of Differential Equations, 249 (2010), 2237-2268.
doi: 10.1016/j.jde.2010.07.002. |
[20] |
M. Gameiro and J.-P. Lessard,
Efficient rigorous numerics for higher-dimensional PDEs via one-dimensional estimates, SIAM Journal on Numerical Analysis, 51 (2013), 2063-2087.
doi: 10.1137/110836651. |
[21] |
M. Grinfeld and A. Novick-Cohen,
Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proceedings of the Royal Society of Edinburgh, 125 (1995), 351-370.
doi: 10.1017/S0308210500028079. |
[22] |
I. Johnson, E. Sander and T. Wanner,
Branch interactions and long-term dynamics for the diblock copolymer model in one dimension, Discrete and Continuous Dynamical Systems. Series A, 33 (2013), 3671-3705.
doi: 10.3934/dcds.2013.33.3671. |
[23] |
S. Maier-Paape, U. Miller, K. Mischaikow and T. Wanner,
Rigorous numerics for the Cahn-Hilliard equation on the unit square, Revista Matematica Complutense, 21 (2008), 351-426.
doi: 10.5209/rev_REMA.2008.v21.n2.16380. |
[24] |
S. Maier-Paape, K. Mischaikow and T. Wanner,
Structure of the attractor of the Cahn-Hilliard equation on a square, International Journal of Bifurcation and Chaos, 17 (2007), 1221-1263.
doi: 10.1142/S0218127407017781. |
[25] |
S. Maier-Paape and T. Wanner,
Solutions of nonlinear planar elliptic problems with triangle symmetry, Journal of Differential Equations, 136 (1997), 1-34.
doi: 10.1006/jdeq.1996.3240. |
[26] |
S. Maier-Paape and T. Wanner,
Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part Ⅰ: Probability and wavelength estimate, Communications in Mathematical Physics, 195 (1998), 435-464.
doi: 10.1007/s002200050397. |
[27] |
S. Maier-Paape and T. Wanner,
Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics, Archive for Rational Mechanics and Analysis, 151 (2000), 187-219.
doi: 10.1007/s002050050196. |
[28] |
J. T. Marti,
Evaluation of the least constant in Sobolev's inequality for $H^{1}(0,\,s)$, SIAM Journal on Numerical Analysis, 20 (1983), 1239-1242.
doi: 10.1137/0720094. |
[29] |
R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia, 2009. |
[30] |
K. Nagatou, Validated computation for infinite dimensional eigenvalue problems, in 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics, IEEE Computer Society, 2007, 3-13. |
[31] |
K. Nagatou, N. Yamamoto and M. T. Nakao,
An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness, Numerical Functional Analysis and Optimization, 20 (1999), 543-565.
doi: 10.1080/01630569908816910. |
[32] |
Y. Nishiura, Far-from-Equilibrium Dynamics, vol. 209 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 2002. |
[33] |
Y. Nishiura and I. Ohnishi,
Some mathematical aspects of the micro-phase separation in diblock copolymers, Physica D, 84 (1995), 31-39.
doi: 10.1016/0167-2789(95)00005-O. |
[34] |
T. Ohta and K. Kawasaki,
Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.
doi: 10.1021/ma00164a028. |
[35] |
M. Plum,
Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems, Journal of Computational and Applied Mathematics, 60 (1995), 187-200.
doi: 10.1016/0377-0427(94)00091-E. |
[36] |
M. Plum,
Existence and multiplicity proofs for semilinear elliptic boundary value problems by computer assistance, Jahresbericht der Deutschen Mathematiker-Vereinigung, 110 (2008), 19-54.
|
[37] |
M. Plum,
Computer-assisted proofs for semilinear elliptic boundary value problems, Japan Journal of Industrial and Applied Mathematics, 26 (2009), 419-442.
doi: 10.1007/BF03186542. |
[38] |
W. Richardson,
Steepest descent and the least $C$ for Sobolev's inequality, Bulletin of the London Mathematical Society, 18 (1986), 478-484.
doi: 10.1112/blms/18.5.478. |
[39] |
S. M. Rump, INTLAB -INTerval LABoratory, in Developments in Reliable Computing (ed. T. Csendes), Kluwer Academic Publishers, Dordrecht, 1999, 77--104, Http://www.ti3.tuhh.de/rump/. |
[40] |
S. M. Rump,
Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, 19 (2010), 287-449.
doi: 10.1017/S096249291000005X. |
[41] |
S. M. Rump,
Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse, BIT. Numerical Mathematics, 51 (2011), 367-384.
doi: 10.1007/s10543-010-0294-0. |
[42] |
E. Sander and T. Wanner,
Monte Carlo simulations for spinodal decomposition, Journal of Statistical Physics, 95 (1999), 925-948.
doi: 10.1023/A:1004550416829. |
[43] |
E. Sander and T. Wanner,
Unexpectedly linear behavior for the Cahn-Hilliard equation, SIAM Journal on Applied Mathematics, 60 (2000), 2182-2202.
doi: 10.1137/S0036139999352225. |
[44] |
T. Stephens and T. Wanner,
Rigorous validation of isolating blocks for flows and their Conley indices, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1847-1878.
doi: 10.1137/140971075. |
[45] |
T. Wanner,
Maximum norms of random sums and transient pattern formation, Transactions of the American Mathematical Society, 356 (2004), 2251-2279.
doi: 10.1090/S0002-9947-03-03480-9. |
[46] |
K. Watanabe, Y. Kametaka, A. Nagai, K. Takemura and H. Yamagishi,
The best constant of Sobolev inequality on a bounded interval, Journal of Mathematical Analysis and Applications, 340 (2008), 699-706.
doi: 10.1016/j.jmaa.2007.08.054. |
[47] |
Y. Watanabe, K. Nagatou, M. Plum and M. T. Nakao,
Verified computations of eigenvalue exclosures for eigenvalue problems in Hilbert spaces, SIAM Journal on Numerical Analysis, 52 (2014), 975-992.
doi: 10.1137/120894683. |
[48] |
E. Zeidler, Nonlinear Functional Analysis and its Applications. Ⅰ: Fixed-Point Theorems, Springer-Verlag, New York -Berlin -Heidelberg, 1986.
doi: 10.1007/978-1-4612-4838-5.![]() ![]() ![]() |
[49] |
P. Zgliczyński and K. Mischaikow,
Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Foundations of Computational Mathematics, 1 (2001), 255-288.
doi: 10.1007/s002080010010. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, San Diego -London, 1975.
![]() ![]() |
[2] |
M. Bahiana and Y. Oono,
Cell dynamical system approach to block copolymers, Physical Review A, 41 (1990), 6763-6771.
doi: 10.1103/PhysRevA.41.6763. |
[3] |
P. W. Bates and P. C. Fife,
Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening, Physica D, 43 (1990), 335-348.
doi: 10.1016/0167-2789(90)90141-B. |
[4] |
P. W. Bates and P. C. Fife,
The dynamics of nucleation for the Cahn-Hilliard equation, SIAM Journal on Applied Mathematics, 53 (1993), 990-1008.
doi: 10.1137/0153049. |
[5] |
D. Blömker, B. Gawron and T. Wanner,
Nucleation in the one-dimensional stochastic Cahn-Hilliard model, Discrete and Continuous Dynamical Systems, Series A, 27 (2010), 25-52.
doi: 10.3934/dcds.2010.27.25. |
[6] |
C.-K. Chen and P. C. Fife,
Nonlocal models of phase transitions in solids, Advances in Mathematical Sciences and Applications, 10 (2000), 821-849.
|
[7] |
R. Choksi, M. A. Peletier and J. F. Williams,
On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM Journal on Applied Mathematics, 69 (2009), 1712-1738.
doi: 10.1137/080728809. |
[8] |
R. Choksi and X. Ren,
On the derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113 (2003), 151-176.
doi: 10.1023/A:1025722804873. |
[9] |
E. B. Davies, Linear Operators and Their Spectra, vol. 106 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2007. |
[10] |
S. Day, J.-P. Lessard and K. Mischaikow,
Validated continuation for equilibria of PDEs, SIAM Journal on Numerical Analysis, 45 (2007), 1398-1424.
doi: 10.1137/050645968. |
[11] |
J. P. Desi, H. Edrees, J. Price, E. Sander and T. Wanner,
The dynamics of nucleation in stochastic Cahn-Morral systems, SIAM Journal on Applied Dynamical Systems, 10 (2011), 707-743.
doi: 10.1137/100801378. |
[12] |
E. Doedel,
AUTO: A program for the automatic bifurcation analysis of autonomous systems, Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, Vol. Ⅰ (Winnipeg, Man., 1980), 30 (1981), 265-284.
|
[13] |
P. C. Fife and M. Kowalczyk,
A class of pattern-forming models, Journal of Nonlinear Science, 9 (1999), 641-669.
doi: 10.1007/s003329900081. |
[14] |
P. C. Fife,
Models for phase separation and their mathematics, Electronic Journal of Differential Equations, 2000 (2000), 1-26.
|
[15] |
P. C. Fife, Pattern formation in gradient systems, in Handbook of dynamical systems, North-Holland, Amsterdam, 2 (2002), 677-722. |
[16] |
P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, 2003,153-191. |
[17] |
P. C. Fife and D. Hilhorst,
The Nishiura-Ohnishi free boundary problem in the 1D case, SIAM Journal on Mathematical Analysis, 33 (2001), 589-606.
doi: 10.1137/S0036141000372507. |
[18] |
P. C. Fife, H. Kielhöfer, S. Maier-Paape and T. Wanner,
Perturbation of doubly periodic solution branches with applications to the Cahn-Hilliard equation, Physica D, 100 (1997), 257-278.
doi: 10.1016/S0167-2789(96)00190-X. |
[19] |
M. Gameiro and J.-P. Lessard,
Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs, Journal of Differential Equations, 249 (2010), 2237-2268.
doi: 10.1016/j.jde.2010.07.002. |
[20] |
M. Gameiro and J.-P. Lessard,
Efficient rigorous numerics for higher-dimensional PDEs via one-dimensional estimates, SIAM Journal on Numerical Analysis, 51 (2013), 2063-2087.
doi: 10.1137/110836651. |
[21] |
M. Grinfeld and A. Novick-Cohen,
Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proceedings of the Royal Society of Edinburgh, 125 (1995), 351-370.
doi: 10.1017/S0308210500028079. |
[22] |
I. Johnson, E. Sander and T. Wanner,
Branch interactions and long-term dynamics for the diblock copolymer model in one dimension, Discrete and Continuous Dynamical Systems. Series A, 33 (2013), 3671-3705.
doi: 10.3934/dcds.2013.33.3671. |
[23] |
S. Maier-Paape, U. Miller, K. Mischaikow and T. Wanner,
Rigorous numerics for the Cahn-Hilliard equation on the unit square, Revista Matematica Complutense, 21 (2008), 351-426.
doi: 10.5209/rev_REMA.2008.v21.n2.16380. |
[24] |
S. Maier-Paape, K. Mischaikow and T. Wanner,
Structure of the attractor of the Cahn-Hilliard equation on a square, International Journal of Bifurcation and Chaos, 17 (2007), 1221-1263.
doi: 10.1142/S0218127407017781. |
[25] |
S. Maier-Paape and T. Wanner,
Solutions of nonlinear planar elliptic problems with triangle symmetry, Journal of Differential Equations, 136 (1997), 1-34.
doi: 10.1006/jdeq.1996.3240. |
[26] |
S. Maier-Paape and T. Wanner,
Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part Ⅰ: Probability and wavelength estimate, Communications in Mathematical Physics, 195 (1998), 435-464.
doi: 10.1007/s002200050397. |
[27] |
S. Maier-Paape and T. Wanner,
Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics, Archive for Rational Mechanics and Analysis, 151 (2000), 187-219.
doi: 10.1007/s002050050196. |
[28] |
J. T. Marti,
Evaluation of the least constant in Sobolev's inequality for $H^{1}(0,\,s)$, SIAM Journal on Numerical Analysis, 20 (1983), 1239-1242.
doi: 10.1137/0720094. |
[29] |
R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia, 2009. |
[30] |
K. Nagatou, Validated computation for infinite dimensional eigenvalue problems, in 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics, IEEE Computer Society, 2007, 3-13. |
[31] |
K. Nagatou, N. Yamamoto and M. T. Nakao,
An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness, Numerical Functional Analysis and Optimization, 20 (1999), 543-565.
doi: 10.1080/01630569908816910. |
[32] |
Y. Nishiura, Far-from-Equilibrium Dynamics, vol. 209 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 2002. |
[33] |
Y. Nishiura and I. Ohnishi,
Some mathematical aspects of the micro-phase separation in diblock copolymers, Physica D, 84 (1995), 31-39.
doi: 10.1016/0167-2789(95)00005-O. |
[34] |
T. Ohta and K. Kawasaki,
Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.
doi: 10.1021/ma00164a028. |
[35] |
M. Plum,
Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems, Journal of Computational and Applied Mathematics, 60 (1995), 187-200.
doi: 10.1016/0377-0427(94)00091-E. |
[36] |
M. Plum,
Existence and multiplicity proofs for semilinear elliptic boundary value problems by computer assistance, Jahresbericht der Deutschen Mathematiker-Vereinigung, 110 (2008), 19-54.
|
[37] |
M. Plum,
Computer-assisted proofs for semilinear elliptic boundary value problems, Japan Journal of Industrial and Applied Mathematics, 26 (2009), 419-442.
doi: 10.1007/BF03186542. |
[38] |
W. Richardson,
Steepest descent and the least $C$ for Sobolev's inequality, Bulletin of the London Mathematical Society, 18 (1986), 478-484.
doi: 10.1112/blms/18.5.478. |
[39] |
S. M. Rump, INTLAB -INTerval LABoratory, in Developments in Reliable Computing (ed. T. Csendes), Kluwer Academic Publishers, Dordrecht, 1999, 77--104, Http://www.ti3.tuhh.de/rump/. |
[40] |
S. M. Rump,
Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, 19 (2010), 287-449.
doi: 10.1017/S096249291000005X. |
[41] |
S. M. Rump,
Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse, BIT. Numerical Mathematics, 51 (2011), 367-384.
doi: 10.1007/s10543-010-0294-0. |
[42] |
E. Sander and T. Wanner,
Monte Carlo simulations for spinodal decomposition, Journal of Statistical Physics, 95 (1999), 925-948.
doi: 10.1023/A:1004550416829. |
[43] |
E. Sander and T. Wanner,
Unexpectedly linear behavior for the Cahn-Hilliard equation, SIAM Journal on Applied Mathematics, 60 (2000), 2182-2202.
doi: 10.1137/S0036139999352225. |
[44] |
T. Stephens and T. Wanner,
Rigorous validation of isolating blocks for flows and their Conley indices, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1847-1878.
doi: 10.1137/140971075. |
[45] |
T. Wanner,
Maximum norms of random sums and transient pattern formation, Transactions of the American Mathematical Society, 356 (2004), 2251-2279.
doi: 10.1090/S0002-9947-03-03480-9. |
[46] |
K. Watanabe, Y. Kametaka, A. Nagai, K. Takemura and H. Yamagishi,
The best constant of Sobolev inequality on a bounded interval, Journal of Mathematical Analysis and Applications, 340 (2008), 699-706.
doi: 10.1016/j.jmaa.2007.08.054. |
[47] |
Y. Watanabe, K. Nagatou, M. Plum and M. T. Nakao,
Verified computations of eigenvalue exclosures for eigenvalue problems in Hilbert spaces, SIAM Journal on Numerical Analysis, 52 (2014), 975-992.
doi: 10.1137/120894683. |
[48] |
E. Zeidler, Nonlinear Functional Analysis and its Applications. Ⅰ: Fixed-Point Theorems, Springer-Verlag, New York -Berlin -Heidelberg, 1986.
doi: 10.1007/978-1-4612-4838-5.![]() ![]() ![]() |
[49] |
P. Zgliczyński and K. Mischaikow,
Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Foundations of Computational Mathematics, 1 (2001), 255-288.
doi: 10.1007/s002080010010. |






Equilibrium Validation for |
|||||||
Number | Index | ||||||
100 | #1 | 0 | 2.020897e-14 | 2.020897e-12 | 7.512488e-06 | 190 | |
#2 | 0 | 2.222804e-14 | 2.222804e-12 | 1.547854e-05 | 106 | ||
#3 | 1 | 4.444820e-14 | 4.444820e-12 | 1.394691e-05 | 112 | ||
#4 | 2 | 3.276268e-15 | 3.276268e-13 | 1.625587e-05 | 59 | ||
150 | #1 | 0 | 3.632609e-14 | 3.632611e-12 | 4.268987e-06 | 332 | |
#2 | 1 | 4.705001e-14 | 4.705003e-12 | 8.092680e-06 | 584 | ||
#3 | 1 | 1.814983e-14 | 1.814984e-12 | 3.929213e-06 | 508 | ||
#4 | 2 | 2.672918e-14 | 2.672918e-12 | 5.189605e-06 | 232 | ||
#5 | 2 | 2.310413e-14 | 2.310413e-12 | 7.824295e-06 | 140 | ||
200 | #1 | 0 | 4.799846e-14 | 4.799850e-12 | 2.932214e-06 | 479 | |
#4 | 1 | 6.153889e-14 | 6.153893e-12 | 5.080967e-06 | 314 | ||
#5 | 2 | 6.674118e-14 | 6.674124e-12 | 3.937197e-06 | 301 | ||
#6 | 2 | 3.750452e-14 | 3.750453e-12 | 6.304030e-06 | 142 | ||
#7 | 3 | 7.652149e-15 | 7.652150e-13 | 3.130001e-06 | 676 |
Equilibrium Validation for |
|||||||
Number | Index | ||||||
100 | #1 | 0 | 2.020897e-14 | 2.020897e-12 | 7.512488e-06 | 190 | |
#2 | 0 | 2.222804e-14 | 2.222804e-12 | 1.547854e-05 | 106 | ||
#3 | 1 | 4.444820e-14 | 4.444820e-12 | 1.394691e-05 | 112 | ||
#4 | 2 | 3.276268e-15 | 3.276268e-13 | 1.625587e-05 | 59 | ||
150 | #1 | 0 | 3.632609e-14 | 3.632611e-12 | 4.268987e-06 | 332 | |
#2 | 1 | 4.705001e-14 | 4.705003e-12 | 8.092680e-06 | 584 | ||
#3 | 1 | 1.814983e-14 | 1.814984e-12 | 3.929213e-06 | 508 | ||
#4 | 2 | 2.672918e-14 | 2.672918e-12 | 5.189605e-06 | 232 | ||
#5 | 2 | 2.310413e-14 | 2.310413e-12 | 7.824295e-06 | 140 | ||
200 | #1 | 0 | 4.799846e-14 | 4.799850e-12 | 2.932214e-06 | 479 | |
#4 | 1 | 6.153889e-14 | 6.153893e-12 | 5.080967e-06 | 314 | ||
#5 | 2 | 6.674118e-14 | 6.674124e-12 | 3.937197e-06 | 301 | ||
#6 | 2 | 3.750452e-14 | 3.750453e-12 | 6.304030e-06 | 142 | ||
#7 | 3 | 7.652149e-15 | 7.652150e-13 | 3.130001e-06 | 676 |
Equilibrium Validation for | |||||||
Number | Index | ||||||
200 | #2 | 0 | 2.642564e-14 | 1.321300e-11 | 4.858832e-07 | 1828 | |
#3 | 1 | 6.084200e-14 | 3.042193e-11 | 4.959956e-07 | 1108 |
Equilibrium Validation for | |||||||
Number | Index | ||||||
200 | #2 | 0 | 2.642564e-14 | 1.321300e-11 | 4.858832e-07 | 1828 | |
#3 | 1 | 6.084200e-14 | 3.042193e-11 | 4.959956e-07 | 1108 |
Equilibrium Validation for | |||||||
Number | Index | ||||||
100 | #1 | 0 | 5.728511e-14 | 5.728512e-12 | 1.047050e-05 | 106 | |
#2 | 0 | 2.129870e-14 | 2.129870e-12 | 8.569644e-06 | 175 | ||
#3 | 1 | 2.129870e-14 | 2.129870e-12 | 8.569644e-06 | 811 | ||
#4 | 1 | 6.466218e-15 | 6.466218e-13 | 4.747448e-05 | 64 | ||
200 | #1 | 0 | 8.153471e-14 | 8.153483e-12 | 2.813889e-06 | 644 | |
#2 | 1 | 7.326258e-14 | 7.326268e-12 | 2.834757e-06 | 552 | ||
#3 | 1 | 1.075251e-13 | 1.075252e-11 | 4.071324e-06 | 180 | ||
#4 | 2 | 8.642886e-14 | 8.642896e-12 | 3.858723e-06 | 271 | ||
#5 | 2 | 2.696057e-14 | 2.696057e-12 | 1.384417e-05 | 113 |
Equilibrium Validation for | |||||||
Number | Index | ||||||
100 | #1 | 0 | 5.728511e-14 | 5.728512e-12 | 1.047050e-05 | 106 | |
#2 | 0 | 2.129870e-14 | 2.129870e-12 | 8.569644e-06 | 175 | ||
#3 | 1 | 2.129870e-14 | 2.129870e-12 | 8.569644e-06 | 811 | ||
#4 | 1 | 6.466218e-15 | 6.466218e-13 | 4.747448e-05 | 64 | ||
200 | #1 | 0 | 8.153471e-14 | 8.153483e-12 | 2.813889e-06 | 644 | |
#2 | 1 | 7.326258e-14 | 7.326268e-12 | 2.834757e-06 | 552 | ||
#3 | 1 | 1.075251e-13 | 1.075252e-11 | 4.071324e-06 | 180 | ||
#4 | 2 | 8.642886e-14 | 8.642896e-12 | 3.858723e-06 | 271 | ||
#5 | 2 | 2.696057e-14 | 2.696057e-12 | 1.384417e-05 | 113 |
Branch Validation for | ||||||||
No. | ||||||||
100 | #1 | 1.0446e-13 | 1.1065e-04 | 1.1079e-04 | 2.6068e-04 | 2.2 | 3.3 | |
#2 | 5.1001e-13 | 8.9555e-05 | 8.9572e-05 | 2.6200e-05 | 5.7 | 8.6 | ||
#3 | 3.0910e-13 | 1.7520e-04 | 1.7527e-04 | 1.2525e-04 | 2.6 | 3.9 | ||
#4 | 8.1585e-14 | 5.3658e-05 | 5.3726e-05 | 9.5155e-05 | 10.0 | 15.1 | ||
150 | #1 | 2.1895e-13 | 5.7574e-05 | 5.7619e-05 | 1.3199e-04 | 2.4 | 3.7 | |
#2 | 5.7821e-12 | 8.6076e-06 | 8.6078e-06 | 4.4425e-07 | 31.3 | 46.9 | ||
#3 | 3.3317e-13 | 1.7562e-05 | 1.7569e-05 | 2.1107e-05 | 7.4 | 11.1 | ||
#4 | 1.5386e-13 | 9.3999e-05 | 9.4056e-05 | 1.5404e-04 | 1.8 | 2.7 | ||
#5 | 1.3800e-13 | 1.3328e-04 | 1.3336e-04 | 1.8173e-04 | 1.9 | 2.9 |
Branch Validation for | ||||||||
No. | ||||||||
100 | #1 | 1.0446e-13 | 1.1065e-04 | 1.1079e-04 | 2.6068e-04 | 2.2 | 3.3 | |
#2 | 5.1001e-13 | 8.9555e-05 | 8.9572e-05 | 2.6200e-05 | 5.7 | 8.6 | ||
#3 | 3.0910e-13 | 1.7520e-04 | 1.7527e-04 | 1.2525e-04 | 2.6 | 3.9 | ||
#4 | 8.1585e-14 | 5.3658e-05 | 5.3726e-05 | 9.5155e-05 | 10.0 | 15.1 | ||
150 | #1 | 2.1895e-13 | 5.7574e-05 | 5.7619e-05 | 1.3199e-04 | 2.4 | 3.7 | |
#2 | 5.7821e-12 | 8.6076e-06 | 8.6078e-06 | 4.4425e-07 | 31.3 | 46.9 | ||
#3 | 3.3317e-13 | 1.7562e-05 | 1.7569e-05 | 2.1107e-05 | 7.4 | 11.1 | ||
#4 | 1.5386e-13 | 9.3999e-05 | 9.4056e-05 | 1.5404e-04 | 1.8 | 2.7 | ||
#5 | 1.3800e-13 | 1.3328e-04 | 1.3336e-04 | 1.8173e-04 | 1.9 | 2.9 |
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