Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 9
Текст из файла (страница 9)
. . . . . . . . . . . . . . . 78410.4 Green functions for perturbation expansions in curvilinear coordinates80510.5 Infinitesimally thin closed current loop L and magnetic field . . . . . 87410.6 Coordinate system q µ and the two sets of local nonholonomic coordinates dxα and dxa . . . . . . . .
. . . . . . . . . . . . . . . . . . . 88613.1 Illustration of associated final points in u-space, to be summed inthe harmonic-oscillator amplitude . . . . . . . . . . . . . . . . . . . 93415.115.215.315.415.5Random chain of N links . . . . . . . . . . . . . . .
. . . . . . . .End-to-end distribution PN (R) of random chain with N links . .Neighboring links for the calculation of expectation values . . . . .Paramters k, β, and m for a best fit of end-to-end distribution . .Structure functions for different persistence lengths following fromthe end-to-end distributions . .
. . . . . . . . . . . . . . . . . . . .15.6 Normalized end-to-end distribution of stiff polymer . . . . . . . . .15.7 Comparison of critical exponent ν in Flory approximation with resultof quantum field theory . . . . . . . . . . . . . . . . . . . . . . . .16.116.216.316.416.516.616.716.816.916.1016.1116.1216.1316.1416.1516.1616.17.1017.1023.1033.1045.1046.1049.1075Second virial coefficient B2 as function of flux µ0 . . .
. . . . . . .1099Lefthanded trefoil knot in polymer . . . . . . . . . . . . . . . . . . .1100Nonprime knot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1101Illustration of multiplication law in knot group . . . . . . . . . . . .1101Inequivalent compound knots possessing isomorphic knot groups . .1102Reidemeister moves in projection image of knot . . . .
. . . . . . .1103Simple knots with up to 8 minimal crossings . . . . . . . . . . . . .1104Labeling of underpasses for construction of Alexander polynomial . .1105Exceptional knots found by Kinoshita and Terasaka, Conway, andSeifert, all with same Alexander polynomial as trivial knot . . . .
.1107Graphical rule for removing crossing in generating Kauffman polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1109Kauffman decomposition of trefoil knot . . . . . . . . . . . . . . . .1109Skein operations relating higher knots to lower ones . .
. . . . . . .1109Skein operations for calculating Jones polynomial of two disjointunknotted loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1110Skein operation for calculating Jones polynomial of trefoil knot . . .1111Skein operation for calculating Jones polynomial of Hopf link . . . .1112Knots with 10 and 13 crossings, not distinguished byJonespolynomials1113Fraction fN of unknotted closed polymers in ensemble of fixed lengthL = Na . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1114H. Kleinert, PATH INTEGRALSxxxvii16.1816.1916.2016.2116.2216.2316.24Idealized view of circular DNA . . . . . . . . . . . . . . . . . . . .Supercoiled DNA molecule . . . . . . . .
. . . . . . . . . . . . . .Simple links of two polymers up to 8 crossings . . . . . . . . . . .Illustration of Calagareau-White relation . . . . . . . . . . . . . .Closed polymers along the contours C1 , C2 respectively . . . . . . .Four diagrams contributing to functional integral . . . . . . . . .Values of parameter ν at which plateaus in fractional quantum Hallresistance h/e2 ν are expected theoretically . . . . . . .
. . . . . .16.25 Trivial windings LT + and LT − . Their removal by means of Reidemeister move of type I decreases or increases writhe w . . . . . . ..1117.1118.1119.1123.1127.113317.1 Plot of symmetric double-well potential . . . . . . . . . . . . .
. .17.2 Classical kink solution in double-well potential connecting two degenerate maxima in reversed potential . . . . . . . . . . . . . . . .17.3 Reversed double-well potential governing motion of position x asfunction of imaginary time τ . . . . . . . . . . . . . . . . . . . . .17.4 Potential for quadratic fluctuations around kink solution . . .
. . .17.5 Vertices and lines of Feynman diagrams for correction factor C inEq. (17.225) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17.6 Positions of extrema xex in asymmetric double-well potential . . .17.7 Classical bubble solution in reversed asymmetric quartic potential17.8 Action of deformed bubble solution as function of deformation parameter .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17.9 Sequence of paths as function of parameter ξ . . . . . . . . . . . .17.10 Lines of constant Re (t2 + t3 ) in complex t-plane and integrationcontours Ci which maintain convergence of fluctuation integral . .17.11 Potential of anharmonic oscillator for small negative coupling . . .17.12 Rosen-Morse Potential for fluctuations around the classical bubblesolution . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17.13 Reduced imaginary part of lowest three energy levels of anharmonicoscillator for negative couplings . . . . . . . . . . . . . . . . . . .17.14 Energies of anharmonic oscillator as function of g 0 ≡ g/ω 3, obtainedfrom the variational imaginary part . . . . . . . . . . . . . . . . .17.15 Reduced imaginary part of ground state energy of anharmonic oscillator from variational perturbation theory .
. . . . . . . . . . .17.16 Cuts in complex ĝ-plane whose moments with respect to inversecoupling constant determine re-expansion coefficients . . . . . . . .17.17 Theoretically obtained convergence behavior of Nth approximantsfor α0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17.18 Theoretically obtained oscillatory behavior around exponentiallyfast asymptotic approach of α0 to its exact value . .
. . . . . . .17.19 Comparison of ratios Rn between successive expansion coefficientsof the strong-coupling expansion with ratios Rnas . . . . . . . . . ..1164.1145.1153.1167.1168.1171.1196.1198.1200.1202.1203.1204.1212.1213.1222.1224.1226.1228.1232.1233.1233xxxviii17.20 Strong-Coupling Expansion of ground state energy in comparisonwith exact values and perturbative results of 2nd and 3rd order . .17.21 Renormalization group trajectories for physically identical superconductors .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17.22 Potential V (ρ) = −ρ2 + ρ4 /2 − j 2 /ρ2 showing barrier in superconducting wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17.23 Condensation energy as function of velocity parameter kn = 2πn/L17.24 Order parameter of superconducting thin circular wire .
. . . . . .17.25 Extremal excursion of order parameter in superconducting wire . .17.26 Infinitesimal translation of the critical bubble yields antisymmetricwave function of zero energy . . . . . . . . . . . . . . . . . . . . .17.27 Logarithmic plot of resistance of thin superconducting wire as function of temperature at current 0.2µA .
. . . . . . . . . . . . . . .17.28 Bubble energy as function of its radius R . . . . . . . . . . . . . .17.29 Qualitative behavior of critical bubble solution as function of itsradius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17.30 Decay of metastable false vacuum in Minkowski space . . . . . . ..1234.1237.1241.1242.1242.1243.1244.1246.1247.1248.125418.1 Closed-time contour in forward–backward path integrals .
. . . . . .128118.2 Behavior of function 6J(z)/π 2 in finite-temperature Lamb shift . .134120.1 Periods of exponential growth of price index averaged over majorindustrial stocks in the United States over 60 years . . . . . . . . . .137020.2 Index S&P 500 for 13-year period Jan. 1, 1984 — Dec. 14, 1996,recorded every minute, and volatility in time intervals 30 minutes. .137120.3 Comparison of best log-normal and Gaussian fits to volatilities over300 min . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .137120.4 Fluctuation spectrum of exchange rate DM/US$ . . . . . . . . . . .137220.5 Behavior of logarithm of stock price following the stochastic differential equation (20.1) . . . . . . . . . . . . . . . . . . . . . . . . . .137320.6 Left: Lévy tails of the S&P 500 index (1 minute log-returns) plottedagainst z/δ. Right: Double-logarithmic plot exhibiting the powerlike falloffs.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .137520.7 Best fit of cumulative versions (20.36) of truncated Lévy distribution 137920.8 Change in shape of truncated Lévy distributions of width σ = 1with increasing kurtoses κ = 0 (Gaussian, solid curve), 1, 2 , 5, 10 .138020.9 Change in shape of truncated Lévy distributions of width σ = 1 andkurtosis κ = 1 with increasing skewness s = 0 (solid curve), 0.4, 0.8 138320.10 Boltzmann distribution of S&P 500 and NASDAQ 100 highfrequency log-returns recorded by the minute. .
. . . . . . . . . . . .138520.11 Market temperatures of S&P 500 and NASDAQ 100 indices from1990 to 2006. The crash of 2000 occurred at the maximal temperatures TSP500 ≈ 0.075 and TNASDAQ100 ≈ 0.15. . . . . . . . . . . . . .1385H. Kleinert, PATH INTEGRALSxxxix20.12 Logarithmic plot of the normalized K -exponential for q ≡ 1+1/K =1 (Gaussian), 1.2, 1.4, 1.6, all for σ = 1, and fit of the log-returnsof20 NYSE top-volume stocks over short time scales from 1 to 3minutes by Student-Tsallis distribution . .
. . . . . . . . . . . . . .138720.13 Comparison of best fit of Meixner distribution to truncated Lévydistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138920.14 Fits of Gaussian distribution to S&P 500 log-returns recorded inintervals of 60 min, 240 min, and 1 day. .
. . . . . . . . . . . . . . .139620.15 Typical Noise of Poisson Process . . . . . . . . . . . . . . . . . . . .139820.16 Cumulative distributions obtained from repeated convolution integrals of distributions of S&P 500 price changes over 15 minutes, andfalloff of kurtosis with time .
. . . . . . . . . . . . . . . . . . . . . .139920.17 Gaussian distributions of S&P 500 and NASDAQ 100 weekly logreturns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140020.18 Variance of S&P 500 and NASDAQ 100 indices as a function of time140120.19 Kurtosis of S&P 500 and NASDAQ 100 indices as a function oftime.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
