Lez.
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Data
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Argomento
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L 1
(1h)
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25.09.24
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- Introduction
- Perturbative Expansions
- Introduction to perturbative expansion methods.
- Regular versus singular perturbation expansions: simple examples.
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L 2
(2h)
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26.09.24
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- Perturbative Expansions
- Regular versus singular perturbation expansions: Integral Representations.
- Iterative approximations.
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L 3
(2h)
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01.10.24
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- Perturbative Expansions
- Nondimensionalization; Example: one dimensional damped mass-spring system.
- Asymptotic expansions
- Order relations: symbols "O", "o" e "∼".
- Asymptotic sequences and asymptotic expansions.
- Generalized asymptotic and Poincaré type expansions.
- Uniqueness of Poincaré type expansions and subdominance.
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L 4
(1h)
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02.10.24
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- Asymptotic expansions
- Uniform versus non uniform asymptotic expansions.
- Convergent series versus asymptotic series.
- Example: The error function
Erf(x).
- Asymptotic power series: Taylor's theorem and Borel's theorem.
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L 5
(2h)
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03.10.24
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- Asymptotic expansions
- Stokes phenomenon.
- Elementary operations on Poincaré type expansions.
- Generalized Stieltjeis Series and Integrals.
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L 6
(2h)
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08.10.24
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- Layer-Type Problems
- Internal and external asymptotic expansions and their domain of validity.
- Intermediate limit and Matched asymptotic expansions.
- Uniform asymptotic expansions.
- Example: uniform asymptotic expansion to
O(1) and O(ε)
of the solution of an ordinary second order differential equation.
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L 7
(1h)
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09.10.24
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- Layer-Type Problems
- Distinguished e undistinguished limits, dominance balance.
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L 8
(2h)
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10.10.24
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- Layer-Type Problems
- Nonlinear boundary layers.
- Boundary layer at either extrema.
- Nested boundary layers.
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L 9
(2h)
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15.10.24
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- Layer-Type Problems
- WKB expansion
- Dispersive and Dissipative behaviors.
- WKB expansion.
- Geometrical optics and physical optics approximantions.
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L 10
(1h)
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16.10.24
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- WKB expansion
- WKB expansion for the Schrödinger equation.
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L 11
(2h)
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17.10.24
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- WKB expansion
- WKB expansion for Boundary Layers.
- Asymptotic behavior of Airy functions for
|x| >> 1.
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L 12
(2h)
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22.10.24
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- WKB expansion
- Asymptotic behavior of the Parabolic Cylinder functions for
|x| >> 1.
- WKB expansion with one linear turning point.
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L 13
(1h)
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23.10.24
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- WKB expansion
- The Liouville-Green transformation and the Langer solution to the linear turning point problem.
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L 14
(2h)
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24.10.24
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- WKB expansion
- WKB expansion with two turning points.
- Eigenvalue condition and semiclassical quantization.
- Multiple Scales Methods
- Resonances and secular terms.
- Strained Coordinates Method: The Duffing's equation.
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L 15
(2h)
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29.10.24
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- Multiple Scales Methods
- Strained Coordinates Method: Limit cicle of the Van der Pol oscillator.
- Derivative Expansion Method: Duffing's equation analysis with two time scales.
- Derivative Expansion Method: Duffing's equation analysis with three time scales.
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L 16
(1h)
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30.10.24
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- Multiple Scales Methods
- Derivative Expansion Method: Duffing's equation analysis with three time scales.
- Derivative Expansion Method: Harmonic oscillator with non linear damping
analysis with two time scales.
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L 17
(2h)
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31.10.24
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- Multiple Scales Methods
- Derivative Expansion Method and WKB Method: Harmonic oscillator with slow
time varying frequency.
- Derivative Expansion Method for Boundary Layer problems.
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L 18
(2h)
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05.11.24
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- Multiple Scales Methods
- Derivative Expansion Method: Van der Pol oscillator with two time scales.
- Cole-Kevorkian Method: Duffing's equation to order
ε2.
- Naive Renormalization Method: Duffing's equation to order
ε2.
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L 19
(1h)
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06.11.24
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- Complements
- Integral representation of the Airy function
Ai(x).
- Asymptotic behavior of
Ai(x) for x >> 1: Steepest descent
and Saddle point methods.
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L 20
(2h)
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07.11.24
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- Renormalization Group and Asymptotic Expansions.
- Ill defined expansion and Renormalizability hypotesis.
- Regularization procedure, Renormalization prescription.
- Perturbative Renormalization.
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L 21
(2h)
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12.11.24
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- Renormalization Group and Asymptotic Expansions.
- Renormalization group.
- Perturbative Renormalization Group and partial resummation.
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L 22
(1h)
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13.11.24
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- Renormalization Group and Asymptotic Expansions.
- Duffing's equation: solution to
O(ε).
- Callan-Symanzik's equation.
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L 23
(2h)
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14.11.24
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- Renormalization Group and Asymptotic Expansions.
- Duffing's equation: solution to
O(ε2).
- Renormalization Group approach to Boundary Layers.
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L 24
(2h)
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19.11.24
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- Renormalization Group and Asymptotic Expansions.
- Schrödinger with one turning point.
- Van der Pol oscillator to
O(ε):
renormalization conditions, minimal subtraction, approach to the limit cicle.
- Van der Pol oscillator: iterative renormalization to
O(ε).
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L 25
(1h)
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20.11.24
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- Renormalization Group and Asymptotic Expansions.
- Mathieu equation: stability analysis to
O(ε2)
near the critical value a = 1/4.
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L 26
(2h)
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21.11.25
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- Diagrammatic Theory
- Generating funtion Z[J] of the n-point functions
<φ1...φn>.
- Perturbative expansion of Z[J].
- Wick and Novikov theorems.
- Pertutbative expansion of
<φ1φ2>
for the φ4 theory and its digrammatic representation.
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L 27
(2h)
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26.11.24
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- Diagrammatic Theory
- Cancellation of vacuum-fluctuations diagrams.
- Dyson-Schwinger equation.
- Perturbative calculation of the moments
<φn>
of the one-dimensional double well in the limit T << 1.
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L 28
(1h)
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27.11.24
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- Diagrammatic Theory
- Perturbative calculation of the moments
<φn>
of the one-dimensional double well in the limit T << 1.
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L 29
(2h)
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28.12.24
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- Diagrammatic Theory
- Connected n-points functions
<φ1...φn>c
and their generating function W[J].
- Perturbative calculation of W[J] for the one-dimensional double well in
the limit
T << 1,
evaluation of <φ2>.
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L 30
(2h)
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03.12.24
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- Diagrammatic Theory
- 1PR and 1PI diagrams: Self-energy Σ and Dyson equation.
- n-points proper vertices Γ(
n)1...n and
their generating function Γ[φ].
- Legendre Transform and diagrams 1PI.
- Connected n-points functions and proper vertices.
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L 31
(1h)
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04.12.23
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- Diagrammatic Theory
- Effective Potential, spontaneous symmetry breaking.
- Γ[
φ] for the one-dimensional double well in the
limit T << 1, evaluation of <φ2>
to order T.
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L 32
(2h)
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05.12.23
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L 33
(2h)
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10.12.24
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- Diagrammatic Theory
- Self-consistent equation for ϕ and Dyson equation.
- 2PR and 2PI diagrams, double Legendre transform and effective Potential
Γ[
φ,G].
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L 34
(1h)
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11.12.24
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- Diagrammatic Theory
- Γ[
φ] and Γ[φ,G] for the
one-dimensional double well: 2-loop and 3-loop calculation.
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L 35
(2h)
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12.12.24
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- Grassmann Variables
- Definition and properties.
- Differentiation and integration.
- Gaussian integrals: Pfaffian and Determinant.
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L 36
(2h)
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17.12.24
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L 37
(1h)
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18.12.24
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L 38
(2h)
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19.12.24
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L 39
(2h)
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07.01.25
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- Grassmann Variables
- Existence of a finite critical temperature of the two-dimensional Ising model
(Pierls' argument).
- Computation of the partition function of the two-dimensional Ising model on
a square lattice using Grassmann variables.
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L 40
(1h)
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08.01.25
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- Grassmann Variables
- Computation of the partition function of the two-dimensional Ising model on
a square lattice using Grassmann variables.
- Crtical temperature of the two-dimensional Ising model
on a square lattice.
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L 41
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L 42
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L 43
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L 44
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