Lez.
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Data
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Argomento
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L 1
(2h)
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25.09.24
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- Introduction
- Motivation
- Brownian Motion: Einstein and Langevin description.
- Birth-Death processes: Master equation.
- Shot Noise: Stochastic Differential Equation.
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L 2
(3h)
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30.09.24
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- Motivation
- Shot Noise: Stochastic Differential Equation.
- Stochastic Processes
- Definitions.
- Bernoulli Trials.
- Markov Processes.
- The Chapman-Kolmogorov equation.
- Continuous Markov processes; examples: Brownian motion and Cauchy process.
- Differential Chapman-Kolmogorov equation.
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L 3
(1h)
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01.10.24
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- Markov Processes
- The master equation: jump processes.
- The Liouville equation: deterministic processes.
- The Fokker-Planck equation: diffusive processes.
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L 4
(2h)
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02.10.24
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- Fokker-Planck Equation
- Backward differential Chapman-Kolmogorov equation.
- Forward Kramers-Moyal equation.
- Stationary and homogeneous Markov Processes.
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L 5
(3h)
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07.10.24
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- Fokker-Planck Equation
- The Wiener Process.
- The Ornstein-Uhlembeck proceess.
- The method of characteristics for quasilinear partial differential equations.
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L 6
(1h)
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08.10.24
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- Fokker-Planck Equation
- Probability current and boundary conditions for the Fokker-Planck equation.
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L 7
(2h)
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09.10.24
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- Fokker-Planck Equation
- Stationary solution for the one-dimensional homegeneous Fokker-Planck equation:
Potential solutions.
- Stationary solution in a gravitational field.
- Stationary solution of the Ornstein-Uhlembeck process.
- Stationary solution for the Brownian motion.
- The eigenvalue method: Wiener process with absorbing and reflectin barriers.
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L 8
(3h)
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14.10.24
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- Fokker-Planck Equation
- First Passage Time for homogeneous processes.
- Escape over a potential barrier, the Arrhenius formula.
- Stochastic Differential Equations
- The Langevin equation and the Wiener process.
- The stochastic integration: the Ito Integral.
- Nonanticipating functions.
dW(t)2 and dW(t)2+N.
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L 9
(1h)
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15.10.24
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- Stochastic Differential Equations
- The Stratonovich Integral.
- The Ito differentiation rule.
- The Dirac delta function and the Heaviside theta function.
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L 10
(2h)
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16.10.24
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- Stochastic Differential Equations
- Numerical integration of SDE: Stochastic second order Runge-Kutta Algorithm.
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L 11
(3h)
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21.10.24
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- Stochastic Differential Equations
- The Ito formula.
- Ito SDE and Fokker-Planck equation.
- Ito and Stratonovich SDE.
- Stratonovich SDE and Fokker-Planck equation.
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L 12
(1h)
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22.10.24
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- Stochastic Differential Equations
- Example: SDE with linear multiplicative noise.
- Example: Linear oscillator with random frequency.
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L 13
(2h)
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23.10.24
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- Path Integral Methods for Stochastic Differential Equations
- Path Integral formulation for additive white noise Langevin equations.
- The Martin-Siggia-Rose-Janssen-de Domicis (MSRJD) Action.
- The Onsager-Machlup-Graham-Bausch-Wegner (OMGBW) Action.
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L 14
(3h)
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28.10.23
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- Path Integral Methods for Stochastic Differential Equations
- Derivation of the Fokker-Planck equation from the path integral.
- Response functions and response fields.
- Quadratic MSRJD action.
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L 15
(1h)
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29.10.24
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- Path Integral Methods for Stochastic Differential Equations
- Example: The one dimensional Ornstein-Uhlembeck process.
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L 16
(2h)
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30.10.24
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- Path Integral Methods for Stochastic Differential Equations
- Path Integral formulation for multiplicative white noise Langevin equations.
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L 17
(3h)
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04.11.24
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- Path Integral Methods for Stochastic Differential Equations
- From Path Integral to Fokker-Planck equation for multiplicative white noise processes:
state dependent diffusion.
- Thermodynamic equilibrium conditions for multiplicative white noise processes.
- Fluctuation-Dissipation Theorem for additive white noise processes.
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L 18
(1h)
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05.11.24
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- Path Integral Methods for Stochastic Differential Equations
- Detailed Balance condition.
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L 19
(2h)
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06.11.24
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- Path Integral Methods for Stochastic Differential Equations
- Fluctuation-Dissipation Theorem for multiplicative white noise processes.
- Dynamical Field Theory
- The Novikov and Wick theorems.
- Perturbation theory in the nonlinearity (interaction).
- Equazione di Dyson dinamica e Terorema FDT.
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L 20
(3h)
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11.11.24
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- Dynamical Field Theory
- Perturbative evaluation of (equilibrium) two points functions.
- The Dyson equation.
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L 21
(1h)
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12.11.24
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- Dynamical Field Theory
- The Dyson equation for the zero-dimensional
λϕ4 model: first order calculation.
- Perturbation theory in the non linearity with non zero external field.
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L 22
(2h)
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13.11.24
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- Dynamical Field Theory
- Dynamical Effective potential
Γ[φ,φ̂].
- Dynamical Effective potential for the Ornstein-Uhlembeck process.
- Loop expansion of the Dynamical Effective potential.
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L 23
(3h)
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18.11.24
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- Dynamical Field Theory
- Dynamical Effective potential for the zero-dimensional
λϕ4 model: zero and one loop calculation.
- Conditional Probability Function: the
φ̂ ≠ 0 solution.
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L 24
(1h)
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19.11.24
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- Dynamical Field Theory
- Dynamical Effective Potential and Conditional Probability Function.
- Arrhenius Law.
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L 25
(2h)
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20.11.24
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- Dynamical Field Theory
- Quantum averages over the Close Time Path (CTP) contour.
- Bosonic Coherent States.
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L 26
(3h)
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25.11.24
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- Dynamical Field Theory
- CTP calculation of Z[0] for
the bosonic system Ĥ = ω0 b̂+ b̂.
- CTP Green function for
the bosonic system Ĥ = ω0 b̂+ b̂.
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L 27
(1h)
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26.11.24
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L 28
(2h)
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27.11.24
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- Dynamical Field Theory
- Quantum Harmonic Oscillator.
- Dynamical Field Theory for disordered systems
- Quenched disorder average in statics and dynamics.
- Random non-symmetric two-body interaction neural network model:
averaged dynamical generating function.
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L 29
(3h)
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02.12.24
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- Dynamical Field Theory for disordered systems
- The spherical p-spin and 2+p-spin spin-glass models: averaged dynamical generating
function and effective dynamics.
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L 30
(1h)
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03.12.24
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L 31
(2h)
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04.12.23
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- Dynamical Field Theory for disordered systems
- Equilibrium dynamics of the spherical 2+p-spin spin-glass models.
- Ergodic dynamics of the 2+3-spin spin-glass model:
critical slowing down and continuous transition.
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L 32
(3h)
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09.12.24
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- Dynamical Field Theory for disordered systems
- Ergodic dynamics of the 2+3-spin spin-glass model:
critical slowing down and continuous transition.
- Ergodic dynamics of the 2+3-spin spin-glass model:
dynamical arrest and discontinuous transition.
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L 33
(1h)
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10.12.24
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L 34
(2h)
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11.12.24
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- Dynamical Field Theory for disordered systems
- Non-ergodic dynamcis of the 2+3-spin spin-glass model:
two-steps relaxation, dynamical marginal condition and 1RSB phase.
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L 35
(3h)
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16.12.24
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L 36
(1h)
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17.12.24
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L 37
(2h)
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18.12.24
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L 38
(1h)
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07.01.25
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L 39
(2h)
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08.01.25
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- Dynamical Field Theory for disordered systems
- Path-Integral approach to a fully connected neural network model with
random independent asymmetric couplings.
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L 40
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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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