Klassifizierung von PDEs

Partial Differential Equations

• 2nd order linear PDEs in two independent variables have a general form: $Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G$, where A, B, C, D, E, F, and G are functions of x and y.
• The discriminant is $\Delta = B^2 - AC$.

Classification Based on Discriminant

• Hyperbolic: $\Delta > 0$
• Parabolic: $\Delta = 0$
• Elliptic: $\Delta < 0$

Wave Equation Example

• For $u_{tt} = c^2 u_{xx}$, $A = c^2$, $B = 0$, $C = -1$, thus $\Delta = c^2 > 0$, classifying it as hyperbolic.

Heat Equation Example

• For $u_t = k u_{xx}$, $A = k$, $B = 0$, $C = 0$, thus $\Delta = 0$, classifying it as parabolic.

Laplace's Equation Example

• For $u_{xx} + u_{yy} = 0$, $A = 1$, $B = 0$, $C = 1$, thus $\Delta = -1 < 0$, classifying it as elliptic.

Well-Posedness Conditions

• Existence: A solution must exist.
• Uniqueness: The solution must be unique.
• Stability: The solution must depend continuously on initial/boundary data.
• A PDE is ill-posed if any of the above conditions are not met.

Common PDEs

• Wave Equation: $u_{tt} = c^2 u_{xx}$, describes vibrations and waves, where $c$ is the wave speed.
• Heat Equation: $u_t = k u_{xx}$, describes heat diffusion, $k$ is the thermal diffusivity.
• Laplace's Equation: $u_{xx} + u_{yy} = 0$ (2D), $u_{xx} + u_{yy} + u_{zz} = 0$ (3D), describes steady-state phenomena at equilibrium.
• Poisson's Equation: $u_{xx} + u_{yy} = f(x, y)$ (2D), a generalization of Laplace's equation.

Boundary Conditions Types

• Dirichlet: $u$ is specified on the boundary.
• Neumann: $\frac{\partial u}{\partial n}$ is specified on the boundary, where n is the normal direction.
• Robin: A linear combination of $u$ and $\frac{\partial u}{\partial n}$ is specified on the boundary.

Solution Methods

• Separation of Variables: Assume $u(x, t) = X(x)T(t)$, substitute into the PDE, separate variables, and solve the resulting ODEs.
• Fourier Transforms: Transform the PDE into Fourier space, solve the resulting equation, and inverse transform to obtain the solution.
• Numerical Methods: Such as Finite Difference Method and Finite Element Method are also used for solving PDEs.

Algorithmic Game Theory

• Game theory: The study of mathematical models of strategic interactions among rational agents, with applications in economics, political science, biology, and computer science.

Prisoner's Dilemma Example

• Two suspects are arrested and face a deal with potential outcomes summarized in a payoff matrix.
• Each suspect's dominant strategy is to confess, even though the outcome is not Pareto efficient.

Algorithmic Game Theory Definition

• Algorithmic game theory: Combines game theory and computer science, dealing with computational aspects like computing solution concepts and analyzing the complexity of games.

Nash Equilibrium

• Nash equilibrium: A set of strategies where no player benefits from changing their strategy, given others' strategies.
• Computing a Nash equilibrium: Polynomial time for two-player zero-sum games using linear programming, PPAD-complete for general games.

Mechanism Design

• Mechanism design: Designing mechanisms to achieve desired outcomes when participants have private information and act strategically.

Auctions Example

• Auctions: A mechanism design example where the goal is to allocate a good to the highest-valuing bidder while maximizing revenue for the seller.

Types of Auctions

• Sealed-bid auction: each bidder submits a bid without knowing the other bids
• First-price auction: the highest bidder wins and pays their bid
• Second-price auction: the highest bidder wins and pays the second-highest bid

Vickrey Auction

• Vickrey auction: A second-price sealed-bid auction where the highest bidder wins but pays the second-highest bid and is incentive compatible and efficient.

Physical Principles of Semiconductor Devices

Band Bending & Non-Equilibrium:

• HW 8 is due today.
• No lab this week. (Thanksgiving Recess)

Band Bending at equilibrium (Review):

• A charge transfer occurs to align the Fermi levels when two materials with different work functions are joined.
• The charge transfer results in a charge density near the junction, and an electric field (related to the slope of the band edges).

Metal-Semiconductor Junction Example (Review):

• Consider a metal and an n-type semiconductor brought into contact.
• Electrons flow from the semiconductor to the metal where $\phi_m > \phi_s$, creates a region depleted of electrons near the junction.
• The depletion region has a positive charge due to ionized donors.

Band Bending equations (Review):

\begin{align*} \frac{d\varepsilon}{dx} &= \frac{\rho(x)}{\varepsilon_s} \ \frac{d^2\psi}{dx^2} &= -\frac{\rho(x)}{\varepsilon_s} \ \varepsilon(x) &= -\frac{d\psi}{dx} \end{align*}

• $\psi$ is the electrostatic potential, and $\varepsilon_s$ is the permittivity of the semiconductor, and $\rho(x)$ is the charge density

Width of Depletion Region (Review):

• The depletion region, $W$, can be found by solving Poisson's equation with the appropriate boundary conditions.

$W = \sqrt{\frac{2\varepsilon_s}{qN_d}(V_{bi})}$

• Where $N_d$ is the donor concentration, $V_{bi}$ is the built-in potential

Non-Equilibrium Conditions:

• the Fermi level is no longer flat across the device when an external voltage is applied to the junction.
• The applied voltage changes the potential difference across the junction, and thus the width of the depletion region.

Forward Bias:

• applied on p-side of the p-n junction as a positive voltage reduces the potential barrier.
• This allows more current to flow across the junction, and the depletion region narrows.

Reverse Bias:

• A negative voltage is applied to the p-side to increase the potential barrier.
• reduces the current flow across the junction and The depletion region widens.

Quasi-Fermi Levels:

• Under non-equilibrium conditions, the Fermi level splits into two quasi-Fermi levels, one for the electrons ($F_n$) and one for the holes ($F_p$).
• Degree of non-equilibrium is indicated by separation between Quasi-Fermi levels.
• Back to equlibrium quasi-Fermi levels recombine

np product:

• We define $n_i^2$ as: $n_i^2 = n_o p_o = N_c N_v e^{-E_g/kT}$
• Under equilibrium conditions we know:

$n_o = N_c e^{-(E_c - E_f)/kT}$ $p_o = N_v e^{-(E_f - E_v)/kT}$

• Under non-equilibrium conditions:

$n = N_c e^{-(E_c - F_n)/kT}$ $p = N_v e^{-(F_p - E_v)/kT}$

np product (continued):

$np = N_c N_v e^{-(E_c - F_n)/kT} e^{-(F_p - E_v)/kT}$

• Since $E_g = E_c - E_v$, we can write:

$np = N_c N_v e^{-(E_g + F_p - F_n)/kT} $ $np = \underbrace{N_c N_v e^{-E_g/kT}}_{n_i^2} e^{(F_n - F_p)/kT} $ $np = n_i^2 e^{(F_n - F_p)/kT} $

Low-Level Injection:

• The condition where the injected minority carrier concentration is much smaller than the majority carrier concentration. e.g. n-type semiconductor under low-level injection: $\delta p ## Heat equation

Definition

• Let $u(x,t)$ be the temperature of a bar at position $x$ and time $t$.
• Heat equation is: $\frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial x^2}$, where $k$ is the thermal diffusivity.

Physical interpretation

• Describes how temperature changes over time in a given region.
• The rate of change of temperature at a point is proportional to the second spatial derivative of the temperature

Boundary conditions

• Dirichlet boundary condition: $u(0,t) = T_1$ and $u(L,t) = T_2$
• Neumann boundary condition: $\frac{\partial u}{\partial x}(0,t) = 0$ and $\frac{\partial u}{\partial x}(L,t) = 0$

Initial condition

• $u(x,0) = f(x)$

Solution

• Using separation of variables, we assume that $u(x,t) = X(x)T(t)$.
• Substituting into the heat equation, we get: $X(x)T'(t) = kX''(x)T(t)$
• Dividing both sides by $X(x)T(t)$, we get: $\frac{T'(t)}{kT(t)} = \frac{X''(x)}{X(x)} = -\lambda$
• Where $\lambda$ is a constant.

Ordinary Differential Equations

• $\qquad T'(t) = -k\lambda T(t)$
• $\qquad X''(x) = -\lambda X(x)$

Solving for $T(t)$

• $\qquad T(t) = Ae^{-k\lambda t}$

Solving for $X(x)$

$\qquad X''(x) = -\lambda X(x)$ The general solution is: $\qquad X(x) = B\cos(\sqrt{\lambda}x) + C\sin(\sqrt{\lambda}x)$

Applying boundary conditions

• Dirichlet boundary condition: $u(0,t) = 0$ and $u(L,t) = 0$ $\qquad X(0) = 0 \Rightarrow B = 0$ $\qquad X(L) = 0 \Rightarrow C\sin(\sqrt{\lambda}L) = 0$ $\qquad \sqrt{\lambda}L = n\pi$ $\qquad \lambda = \frac{n^2\pi^2}{L^2}$ $\qquad X(x) = C\sin(\frac{n\pi x}{L})$
• Neumann boundary condition: $\frac{\partial u}{\partial x}(0,t) = 0$ and $\frac{\partial u}{\partial x}(L,t) = 0$ $\qquad X'(0) = 0 \Rightarrow -B\sqrt{\lambda}\sin(0) + C\sqrt{\lambda}\cos(0) = 0 \Rightarrow C = 0$ $\qquad X'(L) = 0 \Rightarrow -B\sqrt{\lambda}\sin(\sqrt{\lambda}L) + C\sqrt{\lambda}\cos(\sqrt{\lambda}L) = 0$ $\qquad \sqrt{\lambda}L = n\pi$ $\qquad \lambda = \frac{n^2\pi^2}{L^2}$ $\qquad X(x) = B\cos(\frac{n\pi x}{L})$

General solution

• $\qquad u(x,t) = \sum_{n=1}^{\infty} A_n \sin(\frac{n\pi x}{L})e^{-k\frac{n^2\pi^2}{L^2}t}$

Applying initial condition

• $\qquad u(x,0) = f(x) = \sum_{n=1}^{\infty} A_n \sin(\frac{n\pi x}{L})$
• $\qquad A_n = \frac{2}{L}\int_0^L f(x)\sin(\frac{n\pi x}{L})dx$

The Laplace Distribution

Definition

• The Laplace distribution with location parameter $\mu$ and scale parameter $b > 0$ has the probability density function (pdf)
• $f(x) = \frac{1}{2b} \exp \left( -\frac{|x - \mu|}{b} \right)$.

Parameters

• Location: $\mu$
• Scale: $b$

Support

• The support of the Laplace distribution is $x \in (-\infty, \infty)$.

Mean

• The mean of the Laplace distribution is $E[X] = \mu$.

Variance

• The variance of the Laplace distribution is $Var[X] = 2b^2$.

Relation to Exponential Distribution

• If $X \sim \text{Laplace}(\mu, b)$, then $|X - \mu|$ follows an exponential distribution with rate parameter $\lambda = \frac{1}{b}$.

Applications

• Modeling data with heavier tails than the normal distribution.
• Bayesian inference:
• The Laplace distribution is the maximum entropy distribution for a random variable with a fixed first moment.
• Image processing: The Laplace distribution is used in image compression and denoising.

Generating Laplace Random Variables

To generate random variables from the Laplace distribution, you can use the following method:

1. Generate $U$ from a uniform distribution on $(0, 1)$.
2. Compute $X = \mu - b \cdot \text{sgn}(U - 0.5) \cdot \ln(1 - 2|U - 0.5|)$.