Heisenberg Uncertainty Principle

Terms: Heisenberg Uncertainty Principle (182,000),

Terms: phase space (491,000), phase space distribution (8,110), density of phase space (66), density in phase space (297), reaction diffusion (153,000), fisher's equation (1,030), diffusion equation (159,000), dynamical system (272,000),

Wikipedia - Phase space method, Dynamical system,

Terms: canonical coordinates (3,020), canonical transformation (11,800), Hamiltonian mechanics (28,700), Hamiltonian formalism (12,100), Hamilton's equations (13,700),

$\{f,g\} = \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q^{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q^{i}} \right].$

Terms: fourier transform (1,480,000), fourier transforms (368,000), fourier transformation (142,000), Legendre transformation (2,990), Legendre transform (12,000), Legendre transforms (1,160),

Terms: collisionless Boltzman equation (16), Vlasov equation (16,100), Liouville's theorem (20,100), Green-Kubo relations (371),

In the simple case of a non-relativistic particle moving in Euclidean space under a force field $\mathbf{F}$ with coordinates $\mathbf{x}$ and momenta $\mathbf{p}$ , Liouville's theorem can be written

$\frac{\partial\rho}{\partial t}+\frac{\mathbf{p}}{m}\cdot\nabla_\mathbf{x}\rho+\mathbf{F}\cdot\nabla_\mathbf{p}\rho=0.$

In astrophysics this is called the Vlasov equation, or sometimes the Collisionless Boltzmann Equation, and is used to describe the evolution of a large number of collisionless particles moving in a gravitational potential.

$\frac{d\rho}{dt}= \frac{\partial\rho}{\partial t} +\sum_{i=1}^d\left(\frac{\partial\rho}{\partial q^i}\dot{q}^i +\frac{\partial\rho}{\partial p_i}\dot{p}_i\right)=0.$

The Liouville equation is valid for both equilibrium and nonequilibrium systems. It is a fundamental equation of nonequilibrium statistical mechanics. Generalized to collisional systems it is called the Boltzmann equation.
The Liouville equation is integral to the proof of the fluctuation theorem from which the second law of thermodynamics can be derived. It is also the key component of the derivation of Green-Kubo relations for linear transport coefficients such as shear viscosity, thermal conductivity or electrical conductivity.
Virtually any textbook on Hamiltonian mechanics, advanced statistical mechanics, or symplectic geometry will derive the Liouville theorem.

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Terms: fluctuation spectrum (2,210), non-Markov process (89), non-Markov processes (194), langevin equations (5,600),

N. G. van Kampen - Probabilities in Physics

22^nd International Free Electron Laser Conference

Nonequilibrium Statistical Mechanics

Peter Mazur, 1922–2001 - Peter Mazur was one of the founders of non-equilibrium thermodynamics. Together with his teachers S.R. de Groot in Utrecht and I. Prigogine in Brussels, Mazur was able to integrate the reciprocity relations in a comprehensive theory.

http://swc.math.arizona.edu/notes/files/

N. G. van Kampen - Stochastic Processes in Physics and Chemistry

Statistical Mechanics of NonEquilibrium Liquids, Evans & Morriss

Terms: quantum mechanics (3,120,000), product of standard deviations (81), conjugate quantities (402),

Terms: Poisson bracket (20,600), Poisson brackets (14,800), Hamilton-Jacobi (87,500), Liouville equation (15,100), conservation of density in phase space (2), conservation of density (78), conserved current (1,310),

A simple proof of the theorem is to observe that the evolution of $?$ is defined by the continuity equation:

$\frac{\partial\rho}{\partial t}+\sum_{i=1}^d\left(\frac{\partial(\rho\dot{q}^i)}{\partial q^i}+\frac{\partial(\rho\dot{p}_i)}{\partial p_i}\right)=0.$

That is, the tuplet $(\rho, \rho\dot{q}^i,\rho\dot{p}_i)$ is a conserved current.

Wikipedia - Poisson bracket, Momentum in Quantum Mechanics, Liouville's theorem (Hamiltonian),

The definition canonical momentum corresponding to the momentum operator of quantum mechanics when it interacts with the electromagnetic field is, using the principle of least coupling:

In relativistic mechanics, momentum is defined as:
$\mathbf{p} = \gamma m\mathbf{v}$ where $m$ is the invariant mass of the object, $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$ is the Lorentz factor

The four-momentum is defined as: $\left( {E \over c} , p_x , p_y ,p_z \right)$ ,

$(E / c) 2 - p 2 = (m c) 2,$

$E = \gamma mc^2 \;$
$\mathbf P = m\mathbf v + q\mathbf A$ , instead of the customary $\mathbf p = m\mathbf v$ , where: $\mathbf A$ is the electromagnetic vector potential, $m$ the charged particle's invariant mass, $\mathbf v$ its velocity, $q$ its charge.

Suppose that $f (p, q, t)$ is a function on the manifold. Then one has

$\frac {\mathrm{d}}{\mathrm{d}t} f(p,q,t) = \frac{\partial f}{\partial t} + \frac {\partial f}{\partial p} \frac {\mathrm{d}p}{\mathrm{d}t} + \frac {\partial f}{\partial q} \frac {\mathrm{d}q}{\mathrm{d}t}$

Then, by taking $p = p (t)$ and $q = q (t)$ to be solutions to the Hamilton-Jacobi equations $\dot{q}={\partial H}/{\partial p}$ and $\dot{p}=-{\partial H}/{\partial q}$ , one may write

$\frac {\mathrm{d}}{\mathrm{d}t} f(p,q,t) = \frac{\partial f}{\partial t} + \frac {\partial f}{\partial q} \frac {\partial H}{\partial p} - \frac {\partial f}{\partial p} \frac {\partial H}{\partial q} = \frac{\partial f}{\partial t} +\{f,H\}$

$\{f,g\} = \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q^{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q^{i}} \right].$

Boolean: "total differential" (21,600),

Terms: fourier transform duals (8), Pontryagin duality (702), dual group (26,000),

Terms: time and frequency (374,000), time and energy (5,990,000), position and momentum (58,300), angle and angular momentum (67),

Terms: radar ambiguity function (190), radar ambiguity diagram (14),

Section=2007-05-May\
Title=Heisenberg Uncertainty Principle
Author="Richard Collins"
Email=RichardCollins@TheInternetFoundation.Org
Generator=Factor6
Created=5/5/2007 9:12:58 PM
LastUpdate=5/5/2007 10:58:48 PM