Summary of statistical mechanics relations and helpful formulas (cheat sheet fodder)
Posted by peeterjoot on April 29, 2013
Central limit theorem
If and , and , then in the limit
where was something like number of Heads minus number of Tails.
Given the Fourier transform of a probability distribution we have
volume in mD
area of ellipse
Radius of gyration of a 3D polymer
With radius , we have
Velocity random walk
1D Random walk
The diffusion constant relation to the probability current is referred to as Fick’s law
with which we can cast the probability diffusion identity into a continuity equation form
In 3D (with the Maxwell distribution frictional term), this takes the form
Add a frictional term to the velocity space diffusion current
For steady state the continity equation leads to
We also find
Quantum energy eigenvalues
Regardless of whether we have a steady state system, if we sit on a region of phase space volume, the probability density in that neighbourhood will be constant.
A system for which all accessible phase space is swept out by the trajectories. This and Liouville’s threorm allows us to assume that we can treat any given small phase space volume as if it is equally probable to the same time evolved phase space region, and switch to ensemble averaging instead of time averaging.
Example (work on gas): . Adiabatic: . Cyclic: .
Quantum free particle in a box
moment per particle
Grand Canonical ensemble
(so for large temperatures)
For , .
Density of states
This entry was posted on April 29, 2013 at 11:21 pm and is filed under Math and Physics Learning.. Tagged: binomial distribution, Bosons, canonical ensemble, Central limit theorem, cheat sheet, density of states, ergodic, Fermions, Generating function, grand canonical ensemble, Hamilton's equations, Handy mathematics, ideal gas, Liouville's theorem, Maxwell distribution, Microstates, PHY452H1S, Quantum free particle in a box, Radius of gyration of a 3D polymer, random walk, spin, statistical mechanics, thermodynamics, Velocity random walk. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.