## 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

**Binomial distribution**

where was something like number of Heads minus number of Tails.

**Generating function**

Given the Fourier transform of a probability distribution we have

**Handy mathematics**

Heavyside theta

volume in mD

area of ellipse

**Radius of gyration of a 3D polymer**

With radius , we have

**Velocity random walk**

Find

**Random walk**

1D Random walk

leads to

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

**Maxwell distribution**

Add a frictional term to the velocity space diffusion current

For steady state the continity equation leads to

We also find

and identify

**Hamilton’s equations**

SHO

Quantum energy eigenvalues

**Liouville’s theorem**

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.

**Ergodic**

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.

**Thermodynamics**

Example (work on gas): . Adiabatic: . Cyclic: .

**Microstates**

quantum

**Ideal gas**

**Quantum free particle in a box**

**Spin**

magnetization

moment per particle

spin matrices

spin addition

**Canonical ensemble**

classical

quantum

**Grand Canonical ensemble**

**Fermions**

(so for large temperatures)

**Bosons**

For , .

BEC

**Density of states**

Low velocities

relativistic

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.

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