Boltzmann distribution¶

This module initialises particle velocities from a Boltzmann distribution.

The velocity distribution per degree of freedom is a Gaussian with mean $\mu_{v} = 0$ and width $\sigma_{v} = \sqrt{\frac{kT}{m}}$ ,

$f\left(v\right) = \sqrt{\frac{m}{2\pi kT}} \exp\left(\frac{-mv^2}{2 kT}\right)$

To account for the finite size of the system, the velocities are shifted,

$\vec{v}_\text{shifted} \equiv \vec{v} - \vec{V}_\text{cm}$

to yield a centre of mass velocity of zero, and scaled,

$\vec{v}_\text{scaled} \equiv \vec{v}_\text{shifted}\; \sqrt{\frac{\frac{1}{2} kT f N}{E_\text{kin} - \frac{1}{2} M\left(\vec{V}_\text{cm}\right)^2}}$

to yield the temperature $T$ for a system with $f$ positional degrees of freedom, using centre of mass velocity and kinetic energy,

$\vec{V}_\text{cm} &= \frac{\vec{P}_\text{cm}}{M} = \frac{\sum_{n = 1}^N m_n \vec{v}_n}{\sum_{n = 1}^N m_n} \\ E_\text{kin} &= \frac{1}{2} \sum_{n = 1}^N m_n v_n^2$

class halmd.mdsim.velocities.boltzmann(args)¶

Construct boltzmann module.

Parameters:	args (table) – keyword arguments args.particle – instance of `halmd.mdsim.particle` args.temperature (number) – temperature of distribution

set()¶: Initialise particle velocities from Boltzmann distribution.

temperature¶: Temperature of distribution in reduced units.

Navigation

Quick search

Boltzmann distribution¶

Navigation