# Fluctuations

The methods developed allow us to calculate thermodynamic averages. The deviation of a mechanical variable from its mean value is called a fluctuation. The theory of fluctuations is useful for understanding how the different ensembles (NVT, NPT etc.). Fluctuations are also important in theories of light scattering and in the study of transport processes. We will show that the fluctuations about the mean energy are very small for macroscopic systems. Given the fact that the number of particles is much smaller the relative magnitude of fluctuations in a MD simulation are much larger. In part, we study fluctuations to have a criterion for determining the quality of an average property that we calculate in a computer simulation.

The average of a property is also called the first moment of a distribution. The second moment of the distribution is called the variance

$\langle (x- \langle x \rangle)^2 \rangle = \langle x^2 \rangle - \langle x \rangle^2 \label{var}$

The variance is a measure of spread of the probability distribution about the mean value. The mean value is

$\langle x \rangle = \int_{-\infty}^{\infty} x P(x)dx$

or if descritized data are involved

$\langle x \rangle = \sum_i x_i P_i$

where $$P(x)$$ is a normalized probability distribution or $$P_i$$ is a discrete probability distribution. The mean square value is

$\langle x^2 \rangle = \int_{-\infty}^{\infty} x^2 P(x)dx$

or if descritized data are involved

$\langle x^2 \rangle = \sum_i x_i^2 P_i$

We consider the fluctuations in the NVT or canonical ensemble. The number, volume, and temperature are fixed, and we can calculate the fluctuations in the energy in this ensemble. The variance in the energy (Equation \ref{var}) is

\begin{align} \sigma_E^2 & = \langle (E- \langle E \rangle)^2 \rangle = \langle E^2 \rangle - \langle E \rangle^2 \\ & = \sum_i E^2_iP_i - \left( \sum_i E^2_i P_i \right)^2 \end{align}

where

We need to evaluate the mean-square energy term

Thus,

The spread in energies about the mean is given by the ratio of the square root of the variance relative to the energy

For an ideal gas Cv = 3/2Nk and E = 3/2NkT

The fluctuations are proportional to 1/Ö N which is an extremely small number for macroscopic systems where N is of the order of Avagadro's number.