## Statistical quantities ### Arbitrary quantity $A$ $\langle A\rangle = \frac{1}{N_T} \sum_{t=1}^{N_T} A(t)$ $N_T$ is number of time step. $A$ may be potential energy, kinetic energy, total energy, temperature, etc. >[!notice] >This mean quantity is for MD at equilibrium, it's time average! And $A$ should be some instantaneous quantity, and evolving with time. >This would not be conflict with $A$ being [[Retrospect on statistical mechanics#^23b0cd|ensemble average]]. - potential energy: $V(t) = \sum_i\sum_j \Phi(|r_i(t)-r_j(t)|)$ - kinetic energy: $K(t) = \frac{1}{2}\sum_i m_i[v_i(t)]^2$ - total energy: sum of above two - temperature: $\frac{N_f}{2}k_BT = \langle K \rangle$, $N_f$ is total degree of freedom; for 3D case, it becomes the form we're familiar: $\langle K \rangle = \frac{3}{2}Nk_B T$. >[!note] >The expression related to *total degree of freedom* would be true when we're discussing only the kinetic energy (not $H$, which include an additional harmonic term in potential) ### Mean square displacement (MSD) ^010768 $MSD=\langle |\vec{r}(t)-\vec{r}(0)|^2 \rangle = \frac{1}{N} \sum_N (\vec{r}(t)-\vec{r}(0))^2$ Here $N$ is the number of particles. This quantity is related to diffusion process and can be used to calculate the diffusion coefficient $D$. This expression is a direct result of particle movement from [[Fluctuation-Dissipation theorem]], and may link to mobility $\mu$ by [[Einstein relation]]. $D = \lim_{t \to \infty} \frac{1}{6t}\cdot MSD$ >[!notice] >MSD is a property at time $t$, **NOT a time-averaged one**. MSD is time dependent, and it acts as the slope of $D$. ### Pressure Calculated based on Claudins virial function, and can be separated into external force (i.e., container) and internal force (i.e., intermolecular interaction) induced. $pV = Nk_B T+\frac{1}{D} \langle \sum_{i=1}^N \vec{r_i}\cdot \vec{F_i} \rangle$ Here, $D$ is number of dimensionality. >[!note] >$pV=nkT$ -> Classical ideal gas > >$\left( p+a \frac{n^2}{V^2} \right) (V-nb) = nRT$ -> van der Waals gas > >$pV = Nk_BT\left[ 1+B_1\left( \frac{N}{V} \right)+B_2\left( \frac{N^2}{V^2} \right) +\ldots \right]$ -> Virial expansion > >In the expression for pressure, the first term $Nk_B T$ is virial term (or say ideal gas), the second term $\frac{1}{D} \langle \sum_{i=1}^N \vec{r_i}\cdot \vec{F_i} \rangle$ is internal force, can also be expressed as potential function.