Configurational Temperatures and Interactions in Charge-Stabilized Colloid

§ II. Configurational Temperatures

The most general configurational temperature definition, Eq. (3), requires knowledge of the full N-body potential, which typically is not known for experimental systems. A more experimentally accessible form emerges if the particles interact via pairwise additive potentials, u_{{ij}}(\boldsymbol{r}_{i}-\boldsymbol{r}_{j}). In this case, we can interpret gradients of the N-body potential,

\displaystyle\nabla V(\{ q_{i}\}) \displaystyle=\left(\frac{\partial}{\partial q_{1}},\cdots,\frac{\partial}{\partial q_{{3N}}}\right)\,\sum _{{k=1}}^{{N-1}}\sum _{{j=k+1}}^{N}u_{{jk}}(\boldsymbol{r}_{k}-\boldsymbol{r}_{j})
\displaystyle=-(\boldsymbol{F}_{1},\dots,\boldsymbol{F}_{N}) (4)

as components of the net force acting on each of the particles due to their interactions with their neighbors, where

\boldsymbol{F}_{j}=-\nabla _{j}\sum _{{k\neq j}}u_{{jk}}(\boldsymbol{r}_{k}-\boldsymbol{r}_{j}) (5)

is the force acting on the j-th particle. The temperature then may be written as (3)

k_{B}T_{{\text{conF}}}=-\frac{\left<\sum _{{j=1}}^{N}\boldsymbol{F}_{j}^{2}\right>}{\left<-\sum _{{j=1}}^{N}\nabla _{j}\cdot\boldsymbol{F}_{j}\right>}. (6)

Equation (6) can be applied also to non-pairwise-additive systems by including multi-body terms in Eq. (5). It makes sense that the temperature should be reflected in the instantaneous distribution of forces because objects explore more of their potential energy landscape as the temperature increases.

All definitions of the temperature derived from Eq. (1) involve approximations of \mathcal{O}\left(1/N\right), and so only are valid in the thermodynamic limit, N\to\infty. Dropping additional terms of \mathcal{O}\left(1/N\right) in the derivation of Eq. (6) can be justified for systems with short-ranged interactions and leads to another equivalent temperature definition (3):

k_{B}T_{{\text{con1}}}=\left<\frac{\sum _{{j=1}}^{N}\boldsymbol{F}_{j}^{2}}{-\sum _{{j=1}}^{N}\nabla _{j}\cdot\boldsymbol{F}_{j}}\right>. (7)

Neglecting further higher-order terms yields still another form (5),

k_{B}T_{{\text{con2}}}=\left<\frac{-\sum _{{j=1}}^{N}\nabla _{j}\cdot\boldsymbol{F}_{j}}{\sum _{{j=1}}^{N}\boldsymbol{F}_{j}^{2}}\right>^{{-1}}. (8)