2.1: Lattice Models
- Page ID
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Lattice Models
Lattice models provide a minimalist, or coarse-grained, framework for describing the translational, rotational, and conformational degrees of freedom of molecules, and are particularly useful for problems in which entropy of mixing, configurational entropy, or excluded volume are key variables. The lattice forms a basis for enumerating different configurations of the system, or microstates. Each of these microstates may have a different energy, which is then used to calculate a partition function.
\[Q=\sum_{i} e^{-E_{i} / k_{B} T}\]
The thermodynamic quantities then emerge from
\[\begin{array}{l}
F=-k_{B} T \ln Q \\
S=-k_{B} \sum_{i} P_{i} \ln P_{i} \\
U=\sum_{i} P_{i} E_{i}
\end{array}
\nonumber\]
and other internal variables \((X)\) can be statistically described from
\[\langle X\rangle=\sum_{i=1}^{N} P_{i} X_{i} \quad P_{i}\left(E_{i}\right)=\frac{e^{-E_{i} / k_{B} T}}{Q}\nonumber\]
We will typically work with a macroscopic volume broken into cells, typically of a molecular size, which we can fill with the fundamental building blocks in our problem (atoms, molecules, functional groups) subject to certain constraints. In this section we will concern ourselves with the mixing of rigid particles, i.e., translational degrees of freedom. More generally, lattice models can include translational, rotational, and conformational degrees of freedom of molecules.