Exact density functional for hard-rod mixtures derived from Markov chain approach

Using a Markov chain approach we rederive the exact density functional for hard-rod mixtures on a one-dimensional lattice, which forms the basis of the lattice fundamental measure theory. The transition probability in the Markov chain depends on a set of occupation numbers, which reflects the property of a zero-dimensional cavity to hold at most one particle. For given mean occupation numbers (density profile), an exact expression for the equilibrium distribution of microstates is obtained, which means an expression for the unique external potential that generates the density profile in equilibrium. By considering the rod ends to fall onto lattice sites, the mixture is always additive.

Figure 1

Illustration of the set of occupation numbers affecting the occupation of site $k$. Any placement of the left end of a rod of type $\alpha$ at the sites $j$ with $k-l_{\alpha }+1\le j\le k$ means that site $k$ is covered by a part of this rod. This implies (i) that if a left rod end is at site $k$, all occupation numbers in the set ${\left\{n_j^{\alpha }\right\}}_{k-1}=\left\{n_j^{\alpha }\right|1\le \alpha \le q,k-l_{\alpha }+1\le j\le k-1\}$ must be zero, and (ii) that in the set ${\left\{n_j^{\alpha }\right\}}_k=\left\{n_j^{\alpha }\right|1\le \alpha \le q,k-l_{\alpha }+1\le j\le k\}$ there can be at most one occupation number with value 1.