The configurational entropy of random trees

TL;DR

This work develops a rigorous graph-theoretic framework to quantify the configurational entropy of ideal random trees using Prüfer codes, enabling exact counting and efficient sampling of tree ensembles with controlled branching. By linking Prüfer codes to labelled-tree multiplicities and introducing a grand-canonical parameter $\mu$ for branch points, the authors derive exact partition functions $Z_{N,\mu}$ and observables like the mean number of branch-nodes $\langle N_3\rangle$ and the mean-square gyration radius $\langle R_g^2\rangle$, validated against Monte Carlo sampling and established field-theoretic theories. The work provides a fast $\mathcal{O}(N)$ sampling method for generating large random trees, and establishes precise connections to the de Gennes–Daoud–Joanny description of randomly branching polymers, elucidating asymptotic scaling regimes. Overall, the paper bridges combinatorial graph theory with polymer physics, offering exact entropy expressions, efficient sampling, and consistency with continuum theories across linear and highly branched limits.

Abstract

We present a graph theoretical approach to the configurational statistics of random tree-like objects, such as randomly branching polymers. In particular, for ideal trees we show that Prüfer labelling provides: (i) direct access to the exact configurational entropy as a function of the tree composition, (ii) computable exact expressions for partition functions and important experimental observables for tree ensembles with controlled branching activity and (iii) an efficient sampling scheme for corresponding tree configurations and arbitrary static properties.

Figures (7)

• Figure 1: Connectivity entropy per node, $S_N(N_3)/N$, as a function of the fraction of branch-nodes, $N_3/N$. Comparison between numerical results from thermodynamic integration of Monte Carlo simulations of lattice trees on the FCC lattice ($\times$, error bars are smaller than symbols' size) and Eq. \ref{['eq:BoltzmannEntropy']} with Eq. \ref{['eq:OmegaNn3']} ($\square$). Lines serve as a guide for the eye.
• Figure 2: (a) mean number of branch-nodes, $\langle N_3\rangle$, and (b) mean-square gyration radius, $\langle R_g^2\rangle$, as a function of tree nodes $N$. Results for: ($\blacksquare$) exact numerical evaluation, ($\times$) Prüfer-sampled trees, (lines) predictions of the Daoud-Joanny theory DaoudJoanny1981 with branching activity $\lambda^2 = (2+\exp(-(\beta\mu-\log(2))/2))^{-2}$ (Eq. \ref{['eq:LambdaFromMu']}). Different colors correspond to different values of the branching chemical potential $\mu$ (colorcode on the right).
• Figure 3: Mean CPU-time (in seconds, s) to obtain a random tree according to Prüfer sampling ($\bullet$) vs. equilibration time by using the Monte Carlo scheme of Ref. Amoebapaper2024 ($\times$). Lines serve as a guide for the eye. Different colors correspond to different values of the branching chemical potential $\mu$ (colorcode is as in Fig. \ref{['fig:n3+Rg2_usVSDJ']}). The power-law close to MC times has been extensively described in the same reference Amoebapaper2024.
• Figure 4: Trees of different sizes $N$ with Prüfer-sampled connectivities which are randomly embedded on the $3d$ FCC lattice. Bonds are shown with a colorcode indicating when they are defined within the Prüfer code.
• Figure 5: Probability distribution function, $P(R_g^2)$, of tree square gyration radius, $R_g^2$. Different symbols are for different $N$ (see legend), different colors are for different values of effective tree size $\lambda N$ (colorcode on the right) which depends on $\mu$ through Eq. \ref{['eq:LambdaFromMu']}. In the linear chain limit $\lambda N \ll1$ data agree with the exact expression (dashed line) by Fujita and Norisuye FujitaNorisuye1970. (Inset) Same data in log-log representation.