Statistical Mechanics of the Sub-Optimal Transport

Abstract

Statistical mechanics is a powerful framework for analyzing optimization yielding analytical results for matching, optimal transport, and other combinatorial problems. However, these methods typically target the zero-temperature limit, where systems collapse onto optimal configurations, a.k.a. the ground states. Real-world systems often occupy intermediate regimes where entropy and cost minimization genuinely compete, producing configurations that are structured yet sub-optimal. The Sub-Optimal Transport (SOT) model captures this competition through an ensemble of weighted bipartite graphs: a coupling parameter interpolates between entropy-dominated dense configurations and cost-dominated sparse structures. This crossover has been observed numerically but lacked analytical understanding. Here we develop a mean-field theory that characterizes this transition. We show that local fluctuations in Lagrange multipliers become sub-extensive in the thermodynamic limit, reducing the full model with strength constraints to an effective single-constraint problem admitting an exact solution in some intermediate regime. The resulting free energy is analytic in the coupling parameter, confirming a smooth crossover rather than a phase transition. We derive closed-form expressions for thermodynamic observables and weight distributions, validated against numerical simulations. These results establish the first analytical description of the SOT model, extending statistical mechanics methods beyond the zero-temperature regime.

Figures (4)

• Figure 1: Thermodynamic quantities for the single-constraint ensemble Panel (a) Free energy of Eq.\ref{['eq:free_energy_global']} vs $\log\beta\bar{\omega}$ for different mean mass values $\bar{\omega}$. Panel (b) Mean energy in Eq.\ref{['eq:energy']} showing crossover between high-noise ($u = \bar{\omega}/2$) and low-noise ($u\to0$) regimes for different mean mass values $\bar{\omega}$. Panel (c) Energy susceptibility vs $\log\beta\bar{\omega}$. Panel (d) Mass susceptibility in Eq.(\ref{['eq:mass_susceptibility']}) showing system response to $\bar{\omega}$ variations.
• Figure 2: Lagrange multipliers converge to Gaussian distributions in the thermodynamic limit and exhibit non-trivial dependence on $\beta$ and system size. Numerical solutions of the SOT model with uniform cost distribution show that the distributions of individual multipliers $\lambda_i$ (solid lines, layer 1) and $\mu_\alpha$ (dashed lines, layer 2) approach Gaussian shape as system size increases (panel a), with the outer sum $\lambda_i + \mu_\alpha$ exhibiting even stronger convergence (panel b). (c) The mean multiplier values stabilize with increasing system size and become essentially constant at large $\beta$ (inset). (d) The multipliers' standard deviation grows with $\beta$ but decreases with system size, demonstrating that the thermodynamic limit and large-$\beta$ limit do not commute (inset shows the power-law saturation behavior $\text{Std}\sim \text{dim}^{-1}$). In the insets, colors distinguish between the two layers, not system sizes.
• Figure 3: Mean energy at different $\beta$: Numerical results with (a) uniform and (b) Beta cost distributions. Colors encode the linear system size; the solid black line shows the theoretical predictions. The inset of panel (b) shows how the mean energy varies as a function of the system size for the smallest available value of $\beta$; the red dashed line corresponds to the best fit using a saturating power-law $f(x\vert a,b,c) = a-b x^{-c}$, $a=0.45$, $b=0.35$, $c=0.31$.
• Figure 4: Distribution of the weights for the uniform cost: Panel (a) shows the distribution of $w$ from the numerical solution of the SubOT model at different system sizes. Panel (b) shows the distribution of $w$ varying the parameter $\beta$, thus moving from the dense regime at low $\beta$ to the sparse regime at large $\beta$.