Moving probability distributions in the context of stochastic processes efficiently is central to modern generative modeling, uncertainty quantification, and control of large engineered systems. Most current methods generate temporal trajectories independently, leaving open whether the trajectories can cooperate through shared population information to reduce transport cost. Here we show that mean-field path-integral diffusion turns this question into a self-consistent stochastic control problem in which each trajectory responds to the evolving population. In a broad linear-quadratic setting, the problem reduces to a finite system of ordinary differential equations. For quadratic interactions with no background drift, we prove that the optimal population guidance is exactly the straight-line interpolation between the initial and target means, for arbitrary endpoint distributions with finite means. This yields an explicit construction for mixture targets. In demand-response control of multi-zone buildings, the method reduces control energy by 19–24% while maintaining the desired final distribution. These results suggest a practical route to coordinated generative transport for energy and other large-agent systems. Efficiently moving probability distributions in the context of stochastic processes is important in artificial intelligence, physics, and smart-building control. Here, the authors show that temporal trajectories can act like a coordinated population, reducing control energy by 19–24% in demand response while still reaching the desired final distribution.
Quantum advantage in real-time spin dynamics should be assessed against the strongest relevant classical substitutes, not merely against the qubit nature of the microscopic system. We develop a physics-based diagnostic for this boundary by reducing a qubit spin model to a spin-Landau--Lifshitz (LL) classical sector and organizing the residual quantum sector as controlled corrections. The control parameter is graph coordination: we study a spin star with \(d\) leaves and \(O(1/d)\) hub--leaf couplings. In its homogeneous form the star benchmarks the transition from LL-substitutable dynamics to genuinely quantum, discrete-sector interference; in its fully driven form, with time-dependent fields and bilinear couplings, it is the basic message-passing primitive for tree and loopy spin structures. For coherent-state return amplitudes we prove exact leaf elimination and derive a continuous-time \(1/d\) hierarchy. L0 is a driven one-spin weak-mean-field theory, while G1 is a Gaussian nonlocal-in-time influence correction coupling leaf two-time kernels to the hub weak two-point function. On bounded finite-time windows away from zeros of the boundary amplitudes, the hierarchy gives \(\log\mathcal A-\log\mathcal A_{\rm L0}=O(1/d)\) and \(\log\mathcal A-\log\mathcal A_{\rm L0}-\Delta_{\rm G1}=O(1/d^2)\); numerical tests on fully driven anisotropic ensembles give slopes \(-1.05\) and \(-2.03\). Static, inhomogeneous, aligned, and fully driven stars provide validation rungs, and comparison with a temporal matrix-product influence-matrix baseline delineates complementary regimes. Unlike rank compression on a Trotter grid, the hierarchy is ordered by a physical parameter, formulated in continuous time, and each truncation level is itself a physical theory, with the LL sector as the high-coordination limit.
M. Chertkov· 0 citations
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