M2Flow: Macro‐to‐Micro Flow Transformation

• Macro-to-Micro Flow Transformation (M2Flow) is a framework that connects microscopic particle fluctuations to macroscopic deterministic equations through large-deviations analysis and entropy–Wasserstein gradient flows.
• The methodology leverages mathematical techniques including gamma-convergence, optimal transport, and variational time-stepping to rigorously establish the micro–macro passage.
• M2Flow has broad applications across physics, biology, and engineering, providing a foundation for modeling diffusion processes, stochastic perturbations, and emergent dynamics.

Macro-to-Micro Flow Transformation (M2Flow) describes frameworks that systematically bridge the gap between macroscopic (continuum, averaged, or bulk) descriptions of dynamic systems and their microscopic (particle, stochastic, or structure-resolved) origins. Across physical, biological, and engineering domains, M2Flow formalizes how the statistics and variational principles governing ensembles of particles or agents at small scales generate and constrain emergent macroscopic dynamics, serving both as explanatory bridges and computational strategies for multiscale modeling. The following sections detail core principles and methodologies central to M2Flow, focusing on the mathematical and probabilistic connections established between micro and macro levels, especially as realized in the context of Brownian particle systems and the entropy–Wasserstein gradient flows for diffusion (Adams et al., 2010).

1. Large-Deviations Principle and the Statistical Foundation

At the microscopic scale, systems comprising many independent Brownian particles exhibit stochastic fluctuations superimposed on their mean behavior. The large-deviations framework quantifies the probability that the empirical measure $L_n^h$ of particle positions at time $t = h$ deviates from the deterministic profile predicted by the initial distribution $\rho_0$. The key result is the characterization of fluctuation probabilities via a rate functional $J_h$:

$$P(L_n^h \approx \rho \mid L_n^0 \approx \rho_0) \asymp \exp[-n J_h(\rho; \rho_0)]$$

where

$$J_h(\rho; \rho_0) = \inf_{q \in \Gamma(\rho_0, \rho)} H(q \mid q_0)$$

and $H(q | q_0)$ denotes the relative entropy between a candidate coupling $q$ and a reference measure $q_0$, constructed using the Brownian transition kernel $p_h(x, y)$:

$$q_0(dx, dy) = \rho_0(dx) p_h(x, y) dy, \quad p_h(x, y) = (4\pi h)^{-1/2} \exp\left(-\frac{(y-x)^2}{4h}\right)$$

As $h \to 0$, $J_h$ admits an asymptotic expansion in powers of $h$, with the leading term proportional to the squared Wasserstein distance $d(\rho, \rho_0)^2$ and a next-order term involving the entropy difference $E(\rho) - E(\rho_0)$:

$$J_h(\rho; \rho_0) = \frac{1}{4h} d(\rho, \rho_0)^2 + \frac{1}{2}[E(\rho) - E(\rho_0)] + o(1)$$

This expansion provides the statistical underpinning for relating microscopic fluctuations to macroscopic evolutionary equations.

2. Entropy–Wasserstein Gradient Flows and Variational Time Stepping

The deterministic diffusion equation

$\partial_t \rho = \Delta \rho$

can be recast as a gradient flow of the entropy functional $E(\rho) = \int \rho \log \rho dx$ with respect to the Wasserstein metric on probability measures. The discrete-time gradient flow (Jordan–Kinderlehrer–Otto scheme) is given by:

$$\rho^n \in \arg\min_\rho \left[ K_h(\rho; \rho^{n-1}) \right]$$

where

$$K_h(\rho; \rho_0) = \frac{1}{2h} d(\rho, \rho_0)^2 + E(\rho) - E(\rho_0)$$

This variational characterization selects, at each time step, the "most likely" $\rho$ as the solution to a steepest-descent problem balancing entropy dissipation and transport cost.

3. Asymptotic Equivalence and Micro–Macro Passage

A main finding is the rigorous asymptotic equivalence (up to second order in $h$) between the large-deviations rate functional $J_h$ (microscopic level) and the variational time-stepping functional $K_h$ (macroscopic gradient flow):

$$J_h(\rho; \rho_0) - \frac{1}{4h} d(\rho, \rho_0)^2 \to \frac{1}{2}[E(\rho) - E(\rho_0)] \quad \text{as } h \to 0$$

This is established with gamma-convergence arguments. The minimization principle for $K_h$ not only defines the deterministic evolution but also encodes the probability landscape for fluctuations of the underlying particle system. Thus, the macroscopic deterministic variational scheme directly reflects the fluctuation mechanism in the stochastic microscopic dynamics.

4. Microscopic-to-Macroscopic Mapping via Brownian Particle Dynamics

Each particle trajectory $X^{(i)}(t)$ follows standard Brownian motion. The law of large numbers ensures that the empirical measure $L_n^t = (1/n)\sum_{i=1}^n \delta_{X^{(i)}(t)}$ converges, as $n \to \infty$, to the smooth density $\rho(x, t)$ governed by the diffusion partial differential equation. Fluctuation probabilities in finite $n$ systems are controlled by $J_h$, which, in the small $h$ limit, shows that the optimal transport cost structure (Wasserstein metric) and entropy arise naturally from the central limit scaling and Sanov's theorem for empirical measures.

The bridging formula, in LaTeX notation, summarizing the micro–macro passage is:

$$J_h(\rho; \rho_0) = \inf_{q \in \Gamma(\rho_0, \rho)} H(q \mid q_0), \;\; q_0(dx, dy) = \rho_0(dx)p_h(x, y)dy$$

$$J_h(\rho; \rho_0) = \frac{1}{4h} d(\rho, \rho_0)^2 + \frac{1}{2}[E(\rho) - E(\rho_0)] + o(1)$$

This formularizes the passage from stochastic microscopic dynamics and their fluctuation structures to macroscopic gradient-flow evolution.

5. Physical Interpretation: Entropic Gradient Flow and Fluctuation Control

The entropic gradient flow formalism interprets the deterministic macroscopic equation as arising from steepest descent of the entropy $E(\rho)$ in the optimal transport geometry. This construction is physically "natural": both driving force (entropy change) and dissipation mechanism (Wasserstein metric) directly emerge from the statistical mechanical properties of particle motion.

Large-deviations analysis further clarifies that the minimization problem selects not just the deterministic path but also governs the structure of fluctuations about this path. Thus, entropy and optimal transport appear in the macroscopic variational principle due to their presence in microscopic state transitions and the cost structure of the underlying stochastic process.

6. Implications and Significance for M2Flow

This unified framework (M2Flow) consolidates the following points:

• The large-deviations rate functional $J_h$ gives a quantitative measure of atypical events in many-particle systems, and its expansion reveals how entropy and optimal transport govern micro–macro passage.
• Discrete-time Wasserstein gradient flows for macroscopic equations (diffusion PDE) emerge as both the most probable deterministic trajectory and the minimizer of the large-deviations probability cost.
• Physical models for systems with micro–macro structure (e.g., Brownian particles, heat diffusion) are naturally described as entropic gradient flows, with fluctuation behavior encoded in the same variational principles as the deterministic evolution.
• The equivalence between $J_h$ and $K_h$ (up to second order in $h$) provides the mathematical foundation for passing from microscopic stochastic laws, including fluctuation cost, to continuum deterministic equations.

The approach gives a rigorous explanation for why entropy and optimal transport appear in macroscopic modeling and delivers a statistical mechanical foundation for variational PDE discretizations and the analysis of noise-induced effects in diffusion-type processes.

7. Broader Applications and Perspectives

While the case discussed is deterministic diffusion arising from Brownian particles, the underlying principles of M2Flow extend to more general settings: nonlinear PDEs, multiscale stochastic systems, agent-based and continuum limits in traffic and biological flows, and probabilistic interpretations of variational PDE formulations. The method provides conceptual clarity for connecting microscopic randomness, fluctuation behavior, and macroscopic order, and the gamma-convergence framework developed enables rigorous micro–macro passages that are central to modern multiscale analysis.

The M2Flow perspective thus forms a foundational pillar for statistical mechanics-informed modeling of continuum phenomena, optimal transport theory, and the design of physically consistent multiscale numerical methods.