Coarse-Grained Hydrodynamic Assumptions

• Coarse grained hydrodynamics is a framework that defines effective macroscopic fields by statistically averaging microscopic distributions without assuming local thermal equilibrium.
• It partitions phase space into mesoscopic cells to suppress fluctuations through self-averaging, resulting in smooth density and momentum fields.
• The approach delineates between intrinsic diffusion and convective corrections, demonstrating Euler-scale dynamics with zero intrinsic diffusion for systems like hard rods.

Coarse Grained Hydrodynamic Assumptions

Coarse graining in hydrodynamics involves representing the collective dynamics of large ensembles of microscopic degrees of freedom by smooth, effective fields—such as density, momentum, and energy—defined on mesoscopic space-time scales. The foundational assumptions concern the statistical smoothing of the microscopic state, the nature of the emergent conservation laws, the role (or absence) of local equilibrium, and the criteria governing the validity and limitations of hydrodynamic equations. This article synthesizes the precise mathematical and physical assumptions underlying coarse-grained hydrodynamics, with emphasis on results demonstrated in the context of integrable models, notably classical hard rods (Hübner, 23 Jul 2025).

1. Definition and Coarse-Graining Procedure

Coarse graining starts from a microscopic configuration and partitions phase space (e.g., the $(x,p)$ plane for hard rods) into mesoscopic "fluid cells" of size $\Delta x \times \Delta p$. The partitioning is constrained such that $\Delta x, \Delta p \to 0$ as the macroscopic scale $\ell \to \infty$, but $\ell\,\Delta x\,\Delta p \gg 1$, ensuring each cell contains many particles and suppressing local fluctuations.

For a system of $N$ rods with positions and momenta $\{x_i,p_i\}_{i=1}^N$, the coarse-grained, piecewise-constant density is defined: $$\rho_{\ell}(x,p)\equiv \sum_{\alpha,\beta}\rho_{\alpha,\beta}\;1_{(x,p)\in A_\alpha \times B_\beta},\quad \rho_{\alpha,\beta} = \frac{1}{\ell\,\Delta x\,\Delta p}\#\{i: x_i\in A_\alpha,\,p_i\in B_\beta\}$$ where $K$ is a smooth bump function of unit area for explicit cell averaging.

The coarse-grained current is constructed using an effective velocity that depends locally on the density: $$j_\ell(x,t) = \int dp\,v_{\rm eff}[\rho_\ell](x,t;p)\;\rho_\ell(x,t;p)$$ The local coarse-graining ensures the suppression of microscopic "wiggles" by self-averaging within each cell, without invoking any statistical ensemble.

2. Emergence and Structure of Macroscopic Hydrodynamic Equations

At leading order in the $\ell\to\infty$ limit, the coarse-grained density field $\rho_\ell(x,p)$ evolves under closed hydrodynamic equations, which for hard rods take the form: $$\partial_t\rho_{\ell}(x,p) + \partial_x\bigl(v_{\rm eff}[\rho_\ell](x,p)\,\rho_\ell(x,p)\bigr)=0$$ The effective velocity is given by: $$v_{\rm eff}[\rho](x,p) = \frac{p - a\int dq\,q\,\rho(x,q)}{1 - a\int dq\,\rho(x,q)}$$

This equation is structurally identical, at the Euler scale, to the "Bethe–Boltzmann" or "generalized hydrodynamics" (GHD) equation often derived via ensemble averages or local-GGE (generalized Gibbs ensemble) assumptions. Crucially, in the present coarse-grained formulation, no thermal or local equilibrium hypothesis is invoked. The only requirement is that each cell contains sufficiently many particles for local self-averaging.

The current $j_\ell$ is defined as $\int dp\,v_{\rm eff}\,\rho_\ell$, so that continuity holds: $\partial_t\rho_\ell + \partial_x j_\ell = 0$.

3. Diffusion and the Absence of Intrinsic Dissipation

A central question in hydrodynamic theory is the possible emergence of diffusive (Navier–Stokes) corrections to the leading Euler-scale dynamics. In this coarse-grained, "single-sample" formulation for hard rods, all such intrinsic diffusive terms vanish identically:

• Attempting to compute an entropy-increasing $O(1/\ell)$ term via Kubo-type formulas for a locally thermal (GGE) ensemble yields a total error scaling as $\sim \max(\Delta x^2,\,\Delta x/\sqrt{\ell})$.
• With optimal scaling $\Delta x \sim \ell^{-1/2}$, the error is $O(\ell^{-3/2})$—there is no $O(1/\ell)$ contribution.
• Consequently, the variance between the microscopic current $j_{\rm micro}$ and the coarse-grained $j_\ell$ is $o(1/\ell)$, so that no diffusive correction remains at this order.

Thus, for hard rods, "intrinsic diffusion constants" are strictly zero for the properly coarse-grained, deterministic density field. Any entropy production or diffusive broadening does not originate from intrinsic, hydrodynamic-scale dissipation.

4. Distinction Between Intrinsic Diffusion and Diffusion from Convection

If, instead, one averages over an ensemble of initial conditions (e.g., within a local-GGE), the current correction at $O(1/\ell)$ is found to split into two contributions: $$\Delta j(x,p) = \Delta j_{\rm intr}(x,p) + \Delta j_{\rm conv}(x,p)$$ Explicit GHD linearization shows: $\Delta j_{\rm intr}(x,p) \equiv 0$ meaning there is no entropy-producing (Navier–Stokes) "intrinsic" diffusion kernel. All surviving $O(1/\ell)$ terms arise from the transport of initial-state fluctuations, termed "diffusion from convection." This reversible, drift-like correction can be written in closed form involving a known two-body scattering kernel and the symmetrized long-range part of the two-point function of initial data fluctuations.

In summary, all non-trivial corrections to the coarse-grained hydrodynamics in hard rods stem from convective transport of averaged initial inhomogeneities, not from dissipative or stochastic effects intrinsic to the deterministic dynamics.

5. Applicability to Non–Locally–Thermal States

A consequence of the coarse-grained approach is its applicability to initial configurations that are far from local thermal or GGE states. Provided the deterministic packing is generic (examples include Poisson random configurations or random matrix eigenvalue distributions), the fluid-cell coarse-grained field $\rho_\ell(x,p)$ converges algebraically to the GHD prediction, with error $\sim \ell^{-\nu}$ and $\nu > 1$, across broad classes of "non-equilibrium" initial data.

No assumption of local equilibrium is needed at any stage; hydrodynamic behavior emerges solely by virtue of local self-averaging within the coarse cells. GHD-like equations thus govern each individual sample, not just ensemble averages.

6. Summary of Domain of Validity and Essential Assumptions

The above framework relies on the following controlled hierarchy of assumptions:

• Scale Separation: The spatial/temporal resolution $\ell$ satisfies $\ell \gg$ microscopic correlation length, and each cell hosts many degrees of freedom.
• Self-Averaging: The key mechanism replacing statistical equilibrium is the law of large numbers acting within each cell. No ensemble or stochastic hypothesis is required.
• Error Control: The leading error in the evolution of the coarse-grained field is $o(\ell^{-1})$, with no Navier–Stokes (i.e., diffusive) correction surviving at this order.
• Local-GGE/LTE Independence: The derivation and the resulting hydrodynamic equations are equally valid for "atypical," non-locally-thermal microstates, provided coarse local fluctuations are small.

The outcome is a mathematically explicit coarse-grained hydrodynamics whose Euler-scale description applies with high precision at the single-sample level, admits no intrinsic diffusion, and cleanly differentiates between diffusive effects due to initial-state variability (convection) and those that would arise from intrinsic entropy production (which are absent) (Hübner, 23 Jul 2025).