Coarse-Grained Thermodynamic Quantities

• Coarse-Grained Thermodynamic Quantities are observables that reduce complex systems by focusing on slow, essential variables.
• The approach constructs effective dynamics using conditional expectations, utilizing information-theoretic metrics and functional inequalities for error control.
• Numerical examples in molecular dynamics validate the method by accurately reproducing key thermodynamic and kinetic properties.

Coarse-grained thermodynamic quantities are statistical and dynamical observables associated with reduced descriptions of high-dimensional systems in which only a subset of variables (the “coarse-grained” or “slow” variables) are tracked explicitly, while the rest are averaged out or relegated to an effective stochastic influence. Coarse-graining aims to construct models that are computationally tractable yet retain predictive accuracy for thermodynamic and dynamical quantities of interest. This is accomplished through the definition of effective dynamics, quantification of errors with respect to the original system, and the use of information-theoretic measures to control the fidelity of the reduced model.

1. Quantifying Distances Between Probability Measures

A core aspect of analyzing coarse-grained thermodynamic quantities is comparing the probability distributions (measures) associated with the fine-grained (full) system and its coarse-grained representations. Several mathematical tools are used to characterize these differences:

• Total Variation (TV) Norm: For two measures $\nu$ and $\eta$ with densities $\psi_\nu$ and $\psi_\eta$ with respect to Lebesgue measure,

$$\|\nu-\eta\|_{TV} = \int |\psi_\nu(q) - \psi_\eta(q)|\, dq.$$

• Relative Entropy (Kullback–Leibler Divergence):

$$H(\nu|\eta) = \int \ln \left( \frac{d\nu}{d\eta} \right) d\nu.$$

The Csiszár–Kullback inequality relates relative entropy to total variation:

$\|\nu - \eta\|_{TV} \leq \sqrt{2 H(\nu|\eta)}.$

• Wasserstein Distance (Quadratic Cost): On a Riemannian manifold $\Sigma$, with geodesic distance $d_\Sigma(x, y)$,

$$W(\nu, \eta) = \sqrt{ \inf_{\pi \in \Pi(\nu, \eta)} \int_{\Sigma \times \Sigma} d_\Sigma(x, y)^2\, \pi(dx, dy) },$$

where $\Pi(\nu, \eta)$ denotes the set of all couplings of $\nu$ and $\eta$.

These measures allow for rigorous, quantitative statements about the proximity of the reduced (coarse-grained) model distributions to those of the full system, and form the basis for error bounds in effective dynamics.

2. Functional Inequalities and Their Consequences

Functional inequalities provide key tools for assessing the convergence and error control of coarse-grained models:

• Logarithmic Sobolev Inequality (LSI): A probability measure $\eta$ on $\Sigma$ satisfies a logarithmic Sobolev inequality with constant $\rho>0$ if for every $\nu$ absolutely continuous with respect to $\eta$,

$H(\nu|\eta) \leq \frac{1}{2\rho} I(\nu|\eta),$

where the Fisher information is

$$I(\nu|\eta) = \int |\nabla \ln (d\nu/d\eta)|^2 d\nu.$$

• Talagrand Inequality: With the same constant $\rho$,

$$W(\nu, \eta) \leq \sqrt{\frac{2}{\rho} H(\nu|\eta)}.$$

Any measure satisfying an LSI with constant $\rho$ automatically satisfies the corresponding Talagrand inequality.

These inequalities ensure strong concentration properties and stability of measures under perturbation; they play a central role in deriving error estimates for reduced effective dynamics.

3. Effective Dynamics for Reaction Coordinates

To describe the coarse-grained evolution, one typically begins with the full (high-dimensional) overdamped Langevin dynamics:

$$dQ_t = -\nabla V(Q_t)\, dt + \sqrt{2\beta^{-1}}\, dW_t,$$

where $Q_t$ is the configuration, $V$ the potential, and $\beta$ the inverse temperature.

Defining a reaction coordinate $\xi(Q_t)$ as the observed or slow degree of freedom, application of Itô’s formula yields:

$$d\xi(Q_t) = (-\nabla V \cdot \nabla \xi + \beta^{-1} \Delta\xi)(Q_t)\, dt + \sqrt{2\beta^{-1} |\nabla\xi|^2(Q_t)}\, dB_t.$$

The dynamics for $\xi(Q_t)$ are not closed since the right-hand side depends on all microscopic details. Following Gyöngy’s construction, local-in-time closure is achieved by conditional expectation: \begin{align*} b(t, z) &= \mathbb{E}[ (-\nabla V \cdot \nabla \xi + \beta^{-1} \Delta \xi)(Q_t) | \xi(Q_t) = z ], \ \sigma^2(t, z) &= \mathbb{E}[ |\nabla \xi|^2(Q_t) | \xi(Q_t) = z ]. \end{align*} The effective (one-dimensional) SDE for the coarse-grained variable is then:

$$d\hat{z}_t = b(t, \hat{z}_t)\, dt + \sqrt{2\beta^{-1} \sigma(t, \hat{z}_t)}\, dB_t,$$

which reproduces the time marginals of $\xi(Q_t)$ exactly, but is not closed in terms of equilibrium properties alone.

A further approximation, more practical for simulation and analysis, replaces the time-dependent conditional averages with equilibrium (Boltzmann–Gibbs) conditional averages over level sets: \begin{align*} b(z) &= \int [ -\nabla V \cdot \nabla\xi + \beta^{-1} \Delta\xi ] d\mu_{\Sigma_z}, \ \sigma^2(z) &= \int |\nabla \xi|^2 d\mu_{\Sigma_z}, \end{align*} where $\mu_{\Sigma_z}$ is the full equilibrium measure conditioned on $\xi(q) = z$. The resulting autonomous effective dynamic is

$$d\hat{z}_t = b(\hat{z}_t)\, dt + \sqrt{2\beta^{-1} \sigma(\hat{z}_t)}\, dB_t.$$

4. Rigorous Error Estimates

Error between the marginal law of $\xi(Q_t)$ in the full system and that predicted by the effective dynamics can be bounded in relative entropy. Under technical conditions (smooth reaction coordinate with bounded gradient, uniform LSI on level-set conditioned measures, bounded local mean force variation, and controlled gradient variation), the key estimate reads:

$$E(t) \equiv H(\psi^\xi(t, \cdot) \mid \phi(t, \cdot)) \leq \frac{M^2}{4m^2} \left[\lambda^2 + \frac{m^2 \beta^2 \kappa^2}{\rho^2}\right] \left[ H(\psi(0, \cdot) \mid \mu) - H(\psi(t, \cdot) \mid \mu) \right],$$

with $m$ and $M$ bounding $|\nabla\xi|$ and model constants $\lambda, \kappa$ determined by the system. This bound implies uniform-in-time control over the error: as the full system relaxes in entropy, the coarse-grained error decays.

5. Illustrative and Numerical Examples

The theory is illustrated in two main settings:

• Two-Dimensional Specialization: For $Q_t = (X_t, Y_t)$ and $\xi(x, y) = x$, the marginal dynamics for $X_t$ simplifies, and the effective drift and diffusion coefficients can be computed explicitly as averages over $y$. In this ideal case, the marginal distribution of $X_t$ for the full and effective dynamics are identical.
• High-Dimensional Molecular Examples: The methodology is numerically validated with molecules such as a three-atom system in 2D and a butane molecule. Reaction coordinates orthogonal to stiff degrees of freedom (e.g., central angles or dihedral angles) yield effective dynamics that closely reproduce residence times and transition rates of the full system. Poorly chosen coordinates—even when capturing exponential temperature dependencies—lead to sizable pre-exponential factor discrepancies.

6. The Role of Functional Inequalities and Large Deviations

Functional inequalities, especially LSI and Talagrand, guarantee exponential convergence of measures and provide crucial constants for error analysis. The approach also connects directly to large deviation theory: in the examples, the computed residence times conform to Arrhenius-type laws,

$$\tau_{\mathrm{res}} \approx \tau^0_{\mathrm{res}} \exp(s\beta),$$

with $s$ identified as the free energy barrier between relevant regions in the reaction coordinate, and pre-exponential factors linked to the effective diffusion.

In summary, the coarse-graining framework for thermodynamic quantities developed in this work provides:

• Rigorous, explicit methods for constructing effective dynamics for reduced variables from high-dimensional stochastic systems;
• Quantitative error bounds for the approximation, measured using information-theoretic metrics and functional inequalities;
• Practical guidelines for selecting appropriate reaction coordinates to optimize coarse-grained model fidelity;
• Numerical evidence from molecular examples confirming the theoretical predictions.

This structure allows coarse-grained models to faithfully reproduce thermodynamic and kinetic quantities of complex atomistic systems, justifying their use for multiscale modeling and simulation in molecular dynamics (1008.3792).