Deep learning of thermodynamic laws from microscopic dynamics

TL;DR

Problem: can macroscopic thermodynamic laws be discovered from microscopic dynamics without imposing thermodynamic priors? Approach: generate adiabatic MD data of a 2D gas, train a Siamese CNN to predict temporal order of two microscopic states, and interpret the learned scalar as a canonical entropy. Key findings: the DNN-derived order respects the Lieb–Yngvason axioms; the entropy-like representation S̃ maps across system size via affine transformations to the van der Waals entropy, and the representation is extensive and additive; the method demonstrates data-driven emergence of macroscopic physics from microscopic dynamics. Significance: provides a data-driven route to macroscopic thermodynamics and suggests a framework for emergent laws beyond explicit microscopic equations.

Abstract

We numerically show that a deep neural network (DNN) can learn macroscopic thermodynamic laws purely from microscopic data. Using molecular dynamics simulations, we generate the data of snapshot images of gas particles undergoing adiabatic processes. We train a DNN to determine the temporal order of input image pairs. We observe that the trained network induces an order relation between states consistent with adiabatic accessibility, satisfying the axioms of thermodynamics. Furthermore, the internal representation learned by the DNN act as an entropy. These results suggest that machine learning can discover emergent physical laws that are valid at scales far larger than those of the underlying constituents -- opening a pathway to data-driven discovery of macroscopic physics.

Figures (9)

• Figure 1: (a) Schematic illustration of the MD simulation model. (b) An example of the time profile of the piston position $L_x$ for a single cycle. Five snapshots (an extremely short video) are taken at each of the times indicated by the down arrows. Snapshots in (c) and (d) are actual examples obtained from the MD simulation for the case of $N=2000$, $v_{\mathrm{push}} = v_{\mathrm{pull}} = 0.1$, and $\Delta L_{\mathrm{push}} = \Delta L_{\mathrm{pull}} = 50$. The time points at which the snapshots were taken are indicated in (b) by $X_0$ for (c) and $X_1$ for (d).
• Figure 2: (a) Time evolution of the energy $U$ obtained from the MD simulation for cases of $v_{\mathrm{push}} = v_{\mathrm{pull}} = v_{\mathrm{p}}$. (b) Energy $U$ at the end of each piston-stopping period [each of the flat regions in (a)], plotted against the period number. (c) Neural network output $\tilde{S}$ for the state at the end of each piston-stopping period. Results for $v_{\mathrm{p}} = 0.5$ and $v_{\mathrm{p}} = 0.1$ are plotted. The other parameters are $N=2000$ and $\Delta L_{\mathrm{push}} = \Delta L_{\mathrm{pull}} = 15$.
• Figure 3: Schematic illustration of the DNN model. The input is a pair of microscopic states $X_0$ and $X_1$, each of which is an image having $256 \times 256$ pixels and five channels. The five channels correspond to the snapshots composing an extremely short video. Each of the two subnetworks DNN-0 and DNN-1 outputs a representation $\tilde{S}$ of the input image. The two representations are fed forward to the sigmoid unit in the output layer of the whole network to compute the probability of $X_1$ being the later state. The output is the label of the predicted later state.
• Figure 4: Numerical demonstration of \ref{['A1:reflectivity']}. The representation $\tilde{S}$ of a state $X'$ macroscopically equivalent to $X$ is plotted against $\tilde{S}(X)$. The dashed line shows $\tilde{S}(X') = \tilde{S}(X)$.
• Figure 5: (a) Illustration of creating a compound state $(X,X')$. The images of two states, $X$ and $X'$, where the pistons are at $L_x=L/2$ are combined into a single image of $(X,X')$. (b) Illustration of creating a scaled state $\lambda X$. Either the bottom or right part (shaded) in the image of a state $X$ is masked. In the actual data, the masked region is filled with zero.