Hydrodynamics and the Fluctuation Theorem

The fluctuation theorem is a pivotal result of statistical physics. It quantifies the probability of observing fluctuations which are in violation of the second law of thermodynamics. More specifically, it quantifies the ratio of the probabilities of observing entropy-producing and entropy-consuming fluctuations measured over a finite volume and time span in terms of the rate of entropy production in the system, the measurement volume, and time. We study the fluctuation theorem in computer simulations of planar shear flow. The simulations are performed by employing the method of multiparticle collision dynamics, which captures both thermal fluctuations and hydrodynamic interactions. The main outcome of our analysis is that the fluctuation theorem is verified at any averaging time provided that the measurement volume exhibits a specific dependence on a hydrodynamic time scale.

Figure 1

Left: Thermal energy measured in setups 2–5 with ($t<5000$) and without ($t>5000$) a thermostat. Right: Viscosity measured through the rate of temperature increase of the fluid (boxes) compared to the exact analytical results [13] (solid lines).

Figure 2

Top left: Measured probability distributions $P({\sigma }_{xy})$ for setup 6 for one (green), two (red), four (blue), and six (orange) layers over the shortest averaging time $\tau =\delta t$ are not zero-centered and not symmetric. They become increasingly peaked for increasing measurement volumes, as expected. Top right: The corresponding probability functions $\mathcal{P}(\tau ,V)$ normalized by the measurement time $\tau =\delta t$ and the volume of a single layer $L^{2}a/2$ are constant and correctly reflect the number of measurement layers. Bottom: The resulting scaling of the observation volume $V(\tau )$ defined in Eq. (5) with the number of measurement planes $n_l$ is as predicted by the fluctuation theorem for all the simulation setups. Different symbols denote different setups.

Figure 3

The observation volume $V(\tau )$ as a function of the averaging time $\tau$ (top) and the normalized observation volume $V(\tau )/L^3$ as a function of the normalized averaging time $\tau /{\tau }_f$ (bottom) for different setups, denoted by different symbols. The solid lines indicate the total system volumes $L^3$. The dashed lines indicate the single-, two-, and four-layer volumes $L^2{n}_{l}a/2$. Shown are measurements of the viscous stress tensor over one plane (left), two planes (middle), and four planes (right).