Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy Production

Abstract

Many complex fluids can be described by continuum hydrodynamic field equations, to which noise must be added in order to capture thermal fluctuations. In almost all cases, the resulting coarse-grained stochastic partial differential equations carry a short-scale cutoff, which is also reflected in numerical discretisation schemes. We draw together our recent findings concerning the construction of such schemes and the interpretation of their continuum limits, focusing, for simplicity, on models with a purely diffusive scalar field, such as 'Model B' which describes phase separation in binary fluid mixtures. We address the requirement that the steady-state entropy production rate (EPR) must vanish for any stochastic hydrodynamic model in a thermal equilibrium. Only if this is achieved can the given discretisation scheme be relied upon to correctly calculate the nonvanishing EPR for 'active field theories' in which new terms are deliberately added to the fluctuating hydrodynamic equations that break detailed balance. To compute the correct probabilities of forward and time-reversed paths (whose ratio determines the EPR), we must make a careful treatment of so-called 'spurious drift' and other closely related terms that depend on the discretisation scheme. We show that such subtleties can arise not only in the temporal discretisation (as is well documented for stochastic ODEs with multiplicative noise) but also from spatial discretisation, even when noise is additive, as most active field theories assume. We then review how such noise can become multiplicative via off-diagonal couplings to additional fields that thermodynamically encode the underlying chemical processes responsible for activity. In this case, the spurious drift terms need careful accounting, not just to evaluate correctly the EPR but also to numerically implement the Langevin dynamics itself.

Keywords: active field theories; active matter; entropy production; stochastic thermodynamics.

Figure 1

The backward trajectory $x^{\mathrm{R}}\left(t\right)$ (right) is obtained by reflecting the forward trajectory $x\left(t\right)$ (left) around the vertical line $t=(t_N+t_0)/2$. The discretisation parameter a for the forward trajectory (left) becomes $1-a$ for the backward trajectory (right).

Figure 2

Adapted from [9]. (Left) Density map of a fluctuating phase-separated droplet in two-dimensional AMB. (Center) Local contribution to the informatic entropy production $\sigma \left(\mathbf{r}\right)=-{lim}_{t\to \infty }\frac{1}{Tt}{\int }_0^t\langle {\mu }_{\mathrm{A}}\dot{\varphi }\rangle (\mathbf{r},s)\mathrm{d}s$ showing a strong contribution at the interfaces. (Right) Density and entropy production for a 1D system comprising a single domain wall for various temperatures $T\ll a_2^2/4a_4$. The entropy production is strongly inhomogeneous, attaining a finite value as $T\to 0$ at the interface between dense and dilute regions and converging to zero in the bulk in this limit. Values of the parameters used are: $a_2=-0.125$, $a_4=0.125$, $\kappa =8$, $\lambda =2$, $\Delta x=1$, and $\Delta t=0.01$.

Figure 3

Schematic representation of an active system (blue) put in contact with reservoirs of chemical fuel (red) and product (green), which set a constant, homogeneous chemical potential difference $\Delta \mu$ in the active system. Within our framework, $\Delta \mu$ embodies the driving parameter which controls the nonequilibrium terms in the dynamics Equations (85) and (87) for the active density field $\varphi$ and the rate of fuel consumption $\dot{n}$. The active system and the chemical reservoirs are surrounded by the thermostat (yellow), which maintains a fixed temperature T. The fluctuations of $\varphi$ and n lead to the dissipation of heat $\mathcal{Q}$ into the thermostat, which quantifies the energetic cost to maintain the whole system away from equilibrium. Note that the physical separation of the reservoirs from the active system, as illustrated, is conceptually helpful but not necessary: in practice, the fuel, active particles and products can all share the same physical domain. Adapted from [12].

Figure 4

Adapted from [12]. Comparison of the heat production rate and IEPR for AMB. (a,b) The average profile of density $\langle \varphi \left(x\right)\rangle$ shows a separation between dilute ($\langle \varphi \left(x\right)\rangle <0$) and dense ($\langle \varphi \left(x\right)\rangle >0$) phases. The corresponding profiles of heat rate $\dot{q}\left(x\right)$ and the local IEPR $\sigma \left(x\right)$, given, respectively, as: $\dot{\mathcal{Q}}-\gamma V\Delta {\mu }^2={\int }_V\theta \mathrm{d}x$ and $\dot{\mathcal{S}}={\int }_V\sigma \mathrm{d}x$, are flat in bulk regions and vary rapidly across the interface. (c) The non-trivial contribution to heat rate $\dot{\mathcal{Q}}-\gamma V\Delta {\mu }^2$ reaches a finite value at $T=0$, whereas the IEPR measure $T\dot{\mathcal{S}}$ vanishes. (d) $\dot{\mathcal{Q}}-\gamma V\Delta {\mu }^2$ and $T\dot{\mathcal{S}}$, respectively, increase and decrease with the driving parameter $\Delta \mu$, and both scale as $\Delta {\mu }^2$. Parameters used are: $\Gamma =1$, $-a_2=a_4=0.25$, $\kappa =4$, $\overline{\varphi }=0$, $V=128$, $\Delta x=1$, $\Delta t=0.01$, (a,b) $\{\Delta \mu ,T\}=\{2,{10}^{-2}\}$, (c) $\Delta \mu =1$, (d) $T={10}^{-3}$.