Derivation of Stokes’ Law from Kirkwood’s Formula and the Green–Kubo Formula via Large Deviation Theory

Abstract

We study the friction coefficient of a macroscopic sphere in a viscous fluid at low Reynolds number. First, Kirkwood’s formula for the friction coefficient is reviewed on the basis of the Hamiltonian description of particle systems. According to this formula, the friction coefficient is expressed in terms of the stress correlation on the surface of the macroscopic sphere. Then, with the aid of large deviation theory, we relate the surface stress correlation to the stress correlation in the bulk of the fluid, where the latter is characterized by the viscosity in the Green–Kubo formula. By combining Kirkwood’s formula and the Green–Kubo formula in large deviation theory, we derive Stokes’ law without explicitly employing the hydrodynamic equations.

Appendix: Fluctuating Hydrodynamics

We describe a viscous incompressible fluid at low Reynolds number by the fluctuating hydrodynamic equations in spherical coordinates. We denote by $\varvec{u}(\varvec{r},t)$, $p(\varvec{r},t)$, $\overleftrightarrow {\varvec{\sigma }}(\varvec{r},t)$, and $\overleftrightarrow {\varvec{s}}(\varvec{r},t)$ the velocity, pressure, stress tensor, and random stress tensor at position $\varvec{r}$ and time t, respectively. In this case, each component of the stress tensor is written as

$$\begin{aligned} \sigma ^{rr}&= - p + 2\eta \frac{\partial u^{r}}{\partial r} + s^{rr}, \end{aligned}$$

(117)

$$\begin{aligned} \sigma ^{\theta \theta }&= - p + 2\eta \left( \frac{1}{r}\frac{\partial u^{\theta }}{\partial \theta }+\frac{u^{r}}{r}\right) + s^{\theta \theta } , \end{aligned}$$

(118)

$$\begin{aligned} \sigma ^{\varphi \varphi }&= - p + 2\eta \left( \frac{1}{r\sin \theta }\frac{\partial u^{\varphi }}{\partial \varphi }+\frac{u^{r}}{r}+\frac{u^{\theta }}{r\tan \theta }\right) + s^{\varphi \varphi } , \end{aligned}$$

(119)

$$\begin{aligned} \sigma ^{r\theta }&= \eta \left( \frac{1}{r}\frac{\partial u^{r}}{\partial \theta } + \frac{\partial u^{\theta }}{\partial r}-\frac{u^{\theta }}{r}\right) + s^{r\theta }, \end{aligned}$$

(120)

$$\begin{aligned} \sigma ^{\theta \varphi }&= \eta \left( \frac{1}{r\sin \theta }\frac{\partial u^{\theta }}{\partial \varphi }+\frac{1}{r}\frac{\partial u^{\varphi }}{\partial \theta }-\frac{u^{\varphi }}{r\tan \theta }\right) + s^{\theta \varphi }, \end{aligned}$$

(121)

$$\begin{aligned} \sigma ^{\varphi r}&= \eta \left( \frac{1}{r\sin \theta }\frac{\partial u^{r}}{\partial \varphi }+\frac{\partial u^{\varphi }}{\partial r}-\frac{u^{\varphi }}{r}\right) + s^{\varphi r}. \end{aligned}$$

(122)

We assume that the temperature T and density $\rho$ of the fluid are constant and homogeneous. Because we are focusing on incompressible fluid, we assume that the bulk viscosity $\zeta$ is equal to zero and that $\nabla \cdot \varvec{u}=0$. This is written as

$$\begin{aligned} \frac{\partial u^{r}}{\partial r} + \frac{2u^{r}}{r} + \frac{1}{r}\frac{\partial u^{\theta }}{\partial \theta } + \frac{u^{\theta }}{r\tan \theta } + \frac{1}{r\sin \theta }\frac{\partial u^{\varphi }}{\partial \varphi } =0. \end{aligned}$$

(123)

Furthermore, the time evolution equation of $\varvec{u}(\varvec{r},t)$ is assumed to be given by

$$\begin{aligned} \rho \frac{\partial \varvec{u}}{\partial t} = \nabla \cdot \overleftrightarrow {\varvec{\sigma }}, \end{aligned}$$

(124)

because the Reynolds number is sufficiently low. Finally, the random stresses are assumed to be zero-mean Gaussian white noises with covariance

$$\begin{aligned}&\left\langle s^{\alpha \beta }(r,\varOmega ,t)s^{\alpha ' \beta '}(r',\varOmega ',t')\right\rangle \nonumber \\&\qquad = 2k_\mathrm{B}T\eta \Delta ^{\alpha \beta \alpha ' \beta '}\frac{\delta (r-r')\delta (\varOmega -\Omega ')}{r^2}\delta (t-t'), \end{aligned}$$

(125)

$$\begin{aligned} \Delta ^{\alpha \beta \alpha ' \beta '} = \delta ^{\alpha \alpha '}\delta ^{\beta \beta '} + \delta ^{\alpha \beta '}\delta ^{\beta \alpha '} - \frac{2}{3}\delta ^{\alpha \beta }\delta ^{\alpha ' \beta '}. \end{aligned}$$

(126)

where we have used $\zeta = 0$. Note that (125) leads to

$$\begin{aligned} s^{rr}+s^{\theta \theta }+s^{\varphi \varphi }=0. \end{aligned}$$

(127)

By recalling (117), (118), (119), and (123), we also have

$$\begin{aligned} \sigma ^{rr}+\sigma ^{\theta \theta }+\sigma ^{\varphi \varphi } + 3p=0. \end{aligned}$$

(128)

For any time-dependent quantity A(t), we denote its path during the time interval $[0,\tau ]$ by [A]. Then, using (125) and the Gaussian property of $s^{\alpha \beta }$, we obtain the probability density of $\{ [p],[\sigma ^{\alpha \beta }],[u^{\alpha }]\}$ in the form

$$\begin{aligned} \mathcal {P}\left( \{ [p],[\sigma ^{\alpha \beta }],[u^{\alpha }]\} \right) =&\; C_{0} \exp \left[ -\tau \mathcal {I}\left( \{ [p],[\sigma ^{\alpha \beta }],[u^{\alpha }]\} \right) \right] \nonumber \\&\times \prod _{t}\prod _{\varvec{r}}\delta \left( \rho \partial _{t} \varvec{u}-\nabla \cdot \overleftrightarrow {\varvec{\sigma }}\right) \nonumber \\&\times \prod _{t}\prod _{\varvec{r}} \delta \left( \nabla \cdot \varvec{u} \right) \delta \left( \sigma ^{rr}+ \sigma ^{\theta \theta }+ \sigma ^{\varphi \varphi }+3p\right) \end{aligned}$$

(129)

with

$$\begin{aligned} \mathcal {I} \left( \{ [p],[\sigma ^{\alpha \beta }],[u^{\alpha }]\} \right)&= \frac{1}{4k_\mathrm{B}T\eta \tau } \int _{0}^{\tau } dt \int _{\mathcal {R}}^{\infty } dr\; r^2\int d\varOmega \; s^{\alpha \beta }( \tilde{\Delta }^{-1})^{\alpha \beta \alpha ' \beta '}s^{\alpha '\beta '} , \end{aligned}$$

(130)

where $C_0$ is the normalization constant and $s^{\alpha \beta }$ in the right-hand side is related to $(p,\sigma ^{\alpha \beta },u^{\alpha })$ through (117)–(122). By applying the contraction principle to (129), we obtain the probability density of the surface stress fluctuations as

$$\begin{aligned} P(\bar{\sigma }_{*})&= \int _{\bar{\sigma }_{*}\text {:fix}} \mathcal {D}p \mathcal {D} \sigma ^{\alpha \beta } \mathcal {D} u^{\alpha }\; \mathcal {P}\left( \{ [p],[\sigma ^{\alpha \beta }],[u^{\alpha }]\} \right) \nonumber \\&= C_0' \exp \left[ -\tau I(\bar{\sigma }_{*}) \right] \end{aligned}$$

(131)

with

$$\begin{aligned} I(\bar{\sigma }_{*}) = \min _{\begin{array}{c} \{ [\sigma ^{\alpha \beta }], [u^{\alpha }]\} \\ \bar{\sigma }_{*}\text {:fix} \end{array}}\; \frac{1}{4k_\mathrm{B}T\eta \tau }\int _{0}^{\tau }dt \int _{\mathcal {R}}^{\infty }dr \int _{0}^{\pi } d\theta \int _{0}^{2\pi } d\varphi \; \mathcal {L}\left( \{ [\sigma ^{\alpha \beta }],[u^{\alpha }]\} \right) , \end{aligned}$$

(132)

and

$$\begin{aligned} \mathcal {L}\left( \{ [\sigma ^{\alpha \beta }],[u^{\alpha }]\} \right)&= r^2 \sin \theta \; \Big [ s^{{\alpha }{\beta }} (\tilde{\Delta }^{-1})^{{\alpha }{\beta }{\alpha }'{\beta }'} s^{{\alpha }'{\beta }'} \nonumber \\&\quad + \varvec{\lambda }_{1}\cdot \left( \rho \partial _{t}\varvec{u}-\nabla \cdot \overleftrightarrow {\varvec{\sigma }}\right) + \lambda _2 \left( \nabla \cdot \varvec{u}\right) \Big ] , \end{aligned}$$

(133)

where $C_0'$ is the normalization constant, $\varvec{\lambda }_{1}(\varvec{r},t)$ and $\lambda _{2}(\varvec{r},t)$ are the Lagrange multiplier fields, $p=-(\sigma ^{rr}+\sigma ^{\theta \theta }+\sigma ^{\varphi \varphi })/3$, and “$\bar{\sigma }_{*}\text {:fix}$” represents the condition given in

$$\begin{aligned} \bar{\sigma }_{*} = \frac{1}{4\pi \tau } \int _{0}^{\tau } dt \int d\varOmega \; \left[ \cos \theta \; \sigma ^{rr}(\mathcal {R},\varOmega ,t) - \sin \theta \; \sigma ^{r\theta }(\mathcal {R},\varOmega ,t)\right] . \end{aligned}$$

(134)

If a time-independent configuration with $\varvec{u}=\varvec{0}$ is the minimizer of (132), we obtain the same result as in (77)–(79).