Problems
Problem 9.1
In Problems 9.1–9.4, we deal with the 1-D system (9.1). Re-define the state variable to reduce (9.1) to (9.4) in the inertial time scale, and show that its formal solution is
$$ v\left( t\right) =\int _{0}^{t}{e^{m^{-1}\gamma w_{D}\left( {t^{\prime }-t} \right) }m^{-1}\sigma w_{R}\theta \left( {t^{\prime }}\right) dt^{\prime }}. $$
Then calculate the mean square velocity and show that
$$\begin{aligned} \left\langle {v\left( t\right) v\left( t\right) }\right\rangle&=\frac{{ \sigma }^{2}}{m^{2}}\int _{0}^{t}{\int _{0}^{t}{e^{m^{-1}\gamma w_{D}\left( { t^{\prime }-t}\right) }w_{R}^{2}\left\langle {\theta \left( {t^{\prime }} \right) \theta \left( {t}^{\prime \prime }\right) }\right\rangle e^{m^{-1}\gamma w_{D}\left( {t}^{\prime \prime }{-t}\right) }}dt^{\prime }dt^{\prime \prime }} \nonumber \\&=\frac{{\sigma }^{2}}{m^{2}}\int _{0}^{t}{e^{m^{-1}\gamma w_{D}\left( { t^{\prime }-t}\right) }w_{R}e^{m^{-1}\gamma w_{D}\left( {t^{\prime }-t} \right) }w_{R}dt^{\prime }}, \nonumber \\&=\frac{1}{2}m^{-1}\sigma ^{2}w_{R}^{2}\gamma ^{-1}w_{D}^{-1}. \end{aligned}$$
(9.208)
This leads directly to (9.6), assuming (9.3).
Problem 9.2
Show that the drift velocity and the diffusivity of the process (9.12) are given by
$$\begin{aligned} \frac{{\left\langle {\varDelta r}\right\rangle }}{{\varDelta t}}&=\gamma ^{-1}w_{D}^{-1}F_{c}, \nonumber \\ \frac{{\left\langle {\varDelta r\varDelta r}\right\rangle }}{{2\varDelta t}}&=O\left( {\varDelta t}\right) +k_{B}T\gamma ^{-1}w_{D}^{-1}, \end{aligned}$$
(9.209)
leading to the Fokker–Planck equation (9.13).
Problem 9.3
Show that, from (9.4)
$$\begin{aligned} \frac{1}{2}m\frac{d}{{dt}}\left( {\frac{d}{{dt}}\left\langle {r^{2}} \right\rangle }\right) -m\left\langle {v^{2}}\right\rangle +\frac{1}{2} \gamma w_{D}\frac{d}{{dt}}\left\langle {r^{2}}\right\rangle =\sigma w_{R}\left\langle {\theta \left( t\right) r}\right\rangle . \end{aligned}$$
(9.210)
Define
$$ e=d\left\langle {r^{2}}\right\rangle /dt, $$
and show that
$$ \dot{e}+m^{-1}\gamma w_{D}e=2k_{B}Tm^{-1},\,\,\, e\left( 0\right) =0. $$
Show that this has the solution, for the assumed initial condition,
$$ e=\frac{d}{{dt}}\left\langle {r^{2}}\right\rangle =2k_{B}T\gamma ^{-1}w_{D}^{-1}\left[ {1-e^{-m^{-1}\gamma w_{D}t}}\right] . $$
Consequently, if \(\varDelta t\gg \tau _{I}=O\left( {m^{-1}\gamma w_{D}}\right) ,\) show that
$$ \frac{{\left\langle {\varDelta r\varDelta r}\right\rangle }}{{2\varDelta t}} =k_{B}T\gamma ^{-1}w_{D}^{-1}. $$
Problem 9.4
Define the velocity correlation as
$$\begin{aligned} R\left( \tau \right) =\lim _{t\rightarrow \infty }\left\langle {v\left( {t+\tau } \right) v\left( t\right) }\right\rangle , \end{aligned}$$
(9.211)
where the limit refers to large time compared to the inertial time scale, but yet small compared to the relaxation time scale. From the solution (9.5), show that
$$\begin{aligned} R\left( \tau \right)&=\lim _{t\rightarrow \infty }\int _{0}^{t+\tau }{ dt^{\prime }}\int _{0}^{t}e{^{m^{-1}\gamma w_{D}\left( {t^{\prime }-t-\tau } \right) }\left( {m^{-1}\sigma w_{R}}\right) ^{2}} \nonumber \\&{\left\langle {\theta \left( {t^{\prime }}\right) \theta \left( { t^{\prime \prime }}\right) }\right\rangle e^{m^{-1}\gamma w_{D}\left( { t^{\prime \prime }-t}\right) }dt}^{\prime \prime } \end{aligned}$$
(9.212)
$$\begin{aligned}&=e^{-m^{-1}\gamma w_{D}\tau }\lim _{t\rightarrow \infty }\left\langle {v\left( t\right) v\left( t\right) }\right\rangle =e^{-m^{-1}\gamma w_{D}\tau }R\left( 0\right) \nonumber \\&=k_{B}Tm^{-1}e^{-m^{-1}\gamma w_{D}\tau }. \end{aligned}$$
(9.213)
That is, the velocity correlation decays after an inertial time scale, after which the velocity is independent to its previous state.
Next, the diffusivity can also be defined as
$$ D=\lim _{t\rightarrow \infty }\frac{1}{2}\left\langle {v\left( t\right) r\left( t\right) +r\left( t\right) v\left( t\right) }\right\rangle . $$
Show that this leads to
$$\begin{aligned} D&=\lim _{t\rightarrow \infty }\int _{0}^{t}{\left\langle {v\left( t\right) v\left( {t+\tau }\right) }\right\rangle d\tau }=\int _{0}^{\infty }{R\left( \tau \right) d\tau } \\&=k_{B}T\gamma ^{-1}w_{D}^{-1}, \end{aligned}$$
consistent with previous results.
Problem 9.5
Show, with the aid of the Langevin equation (9.14), that the drift and the diffusion of the process are given by
$$\begin{aligned} \left\langle {\frac{{\varDelta \mathbf {v}_{i}}}{{\varDelta t}}}\right\rangle =-m^{-1}\sum \limits _{j}{\left( {\mathbf {F}_{ij}^{C}+\mathbf {F}_{ij}^{D}} \right) =}-m^{-1}\sum \limits _{j}{\left( {\mathbf {F}_{ij}^{C}-\gamma w_{ij}^{D}\mathbf {e}_{ij}\mathbf {e}_{ij}\cdot \mathbf {v}_{ij}}\right) }, \end{aligned}$$
(9.214)
and
$$\begin{aligned} \left\langle {\frac{{\varDelta \mathbf {v}_{\alpha }\varDelta \mathbf {v}_{\beta }}}{ {2\varDelta t}}}\right\rangle =\frac{\gamma k_{B}T}{m^{2}}\left[ {\delta _{\alpha \beta }\sum \limits _{k}{w_{\alpha k}^{D}\mathbf {e}_{\alpha k}\mathbf { e}_{\beta k}}-w_{\alpha \beta }^{D}\mathbf {e}_{\alpha \beta }\mathbf {e} _{\alpha \beta }}\right] . \end{aligned}$$
(9.215)
Thus show that the Fokker–Planck equation is given by (9.32).
Problem 9.6
Show that the equilibrium distribution of the associate system to (9.14) is
$$\begin{aligned} f_{eq}\left( {\chi , t}\right)&=\frac{1}{Z}\exp \left[ {-\frac{1}{{k_{B}T}} \left( {\sum \limits _{i}{\frac{{\mathbf {p}_{i}\cdot \mathbf {p}_{i}}}{{2m}}}+ \frac{1}{2}\sum \limits _{i, j}{\varphi \left( {r_{ij}}\right) }}\right) } \right] \\&=\frac{1}{Z}\exp \left[ {-\frac{\mathcal {H}}{{k_{B}T}}}\right] , \nonumber \end{aligned}$$
(9.216)
where Z is a normalizing constant, and
$$\begin{aligned} \mathcal {H}{=}\sum \limits _{i}{\frac{{\mathbf {p}_{i}\cdot \mathbf {p}_{i}}}{{2m }}}+\frac{1}{2}\sum \limits _{i, j}{\varphi \left( {r_{ij}}\right) } \end{aligned}$$
(9.217)
is the Hamiltonian of the associate system to (9.14). Show that,
$$ \sum \limits _{i}{\mathbf {v}_{i}\cdot \frac{{\partial f_{eq}}}{{\partial \mathbf {r}_{i}}}}=\frac{1}{{k_{B}T}}\sum \limits _{i,j}{\mathbf {v}_{i}\cdot \mathbf {F}_{ij}^{C}f_{eq}},\,\,\,\sum \limits _{i, j}{\mathbf {F}_{ij}^{C}\cdot \frac{{\partial f_{eq}}}{{\partial \mathbf {p}_{i}}}}=-\frac{1}{{k_{B}T}} \sum \limits _{i, j}{\mathbf {F}_{ij}^{C}\cdot \mathbf {v}_{i}f_{eq}}, $$
$$ \gamma \sum \limits _{i,j}{w_{ij}^{D}\mathbf {e}_{ij}\frac{\partial }{{\partial \mathbf {p}_{i}}}\cdot \left( \mathbf {e}{_{ij}\cdot \mathbf {v}_{ij}f_{eq}} \right) }=\gamma \sum \limits _{i, j}{w_{ij}^{D}\left( {-\frac{1}{{k_{B}T}} \mathbf {e}_{ij}\cdot \mathbf {v}_{i}\mathbf {e}_{ij}\cdot \mathbf {v}_{ij}+ \frac{1}{m}}\right) f_{eq}}, $$
$$\begin{aligned}&\gamma k_{B}T\sum \limits _{i,j}{w_{ij}^{D}\mathbf {e}_{ij}\cdot \frac{\partial }{{\partial \mathbf {p}_{i}}}\cdot \left( \mathbf {e}{_{ij}\cdot \left( {\frac{ \partial }{{\partial \mathbf {p}_{i}}}-\frac{\partial }{{\partial \mathbf {p} _{j}}}}\right) f_{eq}}\right) } \\&=\gamma \sum \limits _{i, j}{w_{ij}^{D}\left( {-\frac{1}{m}+\frac{1}{{k_{B}T}} \mathbf {e}_{ij}\cdot \mathbf {v}_{i}\mathbf {e}_{ij}\mathbf {v}_{ij}}\right) ,} \end{aligned}$$
and conclude that \(f_{eq}\) is also a stationary solution (i.e. solution that is independent of time) of the Fokker–Planck equation (9.32).
Problem 9.7
Show the equivalence between (9.61), (9.63), (9.65) and (9.66), by expressing \(f_{2}\left( {\mathbf {r}+\lambda \mathbf {R}, \mathbf {r}-(1-\lambda )\mathbf {R},\mathbf {{v^{\prime }}},\mathbf {{v^{\prime \prime }}}, t}\right) \) as a function of \(\left( {\mathbf {r}-\varepsilon \mathbf {R}}\right) \), where \(\varepsilon =1-\lambda \), then integrating after taking a Taylor’s series in \(\varepsilon \).
Problem 9.8
Show that
$$\begin{aligned} \mathbf {q}_{C}\left( {\mathbf {r}, t}\right)= & {} \frac{1}{4}\int {d\mathbf {R}} \int {d\mathbf {v}}\int {d\mathbf {{v^{\prime }}}}\mathbf {F}^{C}\left( \mathbf { R}\right) \cdot \left( {\mathbf {v}+\mathbf {{v^{\prime }}}}\right) \mathbf {R} \left\{ {1-\frac{1}{2}\mathbf {R}}\cdot {\nabla +\cdots }\right\} \nonumber \\&\,.f_{2}\left( {\mathbf {r}+\mathbf {R},\mathbf {r},\mathbf {v},\mathbf {{v^{\prime }}}, t}\right) . \end{aligned}$$
(9.218)
Problem 9.9
The stress contributed from the damping forces is, from (9.63),
$$\begin{aligned} \mathbf {S}_{D}\left( {\mathbf {r}, t}\right)= & {} -\frac{1}{2}\int {d\mathbf {R}} \int {d\mathbf {v}}\int {d\mathbf {{v^{\prime }}}}\gamma w^{D}\left( R\right) {{\hat{\mathbf{R}}\hat{\mathbf{R}}}}\cdot \left( {\mathbf {v}-\mathbf {{v^{\prime }}}} \right) \mathbf {R}\left\{ {1+O\left( R\right) }\right\} \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .f_{2}\left( {\mathbf {r}+\mathbf {R}, \mathbf {r},\mathbf {v},\mathbf {{v^{\prime }}}, t}\right) . \end{aligned}$$
Show that, for an homogeneously shear flow, \(\mathbf {v}-\mathbf {{v^{\prime }} }=\mathbf {LR},\) where \(\mathbf {L}\) is the velocity gradient, together with Groot and Warren’s approximation, \(f_{2}=n^{2}\left( {1+O\left( R\right) } \right) ,\) the stress contributed by the damping forces is
$$\begin{aligned} S_{D,\alpha \beta }\left( {\mathbf {r}, t}\right)= & {} -\frac{{\gamma n^{2}}}{2} \left\langle {\hat{R}_{\alpha }\hat{R}_{\beta }\hat{R}_{i}\hat{R} _{j}L_{ij}\int {R^{2}w^{D}\left( R\right) 4\pi R^{2}dR}}\right\rangle \\= & {} -\frac{{2\pi \gamma n^{2}}}{{15}}\left( {\delta _{\alpha \beta }\delta _{ij}+\delta _{\alpha i}\delta _{\beta j}+\delta _{\alpha j}\delta _{\beta i} }\right) L_{ij}\int {R^{4}w^{D}\left( R\right) dR}. \end{aligned}$$
For the standard weighting function (9.88) adopted in DPD, show that
$$\begin{aligned} S_{D,\alpha \beta }\left( {\mathbf {r}, t}\right)= & {} -\frac{{2\pi \gamma n^{2}} }{{15}}\left( {L_{\alpha \beta }+L_{\beta \alpha }+L_{ii}\delta _{\alpha \beta }}\right) \int _{0}^{r_{c}}{R^{4}\left( {1-R/r_{c}}\right) ^{2}dR} \nonumber \\= & {} \frac{{2\pi \gamma n^{2}r_{c}^{5}}}{{1575}}\left( {L_{\alpha \beta }+L_{\beta \alpha }+L_{ii}\delta _{\alpha \beta }}\right) , \end{aligned}$$
(9.219)
and consequently the damping-contributed viscosities are given by
$$\begin{aligned} \eta _{D}=\frac{{2\pi \gamma n^{2}r_{c}^{5}}}{{1575}},\,\,\,\,\zeta _{D}= \frac{5}{3}\eta _{D}=\frac{{2\pi \gamma n^{2}r_{c}^{5}}}{{945}}. \end{aligned}$$
(9.220)
Problem 9.10
For the standard DPD weighting function (9.88), show that the viscosities and diffusivity are given as shown in (9.90).
Problem 9.11
For the modified DPD weighting function (9.125), derive the viscosities and diffusivity of the DPD fluid.
Problem 9.12
Write a Matlab routine to compute the radial distribution function (RDF) for DPD particles and integrate it into the DPD main program in Sect. 9.8. Study the effects of \(n, r_c\) and \(k_BT\) on the particle’s exclusion zone for the case of using modified weighting function.
Problem 9.13
Write a Matlab routine to compute the mean square displacements (MSDs) of DPD particles and integrate it into the DPD main program in Sect. 9.8. Study the effects of \(n, r_c\) and \(k_BT\) on the self-diffusion coefficient of the DPD particles for a given noise amplitude \(\sigma =3\).
