Biased Diffusion

Modelling thermal effects is reasonably straightforward for the sinusoidal landscape. In this case, the Langevin equation (Eq. (1)) is most readily solved by transforming it into a Fokker-Planck equation for the probability density $\rho(\ensuremath{{\mathbf{r}}}\xspace ,t)$ of finding particles at position $\ensuremath{{\mathbf{r}}}\xspace$ at time $t$. If inertial effects are negligible, the Fokker-Planck equation for motion transverse to the fringes reduces to a Smoluchowski equation (1),

$$\partial_t \rho(y,t) + \partial_y S(y,t) = 0,$$ (42)

where the probability current is

$$S(y,t) = \xi^{-1} \left( F(r) - k_B T \partial_y \right) \rho (y,t) \, .$$ (43)

In this equation, $F(r)$ is the total force on the particle, including the driving force and the force due to the potential landscape, $T$ is the temperature, and $k_B$ is Boltzmann's constant.

Figure 3: (a) Deflection as a function of orientation at finite temperatures, $\tau = 0.01$, 0.1, 0.2 and 0.3, assuming $\eta = 0.4$. The diagonal dashed line indicates the result with no landscape. (b) Dependence of the travel direction on particle size $a$ for $\eta _0 = 0.4$, $\tan \theta = 0.441$ and the same set of temperatures. Raising the temperature weakens the size dependence of $\psi (a)$, and thus reduces the selectivity.

Following Ref. (1), Eq. (43) can be solved in the steady-state limit by taking $S(y,t) = S(y)$, independent of $t$. The resulting average drift velocity $\langle v_y \rangle$ for the sinusoidal potential of Eq. (30) is given by

$$\frac{\langle v_y \rangle}{v_0 \sin\theta} = 1 + \frac{2 \sin\theta}{\eta} \, \mathrm{Im}\left[S_1\left(\tau,\frac{\sin \theta}{\eta}\right)\right]$$ (44)

where, as before, $\eta = k_0 I_0 \tilde f(0,k_0 a) / F_0$, and we have introduced the normalized temperature $\tau = k_B T/V_0$. The function $S_1(\tau,x)$ is defined recursively in terms of a continued-fraction expansion,

$$S_n\left(\tau,x\right) = \frac{1/4}{\tau + i n \, x + S_{n+1}\left(\tau, x\right)},$$ (45)

which converges rapidly with increasing order $n$.

The average velocity in the $\hat x$ direction is unchanged from the zero-temperature case. The mean deflection angle $\psi$ is thus given by

$$\tan \psi = \tan \theta \left( 1 + \frac{2 \sin\theta}{\eta} \, \mathrm{Im}\left[S_1\left(\tau, \frac{\sin \theta}{\eta}\right)\right] \right) \, .$$ (46)

This is plotted in Fig. 3(a) as a function of the angle $\theta$ of the driving force, for a fixed value of the normalized potential $\eta(a)$, and for various values of the normalized temperature $\tau$. It can be seen that the effect of increasing temperature is to smooth out the transition between the locked-in and freely flowing states of motion. In the zero-temperature limit, the deflection angle is zero for all angles $\theta < \arcsin\eta$. For finite temperatures, the mean deflection angle is non-zero even in this ``locked-in" state: the particles have a finite probability per unit time of being driven over the inter-fringe barrier by thermal fluctuations, and thereby advancing in the $\hat y$ direction.

The benefits of operating in the deterministic regime, in which thermal forces are negligible, can be seen in Fig. 3(b), where the deflection angle is shown as a function of the particle size $a$, for a fixed orientation $\theta$, over a range of temperatures. Contrary to previous assertions that thermal effects can enhance size selectivity (5), the only effect of thermally-assisted hopping in this system is to diminish the sorting resolution.

In other words, achieving high-sensitivity sorting in practice will require that the thermal energy scale be small compared to the landscape's modulation. In a real-world implementation, increasing the depth of modulation $I_0$ of the physical landscape will often be more practical than decreasing the temperature. Retaining the same lock-in conditions then requires that the driving force $F_0$ to be increased proportionately. The practical limit on the achievable sorting efficiency will then be set by the maximum driving force or depth of modulation that can be obtained. For example, the limitation for an optically-implemented landscape is the available laser power.