Landauer's blow-torch effect in systems with entropic potential

We consider local heating of a part of a two-dimensional bilobal enclosure of a varying cross section confining a system of overdamped Brownian particles. Since varying cross section in higher dimension results in an entropic potential in lower dimension, local heating alters the relative stability of the entropic states. We show that this blow-torch effect modifies the entropic potential in a significant way so that the resultant effective entropic potential carries both the features of variation of width of the confinement and variation of temperature along the direction of transport. The reduced probability distribution along the direction of transport calculated by full numerical simulations in two dimensions agrees well with our analytical findings. The extent of population transfer in the steady state quantified in terms of the integrated probability of residence of the particles in either of the two lobes exhibits interesting variation with the mean position of the heated region. Our study reveals that heating around two particular zones of a given lobe maximizes population transfer to the other.

Figure 1

(a) The bilobal system with its geometrical parameters. (b) Schematic diagram of the system subjected to local heating. The shaded region corresponds to the heated zone. $T_0$ is the temperature of the low temperature bath and $T_1$ is the temperature of the high temperature bath. $x_l$ and $x_r$ stand for the position of the left and right boundaries of the heated zone, respectively, and $x_m$ denotes their mean position.

Figure 2

Effective potential of the system subjected to local heating. The system is in contact with the high temperature bath (temperature $T_1)$ within the $x$ range between $x_l$ and $x_r$ and with the low temperature bath (temperature $T_0)$ elsewhere.

Figure 3

Probability distribution curve for the system with four different heated zones in the left lobe. The scaled diffusion coefficients of the low temperature and high temperature regions are $D_0=0.1$ and $D_1=2.0$, respectively.

Figure 4

Probability distribution curve for the system with (a) $D_0=0.1$ and $D_1=0.6$ and (b) $D_0=0.1$ and $D_1=0.2$.

Figure 5

Checking ergodicity: reduced probability density curves calculated from the data of an ensemble of trajectories at the steady state and a single trajectory taking into account its longitudinal space variables during its evolution for the parameter set $D_0=0.1$ and $D_1=2.0$; $x_l=-0.4$ and $x_r=-0.25$.

Figure 6

$P_{\mathrm{left}}\left(t\right)$ and $P_{\mathrm{right}}\left(t\right)$ plotted against time and attainment of the steady state value for the integrated probabilities. The lines with hexagonal and circular symbols correspond to the plot of $P_{\mathrm{left}}\left(t\right)$ and $P_{\mathrm{right}}\left(t\right)$ for the system with a single heat bath with diffusion coefficient $D_0=0.1$. The other three pairs of curves correspond to the case when the system is heated between three different $x$ ranges of same width and for all these cases $D_0=0.1$ and $D_1=2.0$. The curves which saturate below $0.5$ are the integrated probabilities for the left lobe and those saturate above $0.5$ stand for the integrated probabilities for the right lobe.

Figure 7

$P_{\mathrm{right}}^{\mathrm{st}}$ plotted against $x_m$ for four different values of $D_1$. $D_0$ has been kept fixed at $0.1$ and the width $|x_l|-|x_r|$ has value equal to $0.15$ for all four cases.

Figure 8

MFPT plotted against $x_m$ for four different values of $D_1$. $D_0$ has been kept fixed at $0.1$ and the width $|x_l|-|x_r|$ has a value equal to $0.15$ for all four cases.

Figure 9

$P_{\mathrm{right}}^{\mathrm{st}}$ vs $x_m$ plot obtained from the reduced description of the dynamics for $D_0=0.1$ and $D=2.0$.