“window potential” of a chosen functional form. The form of this potential is given by:

\[ W _ { i } ( s ) = \left\{ \begin{array} { l l } { \infty , \qquad s < s _ { i } ^ { \mathrm { m i n } } } & { } \\ { 0 , \qquad s _ { i } ^ { \mathrm { m i n } } < s < s _ { i } ^ { \mathrm { m a x } } } & { } \\ { \infty , \qquad s > s _ { i } ^ { \mathrm { m a x } } } & { } \end{array} \right. \qquad ( 7 ) \]

where \( s _ { i } ^ { \mathrm { m i n } } \) and \( s _ { i } ^ { \mathrm { m a x } } \) are the limits that define the range of s for the i-th window. Within each “window” of s, a probability distribution \( p _ { i } ( s ) \) is calculated in the simu-lation. The window potential width, \( \Delta s \equiv s _ { i } ^ { \mathrm { m a x } } - s _ { i } ^ { \mathrm { m i n } } \), is chosen to be sufficiently small that the variation in F does not exceed a few \( k _ { \mathrm { B } } T \). Adjacent windows overlap, and the SCH algorithm uses the \( p _ { i } ( s ) \) histograms to re-construct the unbiased distribution, \( \mathcal { P } ( s ) \). The details of the histogram reconstruction algorithm are given in Ref. 46. A description for an application to a physical system comparable to that studied here is presented in Ref. 32.

Polymer configurations were generated carrying out single-monomer moves using a combination of trans-lational displacements and crankshaft rotations. The trial moves were accepted with a probability \( \begin{array} { r l } { p _ { \mathrm { a c c } } } & { { } = } \end{array} \)\( \operatorname* { m i n } ( 1 , e ^ { - \Delta E / k _ { \mathrm { B } } T } ) \), where \( \Delta F \) is the energy difference be-tween the trial and current states. Initial polymer con-figurations were generated such that s was within the allowed range for a given window potential. Prior to data sampling, the system was equilibrated. As an il-lustration, for a \( N \) = 121 polymer chain, the system was equilibrated for typically \( \sim \mathrm { 1 0 ^ { 7 } } \) MC cycles, following which a production run of \( \sim 1 0 ^ { 8 } \) MC cycles was carried out. During each MC cycle a move for each monomer is attempted once, on average.

The windows are chosen to overlap with half of the adjacent window, such that \( s _ { i } ^ { \mathrm { m a x } } = s _ { i + 2 } ^ { \mathrm { m i n } } \). The window width was typically \( \Delta s \, = \, \sigma \). Thus, a calculation for \( N \)= 121, where the translocation coordinate spans a range of \( s \, \in \, [ 0 , 1 2 0 ] \), required separate simulations for 239 different window potentials. For each simulation, individual probability histograms were constructed using the binning technique with 10 bins per histogram.

The free energy functions obtained from the MC sim-ulations were used with the procedure summarized in Sec. III to study the translocation dynamics under the as-sumption that conformational quasi-equilibrium is main-tained during the process. The translocation probability \( \mathcal { W } ( s , t ; s _ { 0 } , 0 ) \) was determined by solving Eq. (1) for a cho-sen value of \( s _ { 0 } \). Typically, we used \( s _ { 0 } = L / ( 2 \sigma ) \), at which point the first monomer isj ust on the verge of exiting the pore into the cavity. The equation was solved using standard numerical methods with a “spatial” grid size of \( \Delta s \) = 0.01 and a time increment of \( \Delta t = 0 . 0 0 2 \mathcal { D } ^ { - 1 } \). The distribution \( P ( \tau ) \) was calculated using Eq. (3), where a standard five-point method was used to evaluate the derivatives in Eq. (2). The translocation probability, \( p _ { b } \), and mean first passage time, \( \langle \tau \rangle \), were calculated by nu-merical integration of Eqs. (4) and (6), respectively, using Simpson’s rule.

In the results presented below, quantities of length are measured in units of \( \sigma \), energy in units of \( k _ { \mathrm { B } } T \), force in units of \( k _ { \mathrm { B } } T / \sigma \) and time in units of \( \mathcal { D } ^ { - 1 } \).

# V. RESULTS

We consider first the case of spherical cavities in the absence of a driving force or adsorption potential, i.e. \( r \) = 1, \( f _ { \mathrm { d } } \, = \, 0 \) and \( \epsilon \) = 0. Figure 2 shows free energy functions for cavity volumes of V =150, 250 and 500, each for polymer lengths ranging from N=31 to N=141. As expected, the free energy cost of confining the polymer in the cavity increases as the confinement volume decreases. In addition, the curves all have positive curvature every-where except at near the upper and lower bounds. This indicates that free energy cost of inserting each monomer into the cavity increases as the number (and hence den-sity) of monomers inside increases. This feature is con-sistent with results from previous MC studies27,32,33 and theoretical studies16,26 that explicitly account for repul-sion between monomers, in contrast to the case for ideal polymers.15 Another noteworthy feature is the strong de-gree of overlap between the curves for different N and the same V . Thus, the free energy cost of transferring one monomer from the outside to the inside of the cavity de-pends approximately only on the density of monomers in the cavity. Deviations from this trend are evident where F dips abruptly near \( s = N - \) 1.

FIG. 2. Free energy functions for several different polymer lengths for a cavity aspect ratio of \( r \) = 1. Results for three different cavity volumes are shown.

To provide a quantitative analysis of the free en-ergy functions, we employ a standard scaling theory approach.23 We first recall that the standard form of the entropic free energy barrier for translocation of a polymer through a narrow pore in a flat wall of negligible thick-ness is \( F _ { 0 } / k T = ( 1 - \lambda ) \ln [ m ( N - m ) ] \), where m segments lie on one side of the pore, and \( N - m \) lie on the other