Abstract
A computationally challenging and open problem is how to efficiently generate equilibrated samples of conditioned walks. We present here a general stochastic approach that allows one to produce these samples with their correct statistical weight and without rejections. The method is illustrated for a jump process conditioned to evolve within a cylindrical channel and forced to reach one of its ends. We obtain analytically the exact probability density function of the jumps and offer a direct method for gathering equilibrated samples of a random walk conditioned to stay in a channel with suitable boundary conditions. Unbiased walks of arbitrary length can thus be generated with linear computational complexity—even when the channel width is much smaller than the typical bond length of the unconditioned walk. By profiling the metric properties of the generated walks for various bond lengths we characterize the crossover between weak and strong confinement regimes with great detail.
Notes
For \(m\ne n\) one can use Gradshteyn–Ryzhik, 6.521, to check the orthogonality condition; for \(m=n\) one can integrate twice by parts using \(d(x\,J_1(x))/dx = x\,J_0(x)\).
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We thank Angelo Rosa for useful comments and suggestions.
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Appendices
Appendix
Generating the Conditioned Ensemble by Rejection
Directly sampling a free random walk with exponentially distributed jumps according to Eq. (3) and then rejecting the trajectories that do not satisfy the condition of staying within a cylinder is very inefficient, especially for long chains and large jumps. Figure 6 shows the efficiency of this brute-force method for different number of jumps and their length. The efficiency falls off exponentially fast to zero as expected. Figure 7 presents a summary of the results concerning the runtime to sequentially produce one hundred trajectories using the rejection method and the method described in this work. In the case of small jump length the rejection method is impractical if one wants to explore the limit of long chains (see Fig. 7 right panel). The other limit is shown in the left panel where the jump length is comparable with the confinement. In this case sampling few hundreds of jumps is impractical. Using the rejection method, the runtime grows exponetially with N in agreement with what shown in Fig. 6. The method we described in this work instead presents a runtime that practically grows linearly with N.
Conditioned Random Walks
Let us consider a walker performing independent and identically distributed jumps (of finite size). We denote by \(p(\mathbf {r}'|\mathbf {r})\) the probability density that the walker performs a jump from \(\mathbf {r}\) to \(\mathbf {r}'\). Let us first introduce a terminal domain \(\mathcal{T}\) where the process is absorbed. The joint probability density function of the trajectory \((\mathbf {r}_1, \dots , \mathbf {r}_N)\) starting from \(\mathbf {r}_0\) and with stopping time N reads
Let us now consider a confinement domain \(\mathcal{C}\) disjoint from \(\mathcal{T}\). We want to find the subset of trajectories generated by P that stay inside \(\mathcal{C}\) and terminate in \(\mathcal{T}\). The new joint pdf of the trajectory \((\mathbf {r}_1, \dots , \mathbf {r}_N)\) under the confinement condition can thus be written as
where \(\mathbb {I}_\mathcal{C}(\mathbf {r})\) is the characteristic function of \(\mathcal{C}\), i.e. it is one if \(\mathbf {r}\in \mathcal{C}\) and zero otherwise. The normalisation factor
is the probability that a trajectory sampled from P does not leave \(\mathcal{C}\) and terminates in \(\mathcal{T}\). The new process Q is still Markovian with transition probability
where \(Z(\mathbf {r})\) obeys the backward equation
The equations above are Eq. (1) and Eq. (2) of the main text.
Randomly Sampled Brownian Motion as a Model of a Freely Jointed Chain
In this Appendix we derive the expression for the transition probability (3) that describes the jump process in free space. Consider a Brownian walker starting from \(\mathbf {r}_0\) at time \(t=0\) and diffusing in a three-dimensional space with a diffusion coefficient D; its position \(\mathbf {r}\) after a time t is given by the probability density function (pdf):
We would like to introduce a characteristic length of the jump process. For this purpose, we sample the Brownian motion at exponentially distributed times with mean \(\tau \), thus defining a discrete random walk (see Fig. 1). By definition, this new process is still Markovian and its successive positions \(\mathbf {r}_i\) are given by the transition probability:
where \(\tau \) characterises the coarse-graining. Passing to Fourier space we obtain
where \(m = 1/\sqrt{\tau D}\), which finally leads to Eq. (3). The jumps \((\mathbf {r}_{i+1}-\mathbf {r}_i)\) are identically distributed with mean length \(\ell _f\doteq \langle ||\mathbf {r}_{i+1}-\mathbf {r}_i||\rangle = 2/m\).
Confining the Jump Process Inside a Cylindrical Channel
In this paragraph we show explicitely how to solve Eq. (2) when the domain \(\mathcal{C}\) is an infinite cylinder and the terminal domain has been pushed to \(z \rightarrow \infty \).
The particular choice (3) for the transition probability \(p(\mathbf {r}'\,|\,\mathbf {r})\) in Eq. (2) presents the advantage of satisfying
where \(\delta \) is a Dirac delta function. Thus applying \(\varvec{\nabla }^2 - m^2\) to Eq. (2) gives
that leads to the system of equations
Given that in cylindrical coordinates the problem is separable in z and \((\theta ,\rho )\), the general solution of Eq. (2) reads
where A and B are real constants, \(\lambda \) is a parameter that belongs to (0, m) and satisfies Eq. (6), and \(c_{\lambda }=\sqrt{m^2-\lambda ^2}\). Finally, replacing this solution in Eq. (1) yields the transition probability for the process Eq. (5) conditioned to stay inside the cylinder.
Asymptotic Behavior of \(\lambda R\)
The left panel of Fig. 8 displays the behavior of \(\lambda R\) as a function of \(\ell _f/R\) obtained by solving numerically Eq. (6). In the (diffusive) limit \(\ell _f/R \ll 1\), \(\lambda \) behaves as
where \(z_{0,1}\simeq 2.40483\) is the first zero of the Bessel function \(J_0\). In the limit \(\ell _f/R \gg 1\), Eq. (6) gives the asymptotic behavior
using that, in this limit, max\(\,(\lambda R,\sqrt{m^2-\lambda ^2} R) \le mR \ll 1\) and \(c_\lambda R \sim \exp [-2/(mR)^2]\). Note that \(\lambda \,R\in (0, z_{0,1})\) and is a strictly decreasing function of \(\ell _f/R\).
Monte Carlo Simulation of Conditioned Trajectories
As the transition probability (5) is invariant under translation along the axis z, the distribution of the walker positions in the transverse direction reaches a stationary state:
that verifies, for all \(\rho <R\),
where
Starting from steady state (A.14) in the transverse direction of the channel, we then generate conditioned trajectories using a direct Monte Carlo method with the jump process (5). To sample jumps from (5), we use the acceptance-rejection method with the instrumental pdf
where \(C(\mathbf {r}_0)=m^2 \left[ 1-c_{\lambda }R\;I_0(c_{\lambda }\rho _0)\;K_1(c_{\lambda }R)\right] /c_{\lambda }^2\). The cumulative distribution function (cdf) of Eq. (A.14) can be exactly computed thus sampling \(\rho \) does not represent a computational bottleneck. The cdf for (A.15) cannot be expressed in a closed form but this pdf is indeed simpler to sample due to the absence of the Bessel functions of the first kind. For all \(\mathbf {r}_0\) and \(\mathbf {r}_1\) in the cylinder, \(q(\mathbf {r}_1\,|\,\mathbf {r}_0)\le k\,f(\mathbf {r}_1\,|\,\mathbf {r}_0)\), where \(k = C(\rho _0)/J_0(\lambda \rho _0)\), which sets the rejection threshold to
Observe that this threshold decreases when the walk gets closer to the boundaries (\(\rho _1\rightarrow R\)) and closer to the diffusion limit (\(\lambda R\rightarrow z_{0,1}\)). It finally remains to show how to sample from \(f(\mathbf {r}_1\,|\,\mathbf {r}_0)\). Using the change of variables
we can rewrite the pdf (A.15) as
where \(\ell ^*(\mathbf {r}_0, \xi , \varphi )\) is the maximum length that a walker starting from \(\mathbf {r}_0\) can travel within the cylinder in the direction given by \((\xi , \varphi )\): \(\ell ^*= b(\mathbf {r}_0, \varphi )/\sqrt{1-\xi ^2}\), with \(b(\mathbf {r}_0, \varphi )=\sqrt{R^2 - \rho _0^2\sin ^2(\varphi -\theta _0)} -\rho _0\cos (\varphi -\theta _0)\). The joint probability density can then be decomposed into three probability densities, one for each variable:
where
Starting from \(\mathbf {r}_0\), we thus sample first the angle \(\varphi \) from
then, given the angle \(\varphi \), we then obtain \(\xi \) from the pdf
Finally, the jump length \(\ell \) is sampled directly from
using the Lambert W function.
Theoretical Analysis of the Polymer Extension \(L_z=\langle z_N-z_0\rangle \)
After relaxation to steady state in the transverse direction (see Appendix 5), the generating function of the jump lengths \(z=(z_{i+1}-z_{i})\) along the axis of the cylinder is given by
Integrating, we obtain
where \(c_{\lambda }(s) = \sqrt{m^2-(\lambda +s)^2}\), \(K_i = K_i[c_{\lambda }(s)\, R]\), \(I_i = I_i[c_{\lambda }(s)\, R]\) and \(J_i = J_i(\lambda R)\). Finally expanding to first order at \(s=0\) leads to the expression of the mean jump length along the z-axis, \(\ell _z = \langle z \rangle = G^\prime (s)|_{s=0}\):
where g is a positive and increasing function of \(\ell _f/R\) that vanishes as \(\ell _f/R\) goes to 0:
with \(J_i = J_i(\lambda R)\), \(K_i = K_i(c_{\lambda } R)\) and \(I_i = I_i(c_{\lambda } R)\). We recall that both \(\lambda R\) and \(c_\lambda R\) are functions of \(mR=2R/\ell _f\). In the diffusive limit \(\ell _f\ll 2R\), \(\lambda \ell _f\) behaves as \(O(\ell _f/R)\) (see Eq. (A.12)), and expanding (A.23) to first order thus yields the asymptotic behavior:
This result is in agreement with the direct derivation for the diffusive case, where we found the drift \(2D \lambda \) along the cylinder axis (see Appendix A.9). Since the process is Markovian and is started from steady state, the mean extension of the polymer can be decomposed as \(L_z=N\ell _z\), and the total length of the polymer as \(L=N \ell _c\) (see main text). Moreover, in the diffusive limit, \(\ell _c\sim \ell _f\), so that Eq. (A.25) becomes
which varies linearly with \(\ell _f/R\) (see left panel of Fig. 4). Note that, in the long-jump limit \(mR \ll 1\), the asymptotic expansion of Eq. (A.24) is
Numerical Analysis
Polymer extension \(L_z\)—Perhaps surprisingly, we observe from the numerical simulations that Eq. (A.26) extends to any value of \(\ell _f\),
where \(\lambda \) is now given by Eq. (6), as shown in the Right panel of Fig. 8 where Eq. (A.28) matches almost perfectly the numerical data. In the long-jump regime \(\ell _f\gg R\), using the expansion (A.13) for \(\lambda R\), we obtain
Mean jump length under confinement \(\ell _c\)—Reformulating Eq. (A.28) with Eq. (A.23) we can write:
which stays consistent with the numerical data (see Fig. 4). In the diffusive limit we recover that \(\ell _c\sim \ell _f\), and, in the long-jump limit, using Eq. (A.27), we can note the extremely rapid growth of \(\ell _c\) as \(\ell _f/R\) increases:
End-to-end distance \(R_{ee}=\sqrt{\langle ||\mathbf {r}_n-\mathbf {r}_0||^2\rangle }\)—In Fig. 3, each colored curve (fixed value of \(\ell _f/R\)) displays two distinct regimes: \(R_{ee}\propto \sqrt{L}\) for \(R_{ee}\ll R\) and \(R_{ee}\propto L\) for \(R_{ee}\gg R\). For polymer such that \(R_{ee} \ll R\), the polymer behavior can be modeled by a Brownian walker under confinement, which yields,
with \(t=N\tau =\ell _f L/4\) and where the second term comes from the drift along the z-axis resulting from the confinement (see Appendix A.9). Replacing the value of t, we observe that this second term is negligible with respect to the first one, so that \(R_{ee}\) evolves as \(\sqrt{6Dt}\):
the effect of the confinement on \(R_{ee}\) is not visible in this regime. For long polymers \(R_{ee} \gg R\), we observe that \(R_{ee}\) can be rescaled using \(L\rightarrow \lambda \,\ell _f L/2\) (see Fig. 3). Note that this rescaling is valid only after a large number of jumps, \(N\gg 1\), and is not exact. Indeed, by definition we can write \(R_{ee}=\sqrt{ \langle L_{\perp } ^2 \rangle + \langle \,(z_N-z_0)^2 \rangle }\) where \(L_{\perp }\) in defined in the transverse direction of the cylinder. The rescaling does not apply to \(\langle L_{\perp }^2 \rangle \) in the long-polymer regime (see Fig. 9). However this is not visible on \(R_{ee}\), as, for long polymer (\(R_{ee}\gg R\)), \(\langle \,(z_N-z_0)^2 \rangle \gg \langle L_{\perp } ^2 \rangle \), and therefore \(R_{ee}\sim \sqrt{\langle \,(z_N-z_0)^2 \rangle }\) that rescales for large N.
Now keeping the number of jumps N fixed and varying the effective channel size (see black dots in Fig. 3), we observe three main regimes. They result from the overlap of the two transitions previously described, for \(R_{ee}\) and \(L_z\), and are summarized in Fig. 8.
Brownian Motion Conditioned to Stay Inside a Cylinder
In the continuum (\(l_f/R \rightarrow 0\)) the jump process becomes a controlled Brownian motion for which an analytical description is affordable. The effect of confining a Wiener process in the cylindrical channel \(\mathcal {C}\) is subsumed by an additional drift term [27], \(u(\mathbf {r})\), called control drift (Fig. 10). The Langevin equation for the walker thus reads
where each \(\eta ^i_t\) is an independent white noise. The behaviour of the conditioned Brownian process then corresponds to the optimal (stationary) trajectories of this controlled walker: we look for the optimal control drift using the Hamilton-Jacobi-Bellman equation with a cost that takes into account the boundaries (see [8]). We find that the drift \(\mathbf {u}(\mathbf {r})\) takes the form:
where \(\lambda = z_{0,1}/R\). As a consequence, the mean length travelled by the conditioned Brownian walker in the direction of the z-axis during a time \(\tau \) is
For the process (A.7), where \(\tau D=\ell _f^{\,2}/4\), we thus expected to recover, in the diffusive limit, that \(\ell _z=\lambda \,\ell _f^{\,2}/2\) (Fig. 10). For the same reason, we find that the mean-square distance travelled in the direction of the z-axis during a time t is given by:
The Limit \(H/R \rightarrow \infty \) for the Conditioned Brownian Motion
The Laplace equation in cylindrical coordinates is
The cylinder has a radius R and in the z direction it extends from \(-H\) to H. We impose the following boundary conditions for the Laplace equation:
The equation is separable and, looking for a solution of the kind \(Z(\rho ,\,\theta ,\,z) = P(\rho )\,\varTheta (\theta )\,\zeta (z)\), it can be written as the following equivalent system of coupled ordinary differential equation:
where here \(\lambda \) and \(\mu \) are real parameters. The solution to the equation for \(\zeta \) which satisfies the Dirichlet boundary conditions on the left end of the cylinder is
The equation for \(\varTheta \) satisfying the rotational invariance about the longitudinal axis of the cylinder selects the value \(\mu = 0\) and is just a constant:
Finally, the solution for P is the regular Bessel function of first kind of order zero:
the allowed values of \(\lambda \) are all and only those for which \(P(R) = 0\), so \(\lambda _n = z_{0,n}/R\), where we denote by \(z_{0,n}\) the n-th zero of \(J_0(x)\) (Fig. 11).
Therefore, the solution of the Laplace equation in the cylindrical geometry specified above is, dropping the \(\theta \) dependence,
The vanishing conditions at \(\rho = R\) and \(z = -H\) is already implemented in the solution, while the boundary condition \(Z|_{z=H} = 1\) fixes the coefficients \(c_n\) as the solution of
where \(\tilde{c}_n = c_n\,\sinh [2\,a\,z_{0,n}/R]\).
The set \(\{ J_{0,n}(x)\}_{n=1}^\infty \) is a basis of the set of function in the interval [0, 1) and they are mutually orthogonal therein with respect to the measure \(d\mu (x) = x\, dx\) Footnote 1:
The coefficients \(\tilde{c}_n\) are therefore found to be the (properly normalized) inner products between the function \(f(x) = 1\) and \(J_{0,n}(x)\) within [0, 1):
so that the full solution Z of the Laplace equation is
In the limit \(H/R \rightarrow \infty \) (infinite cylinder) with finite z, only the first term of the expansion can be retained:
The drift in the effective Langevin dynamics of the conditioned Brownian motion is then
where \(\lambda = z_{0,1}/R\) (Fig. 12).
Density Fluctuations
To study the density of beads along the cylinder, we focus on the evolution of the driven Brownian walker (see Appendix A.9) along the z-axis, described by the stochastic process:
where \(\mathrm{d}W_t\) is the standard Wiener process. As described in Appendix A.9 the first term is the drift along the z-axis due to confinement. Consider now the interval \([0,\,\varDelta ]\) along the z-axis, we define the residence time of the walker therein as
where \(\mathbb {I}_\varDelta \) is the characteristic function of \([0,\,\varDelta ]\), equal to 1 within the interval and 0 otherwise. In general, \(\phi _\varDelta \) is a random variable, whose statistics depends on the initial conditions of the process. Its moment generating function is defined as
and satisfies the stationary Feynman–Kac equation
In Eq. (A.40), the average is taken with respect to the measure of the paths generated by the dynamics in Eq. (A.38). The drift in Eq. (A.38), that drives the process towards increasing values of \(z_t\), fixes the boundary conditions of \(G_\varDelta (s,\,z_0)\):
The general solution of Eq. (A.41) then reads
where \(\alpha = \sqrt{\lambda ^2 + s/D}\) and the \(A_i\) and \(B_i\) are constants with respect to \(z_0\). The conditions of Eq. (A.42) then set
The four other constants are uniquely determined by imposing continuity and differentiability of \(G_\varDelta \) at \(z_0=0\) and \(z_0=\varDelta \). Note that, for \(z_0<0\), \(G_\varDelta (s,\,z_0)\) doesn’t depend on \(z_0\), and thus, the statistics of \(\phi _\varDelta \) are independent of the specific value of \(z_0\).
In our estimates of \(\phi _\varDelta \), we are interested only in the case \(z_0<0\), since the initial condition of the process is always to the left of the interval \([0,\,\varDelta ]\). Therefore, the moment generating function of \(\phi _\varDelta \) is given by the amplitude \(B_l\):
where we recall that \(\alpha (s) = \sqrt{\lambda ^2 + s/D}\). Note that \(G_\varDelta \) can be written in the scaling form \(G_\varDelta (s) = \tilde{g}(\varDelta ^2 s / D,\, \varDelta \,\lambda )\), where \(\tilde{g}\) is
In particular, the diffusion constant D can be absorbed in the scaling variable u. Hence it follows that the probability density of the residence time \(\phi _\varDelta \), denoted \(F_\varDelta (\phi _\varDelta )\), is given by the inverse Laplace transform
where \(\tilde{f}\) is the inverse Laplace transform of \(\tilde{g}\) with respect to its first variable. The results are shown in Fig. 13.
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Adorisio, M., Pezzotta, A., de Mulatier, C. et al. Exact and Efficient Sampling of Conditioned Walks. J Stat Phys 170, 79–100 (2018). https://doi.org/10.1007/s10955-017-1911-y
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DOI: https://doi.org/10.1007/s10955-017-1911-y