BetheSF: Efficient computation of the exact tagged-particle propagator in single-file systems via the Bethe eigenspectrum

Abstract

Single-file diffusion is a paradigm for strongly correlated classical stochastic many-body dynamics and has widespread applications in soft condensed matter and biophysics. However, exact results for singlefile systems are sparse and limited to the simplest scenarios. We present an algorithm for computing the non-Markovian time-dependent conditional probability density function of a tagged-particle in a single-file of N particles diffusing in a confining external potential. The algorithm implements an eigenexpansion of the full interacting many-body problem obtained by means of the coordinate Bethe ansatz. While formally exact, the Bethe eigenspectrum involves the generation and evaluation of permutations, which becomes unfeasible for single-files with an increasing number of particles N.Here we exploit the underlying exchange symmetries between the particles to the left and to the right of the tagged-particle and show that it is possible to reduce the complexity of the algorithm from the worst case scenario O(N!) down to O(N). A C++ code to calculate the non-Markovian probability density function using this algorithm is provided. Solutions for simple model potentials are readily implemented including single-file diffusion in a flat and a 'tilted' box, as well as in a parabolic potential. Notably, the program allows for implementations of solutions in arbitrary external potentials under the condition that the user can supply solutions to the respective single-particle eigenspectra.

Ubiquitous Dynamical Time Asymmetry in Measurements on Materials and Biological Systems

“…1d and Appendix D 4 b) [39], as well as two theoretical examples: the end-to-end distance fluctuations of a Rouse polymer chain [100] (Fig. 1e and Appendix D8) and tracer particle dynamics in a single file of impenetrable diffusing particles, where q(t) reflects the position of the tracer particle [19,85,101] (Fig. 1f and Appendix D 3).…”

“…The single file model refers to the overdamped Brownian motion of a system of N particles with hard core exclusion interactions, which for simplicity (and because the finite-size scenario is obtained by a simple re-scaling of space) we assume to be point-like and confined to an interval of unit length L = 1 [101,124,125]. We express length in units of L and time in units of τ = D/L 2 , where D corresponds to the diffusion coefficient which is assumed to be equal for all particles.…”

“…The system is exactly solvable with the coordinate Bethe ansatz, which yields explicit results for the spectral expansion of Q(x, t|x 0 ) [101,125], i.e. Q(x, t|x 0 ) = k ψ R k (x)ψ L k (x 0 )e −λ k t according to Eq.…”

“…, k N ), k i ∈ N, ∀i. Expressions for the eigenfunctions ψ R k (x), ψ L k (x) are given in [101,125] and the eignevalues corresponding to λ k = N i=1 k i π 2 . Note that the relaxation time t rel = 1/λ 1 once re-scaled to natural units in terms of the collision time (i.e.…”

Toolbox for quantifying memory in dynamics along reaction coordinates

“…Memory effects can have intriguing manifestations in the evolution of both, ensemble- [15,[25][26][27][28] and timeaverage observables [15,29], and are often particularly well-pronounced in observations that reflect, or couple to, intra-molecular distances in conformationally flexible biomolecules [16, 18, 20-22, 25, 30-38]. Moreover, if the dynamics is ergodic in the sense that the system relaxes to a unique equilibrium probability density function from any initial condition (i.e.…”