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

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 single-file 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 . 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.单文件扩散（Single-file diffusion）是一类描述强关联经典随机多体动力学的范式，在软凝聚态物理与生物物理学领域拥有广泛应用。然而，目前针对单文件系统的精确解析结果十分有限，且仅局限于最为简单的场景。本文提出一种算法，可用于计算由N个粒子组成、在约束外势中扩散的单文件系统内，示踪粒子（tagged-particle）的非马尔可夫时变条件概率密度函数。该算法通过坐标贝特拟设（coordinate Bethe ansatz）对完整的相互作用多体问题进行本征展开。尽管该算法在形式上严格精确，但贝特本征谱涉及排列的生成与求值，随着粒子数增加，其计算复杂度会变得难以承受。本文利用示踪粒子两侧粒子间的固有交换对称性，证明可将算法的最坏情形复杂度从O(N!)降至O(N)。本文附带了基于该算法计算非马尔可夫概率密度函数的C++代码。该代码已内置多种简单模型势的求解方案，包括平直箱、“倾斜”箱以及抛物势中的单文件扩散场景。值得注意的是，若用户能够提供对应单粒子本征谱的求解结果，该程序还可支持任意外势下的求解方案实现。