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A Bristol academic has achieved a milestone in statistical/mathematical physics by solving a 100-year-old physics problem -- the discrete diffusion equation in finite space.

布里斯托尔的一位学者解决了一个有百年历史的物理问题--有限空间中的离散扩散方程，从而在统计/数学物理学领域取得了里程碑的成就。

The long-sought-after solution could be used to accurately predict encounter and transmission probability between individuals in a closed environment, without the need for time-consuming computer simulations.

这一备受追捧的解决方案可以用来准确预测在封闭环境中个体之间的相遇和传播概率，而不需要耗时的计算机模拟。

In his paper, published in Physical Review X, Dr Luca Giuggioli from the Department of Engineering Mathematics at the University of Bristol describes how to analytically calculate the probability of occupation (in discrete time and discrete space) of a diffusing particle or entity in a confined space -- something that until now was only possible computationally.

布里斯托尔大学工程数学系的Luca Giuggioli博士在他发表在“物理评论X”上的论文中描述了如何分析计算在有限空间中扩散的粒子或实体被占据的概率(在离散时间和离散空间中)--到目前为止，只有通过计算才能做到这一点。

Dr Giuggioli said: "The diffusion equation models random movement and is one of the fundamental equations of physics. The analytic solution of the diffusion equation in finite domains, when time and space is continuous, has been known for a long time.

Giuggioli博士说：“扩散方程模拟随机运动，是物理学的基本方程之一。当时间和空间连续时，扩散方程在有限域中的解析解早已为人所知。”

"However, to compare model predictions with empirical observations, one needs to study the diffusion equation in finite space. Despite the work of illustrious scientists such as Smoluchowski, Pólya, and other investigators of yore, this has remained an outstanding problem for over a century -- until now.

“然而，要将模型预测与经验观测相比较，人们需要研究有限空间中的扩散方程。尽管有斯莫卢乔夫斯基、波利亚等杰出科学家的工作，这一问题在一个多世纪以来一直是一个突出的问题--直到现在。

"Excitingly, the discovery of this exact analytic solution allows us to tackle problems that were almost impossible in the past because of the prohibitive computational costs."

“令人兴奋的是，这种精确解析解的发现使我们能够解决过去由于高昂的计算成本而几乎不可能解决的问题。”

The finding has far-reaching implications across a range of disciplines and possible applications include predicting molecules diffusing inside cells, bacteria roaming in a petri dish, animals foraging within their home ranges, or robots searching in a disaster area.

这一发现对一系列学科都有深远的影响，可能的应用包括预测分子在细胞内的扩散，细菌在培养皿中漫游，在它们的家园范围内觅食的动物，或者在灾区搜索的机器人。

It could even be used to predict how a pathogen is transmitted in a crowd between individuals.

它甚至可以用来预测病原体是如何在人群中在个体之间传播的。

Solving the conundrum involved the joint use of two techniques: special mathematical functions known as Chebyshev polynomials, and a technique invented to tackle electrostatic problems, the so-called method of images.

解决这个难题涉及到两种技术的联合使用：一种是被称为切比雪夫多项式的特殊数学函数，另一种是为解决静电问题而发明的技术，即所谓的图像法。

This approach allowed Dr Giuggioli to construct hierarchically the solution to the discrete diffusion equation in higher dimension from the one in lower dimensions.

这种方法允许Giuggioli博士从低维的离散扩散方程到高维的离散扩散方程分层构造解。。