Exponential, and not Gaussian, decay of probability density functions was studied by Laplace in the context of his analysis of errors. Such Laplace propagators for the diffusive motion of single particles in disordered media were recently observed in numerous experimental systems. What will happen to this universality when an external driving force is applied? Using the ubiquitous continuous time random walk with bias, and the Crooks relation in conjunction with large deviations theory, we derive two properties of the positional probability density function $P_F(x,t)$ that hold for a wide spectrum of random walk models: (I) Universal asymmetric exponential decay of $P_F(X,t)$ for large $|X|$, and (II) Existence of a time transformation that for large $|X|$ allows to express $P_F(X,t)$ in terms of the propagator of the unbiased process (measured at a shorter time). These findings allow us to establish how the symmetric exponential-like tails, measured in many unbiased processes, will transform into asymmetric Laplace tails when an external force is applied.

中文翻译：

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有偏随机游走传播的指数尾和不对称关系

拉普拉斯在误差分析的背景下研究了概率密度函数的指数而非高斯衰减。最近在许多实验系统中观察到了这种用于单粒子在无序介质中的扩散运动的拉普拉斯传播子。当施加外部驱动力时，这种普遍性会发生什么？使用普遍存在的带偏差的连续时间随机游走，以及结合大偏差理论的 Crooks 关系，我们推导出位置概率密度函数 $P_F(x,t)$ 的两个属性，它们适用于广泛的随机游走模型： (I) 对于大 $|X|$，$P_F(X,t)$ 的通用非对称指数衰减，以及 (II) 对于大 $|X|$，存在时间变换允许表达 $P_F(X, t)$ 表示无偏过程的传播者（在更短的时间测量）。这些发现使我们能够确定在许多无偏过程中测量的对称指数状尾如何在施加外力时转变为不对称拉普拉斯尾。