We introduce a simple model of diffusive jump process where a fee is charged for each jump. The nonlinear cost function is such that slow jumps incur a flat fee, while for fast jumps the cost is proportional to the velocity of the jump. The model—inspired by the way taxi meters work—exhibits a very rich behavior. The cost for trajectories of equal length and equal duration exhibits giant fluctuations at a critical value of the scaled distance traveled. Furthermore, the full distribution of the cost until the target is reached exhibits an interesting “freezing” transition in the large-deviation regime. All the analytical results are corroborated by numerical simulations. Our results also apply to elastic systems near the depinning transition, when driven by a random force.

中文翻译：

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扩散成本：非线性和巨大波动

我们引入一个简单的扩散跳跃过程模型，其中每次跳跃都会收取费用。非线性成本函数使得慢速跳跃产生固定费用，而对于快速跳跃，成本与跳跃的速度成正比。该模型的灵感来自于出租车计价器的工作方式，展示了非常丰富的行为。等长度和等持续时间的轨迹的成本在行进距离的临界值处表现出巨大的波动。此外，达到目标之前成本的完全分配在大偏差状态下表现出有趣的“冻结”转变。所有分析结果均通过数值模拟得到证实。我们的结果也适用于由随机力驱动时接近脱钉转变的弹性系统。