论文标题
由不对称示踪粒子驱动的简单对称排除过程
A simple symmetric exclusion process driven by an asymmetric tracer particle
论文作者
论文摘要
我们考虑在周期性的一维晶格上进行排除过程，其中所有粒子以$ 1 $的价格执行简单的对称排除，除了单个示踪剂粒子，该粒子在右侧的速度$ $ p $中执行了部分简单的不对称排除，对左侧的费率$ p $。 Ferrari，Goldstein和Lebowitz（Progr。Phys。，1985）首先考虑了该模型，以测试爱因斯坦关系在微观系统中的有效性。 这项工作的主要目的是对于此排除过程的稳定状态的精确解决方案。我们表明，固定概率分解并给出非平衡分区函数的确切公式。也许令人惊讶的是，我们发现，如果$ p $ $ p $和$ p \ p \ neq q $，则在热力学限制下，该系统的非平衡自由能在热力学限制中没有很好地定义。我们为当前和两点相关提供公式。当示踪剂粒子执行不对称排除（$ q = 0 $）时，结果显示出可显着简化的，我们发现与设定分区的组合设备的意外连接。最后，我们从示踪剂粒子的角度研究系统，即所谓的环境过程。在环境过程中，我们表明颗粒的密度呈指数衰减，而热力学极限中示踪剂粒子前的缩放位置呈指数衰减。
We consider an exclusion process on a periodic one-dimensional lattice where all particles perform simple symmetric exclusion at rate $1$ except for a single tracer particle, which performs partially simple asymmetric exclusion with rate $p$ to the right and rate $q$ to the left. This model was first considered by Ferrari, Goldstein and Lebowitz (Progr. Phys., 1985) as a test for the validity of the Einstein relation in microscopic systems. The main thrust of this work is an exact solution for the steady state of this exclusion process. We show that the stationary probabilities factorize and give an exact formula for the nonequilibrium partition function. Perhaps surprisingly, we find that the nonequilibrium free energy in the steady state is not well-defined for this system in the thermodynamic limit for any values of $p$ and $q$ if $p \neq q$. We provide formulas for the current and two-point correlations. When the tracer particle performs asymmetric exclusion ($q=0$), the results are shown to simplify significantly and we find an unexpected connection with the combinatorics of set partitions. Finally, we study the system from the point of view of the tracer particle, the so-called environment process. In the environment process, we show that the density of particles decays exponentially with the scaled position in front of the tracer particle in the thermodynamic limit.
