We study a model of active particles that perform a simple random walk and on top of that have a preferred direction determined by an internal state which is modelled by a stationary Markov process. First we calculate the limiting diffusion coefficient. Then we show that the ‘active part’ of the diffusion coefficient is in some sense maximal for reversible state processes. Further, we obtain a large deviations principle for the active particle in terms of the large deviations rate function of the empirical process corresponding to the state process. Again we show that the rate function and free energy function are (pointwise) optimal for reversible state processes. Finally, we show that in the case with two states, the Fourier–Laplace transform of the distribution, the moment generating function and the free energy function can be computed explicitly. Along the way we provide several examples.

我们研究了一个活动粒子模型，它执行简单的随机游走，并且在其之上具有由内部状态确定的首选方向，该内部状态由固定马尔可夫过程建模。首先我们计算极限扩散系数。然后我们证明扩散系数的“活性部分”在某种意义上对于可逆状态过程是最大的。进一步，根据状态过程对应的经验过程的大偏差率函数，我们得到了活性粒子的大偏差原理。我们再次表明速率函数和自由能函数对于可逆状态过程是（逐点）最优的。最后，我们证明了在有两种状态的情况下，分布的傅立叶-拉普拉斯变换，矩生成函数和自由能函数可以明确计算。在此过程中，我们提供了几个示例。