We extend stochastic thermodynamics by relaxing the two assumptions that the Markovian dynamics must be linear and that the equilibrium distribution must be a Boltzmann distribution. We show that if we require the second law to hold when those assumptions are relaxed, then it cannot be formulated in terms of Shannon entropy. However, thermodynamic consistency is salvaged if we reformulate the second law in terms of generalized entropy; our first result is an equation relating the precise form of the non-linear master equation to the precise associated generalized entropy which results in thermodynamic consistency. We then build on this result to extend the usual trajectory-level definitions of thermodynamic quantities that are appropriate even when the two assumptions are relaxed. We end by using these trajectory-level definitions to derive extended versions of the Crooks fluctuation theorem and Jarzynski equality which apply when the two assumptions are relaxed.

我们通过放宽马尔可夫动力学必须是线性的和平衡分布必须是玻尔兹曼分布这两个假设来扩展随机热力学。我们表明，如果我们要求在放松这些假设时第二定律成立，那么它就不能用香农熵来表述。然而，如果我们根据广义熵重新表述第二定律，热力学一致性就可以挽救。我们的第一个结果是一个方程，它将非线性主方程的精确形式与精确的相关广义熵相关联，从而导致热力学一致性。然后，我们在此结果的基础上扩展了热力学量的通常轨迹级定义，即使放宽这两个假设也是合适的。