We analyze the stochastic thermodynamics of systems with a continuous space of states. The evolution equation, the rate of entropy production, and other results are obtained by a continuous time limit of a discrete time formulation. We point out the role of time reversal and of the dissipation part of the probability current on the production of entropy. We show that the rate of entropy production is a bilinear form in the components of the dissipation probability current with coefficients being the components of the precision matrix related to the Gaussian noise. We have also analyzed a type of noise that makes the energy function to be strictly constant along the stochastic trajectory, being appropriate to describe an isolated system. This type of noise leads to nonzero entropy production and thus to an increase of entropy in the system. This result contrasts with the invariance of the entropy predicted by the Liouville equation, which also describes an isolated system.

中文翻译：

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具有连续状态空间的系统的随机热力学

我们分析具有连续状态空间的系统的随机热力学。通过离散时间公式的连续时间限制可以获得演化方程，熵产生率和其他结果。我们指出时间逆转和概率电流耗散部分对熵产生的作用。我们表明，在耗散概率电流的分量中，熵的产生率是双线性形式的，系数是与高斯噪声有关的精度矩阵的分量。我们还分析了一种噪声，该噪声使能量函数沿随机轨迹严格恒定，适合描述一个孤立的系统。这种类型的噪声会导致产生非零的熵，从而导致系统中熵的增加。