标题： 关于不可逆性和随机系统：第一部分 显示英文标题

标题： On Irreversibility and Stochastic Systems: Part One

摘要： 我们试图从时间箭头方向不同而存在的正向和反向数学表示的差异来表征动力系统的不可逆性。 这种不同的表示已经被深入研究，并被证明存在于随机扩散模型中。 在这种情况下，却需要首先对物理系统进行随机描述的初步验证，这些物理系统由经典力学描述，本质上是确定性和保守的。 在本文的第一部分，我们首先在确定性背景下解决线性系统的建模问题。 我们表明，当与无限维的保守热浴耦合时，正向-反向表示也可以描述保守的有限维确定性系统。 一个新颖的关键观察是，热浴通过{\em 状态反馈}对有限维保守系统起作用，并可以将其本征值移动，使系统变得耗散，但也可能生成另一个完全不稳定的模型，该模型自然地随时间反向演化。 在第二部分，我们讨论这两种表示的随机描述。 在自然的不变测度族下，可以证明热浴会诱导一个白噪声输入作用于系统，使其看起来像一个真正的耗散扩散。 显示英文摘要

摘要： We attempt to characterize irreversibility of a dynamical system from the existence of different forward and backward mathematical representations depending on the direction of the time arrow. Such different representations have been studied intensively and are shown to exist for stochastic diffusion models. In this setting one has however to face the preliminary justification of stochastic description for physical systems which are described by classical mechanics as inherently deterministic and conservative. In part one of this paper we first address this modeling problem for linear systems in a deterministic context. We show that forward-backward representations can also describe conservative finite dimensional deterministic systems when they are coupled to an infinite-dimensional conservative heat bath. A novel key observation is that the heat bath acts on the finite-dimensional conservative system by {\em state-feedback} and can shift its eigenvalues to make the system dissipative but may also generate another totally unstable model which naturally evolves backward in time. In the second part, we address the stochastic description of these two representations. Under a natural family of invariant measures the heat bath can be shown to induce a white noise input acting on the system making it look like a true dissipative diffusion.