Deterministic and time-reversible nonequilibrium molecular dynamics simulations typically generate “fractal” (fractional-dimensional) phase-space distributions. Because these distributions and their time-reversed twins have zero phase volume, stable attractors “forward in time” and unstable (unobservable) repellors when reversed, these simulations are consistent with the second law of thermodynamics. These same reversibility and stability properties can also be found in compressible baker maps, or in their equivalent random walks, motivating their careful study. We illustrate these ideas with three examples: a Cantor set map and two linear compressible baker maps, N2\((q,p)\) and N3\((q,p)\). The two baker maps’ information dimensions estimated from sequential mappings agree, while those from pointwise iteration do not, with the estimates dependent upon details of the approach to the maps’ nonequilibrium steady states.

确定性和时间可逆的非平衡分子动力学模拟通常会生成“分形”（分数维）相空间分布。由于这些分布及其时间反向的孪晶的相体积为零，反向时稳定的吸引子“向前”，而反向时的不稳定（无法观察到的）排斥器，因此这些模拟与热力学第二定律是一致的。这些相同的可逆性和稳定性也可以在可压缩的贝克贴图或等效的随机游走图中找到，从而激发了他们的仔细研究的兴趣。我们用三个示例说明这些想法：一个Cantor集图和两个线性可压缩贝克图，N2 \（（（q，p）\）和N3 \（（q，p）\）。从顺序映射估计的两个贝克图的信息维数一致，而从逐点迭代估计的两个贝克图的信息维不同，其估计值取决于映射的非平衡稳态方法的细节。