Jaynes最大熵原理、黎曼流形和广义最小Action界
Jaynes' Maximum Entropy Principle, Riemannian Metrics and Generalised Least Action Bound
摘要：
由Jaynes的一般最大熵（MaxEnt）或最大相对熵（MaxREnt）原理推断的解集-被认为是力矩约束或其共轭拉格朗日乘子的函数-被赋予黎曼几何描述，基于熵的二阶微分张量或其勒让德变换（负Massieu函数）。分析提供了一个广义的{ \it最小动作界}适用于所有的Jaynesian系统，它提供了一个下界的成本（在通用熵单位）之间的过渡推断位置沿指定的路径，在指定的控制参数的变化率。因此，该分析将“有限时间热力学”的概念扩展到通用的Jaynes域，提供了系统的纯静态（静止）推断位置之间的联系，以及这些位置之间的动态转换（作为时间或其他坐标的函数）。如果路径是未指定的，分析给出了一个绝对下界的过渡成本，对应的黎曼超曲面的测地线。该分析适用于（i）平衡热力学系统的平均内能和体积的限制，和（ii）在稳态的流动系统，受到限制的平均热量，质量和动量通量和化学反应速率。第一个例子恢复了平衡位置之间的转变的{ \it最小熵成本}，这是有限时间热力学的一个广泛使用的结果。第二个例子导致一个新的{ \it最小熵产生原理}，为流动系统的稳态位置之间的过渡的成本。
英文摘要：
The set of solutions inferred by the generic maximum entropy (MaxEnt) or maximum relative entropy (MaxREnt) principles of Jaynes - considered as a function of the moment constraints or their conjugate Lagrangian multipliers - is endowed with a Riemannian geometric description, based on the second differential tensor of the entropy or its Legendre transform (negative Massieu function). The analysis provides a generalised {\it least action bound} applicable to all Jaynesian systems, which provides a lower bound to the cost (in generic entropy units) of a transition between inferred positions along a specified path, at specified rates of change of the control parameters. The analysis therefore extends the concepts of "finite time thermodynamics" to the generic Jaynes domain, providing a link between purely static (stationary) inferred positions of a system, and dynamic transitions between these positions (as a function of time or some other coordinate). If the path is unspecified, the analysis gives an absolute lower bound for the cost of the transition, corresponding to the geodesic of the Riemannian hypersurface. The analysis is applied to (i) an equilibrium thermodynamic system subject to mean internal energy and volume constraints, and (ii) a flow system at steady state, subject to constraints on the mean heat, mass and momentum fluxes and chemical reaction rates. The first example recovers the {\it minimum entropy cost} of a transition between equilibrium positions, a widely used result of finite-time thermodynamics. The second example leads to a new {\it minimum entropy production principle}, for the cost of a transition between steady state positions of a flow system.
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