标题： 统计力学的概率基础：贝叶斯方法 显示英文标题

标题： Probabilistic Foundations of Statistical Mechanics: A Bayesian Approach

摘要： 我们研究统计力学的基本方面，将问题分为纯粹关于概率的讨论，我们从贝叶斯的角度对其进行分析。 我们认为，在给定数据$K$的情况下，对于标有$j$的状态，存在一个唯一的最大概率分布$\{p(j\vert K)\}$，这表明信息熵$\sigma(\{(p_j\vert K)\}) = -\sum_j (p_j \vert K)\ln{(p_j\vert K)}$的对应最大值在平衡时明确依赖于数据和系统的哈密顿量。 因此，它是对我们关于物体精确状态的不确定性的直接度量，并可以与传统的热力学熵相联系。 我们考虑众所周知的微正则、正则和大正则方法，并确保在将这些平均值与实验可观测量相联系之前，宏观系统关于平均值的涨落通常非常微小，从而连接到热力学中的许多标准结果。 令人意外的是，我们发现一般来说，量子过程不可能同时是等熵的和可逆绝热的。 这与传统热力学形成鲜明对比，在传统热力学中假设等熵的可逆绝热过程可以按需召唤并容易实现。 相比之下，我们发现压力$P_j$和能量$E_j$之间的线性关系是准静态和绝热变化为等熵的必要且充分条件，但当然，这种关系仅适用于少数特别简单的系统，如理想气体，并不普遍适用于更复杂的系统。 通过考虑小体积变化的二阶相关熵增，我们认为其后果在实践中可以忽略不计。 显示英文摘要

摘要： We examine the fundamental aspects of statistical mechanics, dividing the problem into a discussion purely about probability, which we analyse from a Bayesian standpoint. We argue that the existence of a unique maximising probability distribution $\{p(j\vert K)\}$ for states labelled by $j$ given data $K$ implies that the corresponding maximal value of the information entropy $\sigma(\{(p_j\vert K)\}) = -\sum_j (p_j \vert K)\ln{(p_j\vert K)}$ depends explicitly on the data at equilibrium and on the Hamiltonian of the system. As such, it is a direct measure of our uncertainty about the exact state of the body and can be identified with the traditional thermodynamic entropy. We consider the well known microcanonical, canonical and grand canonical methods and ensure that the fluctuations about mean values are generally minuscule for macroscopic systems before identifying these mean values with experimental observables and thereby connecting to many standard results from thermodynamics. Unexpectedly, we find that it is not generally possible for a quantum process to be both isentropic and reversibly adiabatic. This is in sharp contrast to traditional thermodynamics where it is assumed that isentropic, reversible adiabatic processes can be summoned up on demand and easily realised. By contrast, we find that linear relations between pressures $P_j$ and energies $E_j$ are necessary and sufficient conditions for a quasi-static and adiabatic change to be isentropic, but, of course, this relationship only holds for a few especially simple systems, such as the perfect gas, and is not generally true for more complicated systems. By considering the associated entropy increases up to second order in small volume changes we argue that the consequences are in practice negligible.