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→Principles: mechanics and ensembles
The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics).
The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics).
力学系统在给定时间内的完整状态，用数学表示为相空间中的点(经典力学)或纯量子态矢量(量子力学)。
# An equation of motion which carries the state forward in time: [[Hamilton's equations]] (classical mechanics) or the [[time-dependent Schrödinger equation]] (quantum mechanics)
# An equation of motion which carries the state forward in time: [[Hamilton's equations]] (classical mechanics) or the [[time-dependent Schrödinger equation]] (quantum mechanics)
An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the time-dependent Schrödinger equation (quantum mechanics)
An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the time-dependent Schrödinger equation (quantum mechanics)
一个运动方程描述状态在时间上的演化: 哈密尔顿方程(经典力学)或含时薛定谔方程(量子力学)
Using these two concepts, the state at any other time, past or future, can in principle be calculated.
Using these two concepts, the state at any other time, past or future, can in principle be calculated.
Using these two concepts, the state at any other time, past or future, can in principle be calculated.
Using these two concepts, the state at any other time, past or future, can in principle be calculated.
使用这两个概念，系统在任何时间的状态，无论过去或未来，原则上都可以计算出来。
There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.
There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.
There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.
There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.
然而，这些定律与日常生活经验之间存在脱节。因为对于在人类尺度上进行过程(例如化学反应)，我们没有必要(甚至在理论上也不可能)在微观层面上准确地知道每个分子所在的位置及其速度。统计力学通过增加一些对于系统状态的不确定性，填补了力学定律和不完全知识的实践经验之间的这种脱节，。
Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinates. In quantum statistical mechanics, the ensemble is a probability distribution over pure states, and can be compactly summarized as a density matrix.
Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinates. In quantum statistical mechanics, the ensemble is a probability distribution over pure states, and can be compactly summarized as a density matrix.
普通力学只考虑单一状态的行为，而统计力学引入了统计系综，它是系统在各种状态下的大量虚拟、独立副本的集合。系综是一个覆盖系统所有可能状态的概率分布。在经典的统计力学中，系综是相点上的概率分布(与普通力学中的单相点相反) ，通常表现为正则坐标下相空间中的分布。在量子统计力学中，系综是纯态上的概率分布，可以简单地概括为密度矩阵。
As is usual for probabilities, the ensemble can be interpreted in different ways:
As is usual for probabilities, the ensemble can be interpreted in different ways:
与通常的概率一样，系综可以用不同的方式来解释:
* an ensemble can be taken to represent the various possible states that a ''single system'' could be in ([[epistemic probability]], a form of knowledge), or
* the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner ([[empirical probability]]), in the limit of an infinite number of trials.
* an ensemble can be taken to represent the various possible states that a ''single system'' could be in ([[epistemic probability]], a form of knowledge), or
* an ensemble can be taken to represent the various possible states that a ''single system'' could be in ([[epistemic probability]], a form of knowledge), or
合奏的一个特殊类别是那些不随时间演变的合奏。这些系综称为平衡系综，它们的状态称为统计平衡。如果对于集合中的每个状态，集合也包含其所有的未来和过去状态，其概率等于处于该状态的概率，则出现统计平衡。孤立系统的平衡系综是统计热力学研究的重点。非平衡统计力学解决了更一般的情况下的系综，随着时间的推移而改变，和 / 或非孤立系统的系综。
合奏的一个特殊类别是那些不随时间演变的合奏。这些系综称为平衡系综，它们的状态称为统计平衡。如果对于集合中的每个状态，集合也包含其所有的未来和过去状态，其概率等于处于该状态的概率，则出现统计平衡。孤立系统的平衡系综是统计热力学研究的重点。非平衡统计力学解决了更一般的情况下的系综，随着时间的推移而改变，和 / 或非孤立系统的系综。
== Statistical thermodynamics ==
== Statistical thermodynamics ==